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Structure and Interpretation of Computer Programs
=================================================

Second Edition
by Harold Abelson and Gerald Jay Sussman, with Julie Sussman
foreword by Alan J. Perlis
(C) 1996 Massachusetts Institute of Technology

Unofficial Texinfo Format version 2.neilvandyke4 (January 10, 2007)

* Menu:

* UTF::              Unofficial Texinfo Format
* Dedication::       Dedication
* Foreword::         Foreword
* Preface::          Preface to the Second Edition
* Preface 1e::       Preface to the First Edition
* Acknowledgements:: Acknowledgements
* Chapter 1::        Building Abstractions with Procedures
* Chapter 2::        Building Abstractions with Data
* Chapter 3::        Modularity, Objects, and State
* Chapter 4::        Metalinguistic Abstraction
* Chapter 5::        Computing with Register Machines
* References::       References
* Index::            Index

 --- The Detailed Node Listing ---

Programming in Lisp

* 1-1::              The Elements of Programming
* 1-2::              Procedures and the Processes They Generate
* 1-3::              Formulating Abstractions with Higher-Order Procedures

The Elements of Programming

* 1-1-1::            Expressions
* 1-1-2::            Naming and the Environment
* 1-1-3::            Evaluating Combinations
* 1-1-4::            Compound Procedures
* 1-1-5::            The Substitution Model for Procedure Application
* 1-1-6::            Conditional Expressions and Predicates
* 1-1-7::            Example: Square Roots by Newton's Method
* 1-1-8::            Procedures as Black-Box Abstractions

Procedures and the Processes They Generate

* 1-2-1::            Linear Recursion and Iteration
* 1-2-2::            Tree Recursion
* 1-2-3::            Orders of Growth
* 1-2-4::            Exponentiation
* 1-2-5::            Greatest Common Divisors
* 1-2-6::            Example: Testing for Primality

Formulating Abstractions with Higher-Order Procedures

* 1-3-1::            Procedures as Arguments
* 1-3-2::            Constructing Procedures Using `Lambda'
* 1-3-3::            Procedures as General Methods
* 1-3-4::            Procedures as Returned Values

Building Abstractions with Data

* 2-1::              Introduction to Data Abstraction
* 2-2::              Hierarchical Data and the Closure Property
* 2-3::              Symbolic Data
* 2-4::              Multiple Representations for Abstract Data
* 2-5::              Systems with Generic Operations

Introduction to Data Abstraction

* 2-1-1::            Example: Arithmetic Operations for Rational Numbers
* 2-1-2::            Abstraction Barriers
* 2-1-3::            What Is Meant by Data?
* 2-1-4::            Extended Exercise: Interval Arithmetic

Hierarchical Data and the Closure Property

* 2-2-1::            Representing Sequences
* 2-2-2::            Hierarchical Structures
* 2-2-3::            Sequences as Conventional Interfaces
* 2-2-4::            Example: A Picture Language

Symbolic Data

* 2-3-1::            Quotation
* 2-3-2::            Example: Symbolic Differentiation
* 2-3-3::            Example: Representing Sets
* 2-3-4::            Example: Huffman Encoding Trees

Multiple Representations for Abstract Data

* 2-4-1::            Representations for Complex Numbers
* 2-4-2::            Tagged data
* 2-4-3::            Data-Directed Programming and Additivity

Systems with Generic Operations

* 2-5-1::            Generic Arithmetic Operations
* 2-5-2::            Combining Data of Different Types
* 2-5-3::            Example: Symbolic Algebra

Modularity, Objects, and State

* 3-1::              Assignment and Local State
* 3-2::              The Environment Model of Evaluation
* 3-3::              Modeling with Mutable Data
* 3-4::              Concurrency: Time Is of the Essence
* 3-5::              Streams

Assignment and Local State

* 3-1-1::            Local State Variables
* 3-1-2::            The Benefits of Introducing Assignment
* 3-1-3::            The Costs of Introducing Assignment

The Environment Model of Evaluation

* 3-2-1::            The Rules for Evaluation
* 3-2-2::            Applying Simple Procedures
* 3-2-3::            Frames as the Repository of Local State
* 3-2-4::            Internal Definitions

Modeling with Mutable Data

* 3-3-1::            Mutable List Structure
* 3-3-2::            Representing Queues
* 3-3-3::            Representing Tables
* 3-3-4::            A Simulator for Digital Circuits
* 3-3-5::            Propagation of Constraints

Concurrency: Time Is of the Essence

* 3-4-1::            The Nature of Time in Concurrent Systems
* 3-4-2::            Mechanisms for Controlling Concurrency

Streams

* 3-5-1::            Streams Are Delayed Lists
* 3-5-2::            Infinite Streams
* 3-5-3::            Exploiting the Stream Paradigm
* 3-5-4::            Streams and Delayed Evaluation
* 3-5-5::            Modularity of Functional Programs and Modularity of
                     Objects

Metalinguistic Abstraction

* 4-1::              The Metacircular Evaluator
* 4-2::              Variations on a Scheme -- Lazy Evaluation
* 4-3::              Variations on a Scheme -- Nondeterministic Computing
* 4-4::              Logic Programming

The Metacircular Evaluator

* 4-1-1::            The Core of the Evaluator
* 4-1-2::            Representing Expressions
* 4-1-3::            Evaluator Data Structures
* 4-1-4::            Running the Evaluator as a Program
* 4-1-5::            Data as Programs
* 4-1-6::            Internal Definitions
* 4-1-7::            Separating Syntactic Analysis from Execution

Variations on a Scheme -- Lazy Evaluation

* 4-2-1::            Normal Order and Applicative Order
* 4-2-2::            An Interpreter with Lazy Evaluation
* 4-2-3::            Streams as Lazy Lists

Variations on a Scheme -- Nondeterministic Computing

* 4-3-1::            Amb and Search
* 4-3-2::            Examples of Nondeterministic Programs
* 4-3-3::            Implementing the `Amb' Evaluator

Logic Programming

* 4-4-1::            Deductive Information Retrieval
* 4-4-2::            How the Query System Works
* 4-4-3::            Is Logic Programming Mathematical Logic?
* 4-4-4::            Implementing the Query System

Implementing the Query System

* 4-4-4-1::          The Driver Loop and Instantiation
* 4-4-4-2::          The Evaluator
* 4-4-4-3::          Finding Assertions by Pattern Matching
* 4-4-4-4::          Rules and Unification
* 4-4-4-5::          Maintaining the Data Base
* 4-4-4-6::          Stream Operations
* 4-4-4-7::          Query Syntax Procedures
* 4-4-4-8::          Frames and Bindings

Computing with Register Machines

* 5-1::              Designing Register Machines
* 5-2::              A Register-Machine Simulator
* 5-3::              Storage Allocation and Garbage Collection
* 5-4::              The Explicit-Control Evaluator
* 5-5::              Compilation

Designing Register Machines

* 5-1-1::            A Language for Describing Register Machines
* 5-1-2::            Abstraction in Machine Design
* 5-1-3::            Subroutines
* 5-1-4::            Using a Stack to Implement Recursion
* 5-1-5::            Instruction Summary

A Register-Machine Simulator

* 5-2-1::            The Machine Model
* 5-2-2::            The Assembler
* 5-2-3::            Generating Execution Procedures for Instructions
* 5-2-4::            Monitoring Machine Performance

Storage Allocation and Garbage Collection

* 5-3-1::            Memory as Vectors
* 5-3-2::            Maintaining the Illusion of Infinite Memory

Registers and operations

* 5-4-1::            The Core of the Explicit-Control Evaluator
* 5-4-2::            Sequence Evaluation and Tail Recursion
* 5-4-3::            Conditionals, Assignments, and Definitions
* 5-4-4::            Running the Evaluator

An overview of the compiler

* 5-5-1::            Structure of the Compiler
* 5-5-2::            Compiling Expressions
* 5-5-3::            Compiling Combinations
* 5-5-4::            Combining Instruction Sequences
* 5-5-5::            An Example of Compiled Code
* 5-5-6::            Lexical Addressing
* 5-5-7::            Interfacing Compiled Code to the Evaluator


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Unofficial Texinfo Format
*************************

This is the second edition SICP book, from Unofficial Texinfo Format.

   You are probably reading it in an Info hypertext browser, such as
the Info mode of Emacs.  You might alternatively be reading it
TeX-formatted on your screen or printer, though that would be silly.
And, if printed, expensive.

   The freely-distributed official HTML-and-GIF format was first
converted personually to Unofficial Texinfo Format (UTF) version 1 by
Lyssa Ayth during a long Emacs lovefest weekend in April, 2001.

   The UTF is easier to search than the HTML format.  It is also much
more accessible to people running on modest computers, such as donated
'386-based PCs.  A 386 can, in theory, run Linux, Emacs, and a Scheme
interpreter simultaneously, but most 386s probably can't also run both
Netscape and the necessary X Window System without prematurely
introducing budding young underfunded hackers to the concept of "thrashing".
UTF can also fit uncompressed on a 1.44MB floppy diskette, which may
come in handy for installing UTF on PCs that do not have Internet or
LAN access.

   The Texinfo conversion has been a straight transliteration, to the
extent possible.  Like the TeX-to-HTML conversion, this was not without
some introduction of breakage.  In the case of Unofficial Texinfo
Format, figures have suffered an amateurish resurrection of the lost
art of ASCII art.  Also, it's quite possible that some errors of
ambiguity were introduced during the conversion of some of the copious
superscripts (`^') and subscripts (`_').  Divining _which_ has been
left as an exercise to the reader.  But at least we don't put our brave
astronauts at risk by encoding the _greater-than-or-equal_ symbol as
`<u>&gt;</u>'.

   If you modify `sicp.texi' to correct errors or improve the ASCII
art, then update the `@set utfversion 2.neilvandyke4' line to reflect
your delta.  For example, if you started with Lytha's version `1', and
your name is Bob, then you could name your successive versions
`1.bob1', `1.bob2', ... `1.bobn'.  Also update `utfversiondate'.  If
you want to distribute your version on the Web, then embedding the
string "sicp.texi" somewhere in the file or Web page will make it
easier for people to find with Web search engines.

   It is believed that the Unofficial Texinfo Format is in keeping with
the spirit of the graciously freely-distributed HTML version.  But you
never know when someone's armada of lawyers might need something to do,
and get their shorts all in a knot over some benign little thing, so
think twice before you use your full name or distribute Info, DVI,
PostScript, or PDF formats that might embed your account or machine
name.

Peath,

Lytha Ayth

   Addendum: See also the SICP video lectures by Abelson and Sussman:
`http://www.swiss.ai.mit.edu/classes/6.001/abelson-sussman-lectures/'


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Dedication
**********

This book is dedicated, in respect and admiration, to the spirit that
lives in the computer.

     "I think that it's extraordinarily important that we in computer
     science keep fun in computing.  When it started out, it was an
     awful lot of fun.  Of course, the paying customers got shafted
     every now and then, and after a while we began to take their
     complaints seriously.  We began to feel as if we really were
     responsible for the successful, error-free perfect use of these
     machines.  I don't think we are.  I think we're responsible for
     stretching them, setting them off in new directions, and keeping
     fun in the house.  I hope the field of computer science never
     loses its sense of fun.  Above all, I hope we don't become
     missionaries.  Don't feel as if you're Bible salesmen.  The world
     has too many of those already.  What you know about computing
     other people will learn.  Don't feel as if the key to successful
     computing is only in your hands.  What's in your hands, I think
     and hope, is intelligence: the ability to see the machine as more
     than when you were first led up to it, that you can make it more."

     --Alan J. Perlis (April 1, 1922  February 7, 1990)


File: sicp.info,  Node: Foreword,  Next: Preface,  Prev: Dedication,  Up: Top

Foreword
********

Educators, generals, dieticians, psychologists, and parents program.
Armies, students, and some societies are programmed.  An assault on
large problems employs a succession of programs, most of which spring
into existence en route.  These programs are rife with issues that
appear to be particular to the problem at hand.  To appreciate
programming as an intellectual activity in its own right you must turn
to computer programming; you must read and write computer
programs--many of them.  It doesn't matter much what the programs are
about or what applications they serve.  What does matter is how well
they perform and how smoothly they fit with other programs in the
creation of still greater programs.  The programmer must seek both
perfection of part and adequacy of collection.  In this book the use of
"program" is focused on the creation, execution, and study of programs
written in a dialect of Lisp for execution on a digital computer.
Using Lisp we restrict or limit not what we may program, but only the
notation for our program descriptions.

   Our traffic with the subject matter of this book involves us with
three foci of phenomena: the human mind, collections of computer
programs, and the computer.  Every computer program is a model, hatched
in the mind, of a real or mental process.  These processes, arising
from human experience and thought, are huge in number, intricate in
detail, and at any time only partially understood.  They are modeled to
our permanent satisfaction rarely by our computer programs.  Thus even
though our programs are carefully handcrafted discrete collections of
symbols, mosaics of interlocking functions, they continually evolve: we
change them as our perception of the model deepens, enlarges,
generalizes until the model ultimately attains a metastable place
within still another model with which we struggle.  The source of the
exhilaration associated with computer programming is the continual
unfolding within the mind and on the computer of mechanisms expressed
as programs and the explosion of perception they generate.  If art
interprets our dreams, the computer executes them in the guise of
programs!

   For all its power, the computer is a harsh taskmaster.  Its programs
must be correct, and what we wish to say must be said accurately in
every detail.  As in every other symbolic activity, we become convinced
of program truth through argument.  Lisp itself can be assigned a
semantics (another model, by the way), and if a program's function can
be specified, say, in the predicate calculus, the proof methods of
logic can be used to make an acceptable correctness argument.
Unfortunately, as programs get large and complicated, as they almost
always do, the adequacy, consistency, and correctness of the
specifications themselves become open to doubt, so that complete formal
arguments of correctness seldom accompany large programs.  Since large
programs grow from small ones, it is crucial that we develop an arsenal
of standard program structures of whose correctness we have become
sure--we call them idioms--and learn to combine them into larger
structures using organizational techniques of proven value.  These
techniques are treated at length in this book, and understanding them
is essential to participation in the Promethean enterprise called
programming.  More than anything else, the uncovering and mastery of
powerful organizational techniques accelerates our ability to create
large, significant programs.  Conversely, since writing large programs
is very taxing, we are stimulated to invent new methods of reducing the
mass of function and detail to be fitted into large programs.

   Unlike programs, computers must obey the laws of physics.  If they
wish to perform rapidly--a few nanoseconds per state change--they must
transmit electrons only small distances (at most 11 over 2 feet).  The
heat generated by the huge number of devices so concentrated in space
has to be removed.  An exquisite engineering art has been developed
balancing between multiplicity of function and density of devices.  In
any event, hardware always operates at a level more primitive than that
at which we care to program.  The processes that transform our Lisp
programs to "machine" programs are themselves abstract models which we
program.  Their study and creation give a great deal of insight into
the organizational programs associated with programming arbitrary
models.  Of course the computer itself can be so modeled.  Think of it:
the behavior of the smallest physical switching element is modeled by
quantum mechanics described by differential equations whose detailed
behavior is captured by numerical approximations represented in
computer programs executing on computers composed of ...!

   It is not merely a matter of tactical convenience to separately
identify the three foci.  Even though, as they say, it's all in the
head, this logical separation induces an acceleration of symbolic
traffic between these foci whose richness, vitality, and potential is
exceeded in human experience only by the evolution of life itself.  At
best, relationships between the foci are metastable.  The computers are
never large enough or fast enough.  Each breakthrough in hardware
technology leads to more massive programming enterprises, new
organizational principles, and an enrichment of abstract models.  Every
reader should ask himself periodically "Toward what end, toward what
end?"--but do not ask it too often lest you pass up the fun of
programming for the constipation of bittersweet philosophy.

   Among the programs we write, some (but never enough) perform a
precise mathematical function such as sorting or finding the maximum of
a sequence of numbers, determining primality, or finding the square
root.  We call such programs algorithms, and a great deal is known of
their optimal behavior, particularly with respect to the two important
parameters of execution time and data storage requirements.  A
programmer should acquire good algorithms and idioms.  Even though some
programs resist precise specifications, it is the responsibility of the
programmer to estimate, and always to attempt to improve, their
performance.

   Lisp is a survivor, having been in use for about a quarter of a
century.  Among the active programming languages only Fortran has had a
longer life.  Both languages have supported the programming needs of
important areas of application, Fortran for scientific and engineering
computation and Lisp for artificial intelligence.  These two areas
continue to be important, and their programmers are so devoted to these
two languages that Lisp and Fortran may well continue in active use for
at least another quarter-century.

   Lisp changes.  The Scheme dialect used in this text has evolved from
the original Lisp and differs from the latter in several important
ways, including static scoping for variable binding and permitting
functions to yield functions as values.  In its semantic structure
Scheme is as closely akin to Algol 60 as to early Lisps.  Algol 60,
never to be an active language again, lives on in the genes of Scheme
and Pascal.  It would be difficult to find two languages that are the
communicating coin of two more different cultures than those gathered
around these two languages.  Pascal is for building pyramids--imposing,
breathtaking, static structures built by armies pushing heavy blocks
into place.  Lisp is for building organisms--imposing, breathtaking,
dynamic structures built by squads fitting fluctuating myriads of
simpler organisms into place.  The organizing principles used are the
same in both cases, except for one extraordinarily important
difference: The discretionary exportable functionality entrusted to the
individual Lisp programmer is more than an order of magnitude greater
than that to be found within Pascal enterprises.  Lisp programs inflate
libraries with functions whose utility transcends the application that
produced them.  The list, Lisp's native data structure, is largely
responsible for such growth of utility.  The simple structure and
natural applicability of lists are reflected in functions that are
amazingly nonidiosyncratic.  In Pascal the plethora of declarable data
structures induces a specialization within functions that inhibits and
penalizes casual cooperation.  It is better to have 100 functions
operate on one data structure than to have 10 functions operate on 10
data structures.  As a result the pyramid must stand unchanged for a
millennium; the organism must evolve or perish.

   To illustrate this difference, compare the treatment of material and
exercises within this book with that in any first-course text using
Pascal.  Do not labor under the illusion that this is a text digestible
at MIT only, peculiar to the breed found there.  It is precisely what a
serious book on programming Lisp must be, no matter who the student is
or where it is used.

   Note that this is a text about programming, unlike most Lisp books,
which are used as a preparation for work in artificial intelligence.
After all, the critical programming concerns of software engineering
and artificial intelligence tend to coalesce as the systems under
investigation become larger.  This explains why there is such growing
interest in Lisp outside of artificial intelligence.

   As one would expect from its goals, artificial intelligence research
generates many significant programming problems.  In other programming
cultures this spate of problems spawns new languages.  Indeed, in any
very large programming task a useful organizing principle is to control
and isolate traffic within the task modules via the invention of
language.  These languages tend to become less primitive as one
approaches the boundaries of the system where we humans interact most
often.  As a result, such systems contain complex language-processing
functions replicated many times.  Lisp has such a simple syntax and
semantics that parsing can be treated as an elementary task.  Thus
parsing technology plays almost no role in Lisp programs, and the
construction of language processors is rarely an impediment to the rate
of growth and change of large Lisp systems.  Finally, it is this very
simplicity of syntax and semantics that is responsible for the burden
and freedom borne by all Lisp programmers.  No Lisp program of any size
beyond a few lines can be written without being saturated with
discretionary functions.  Invent and fit; have fits and reinvent!  We
toast the Lisp programmer who pens his thoughts within nests of
parentheses.

Alan J. Perlis
New Haven, Connecticut


File: sicp.info,  Node: Preface,  Next: Preface 1e,  Prev: Foreword,  Up: Top

Preface to the Second Edition
*****************************

     Is it possible that software is not like anything else, that it is
     meant to be discarded: that the whole point is to always see it as
     a soap bubble?

     --Alan J. Perlis

   The material in this book has been the basis of MIT's entry-level
computer science subject since 1980.  We had been teaching this
material for four years when the first edition was published, and
twelve more years have elapsed until the appearance of this second
edition.  We are pleased that our work has been widely adopted and
incorporated into other texts.  We have seen our students take the
ideas and programs in this book and build them in as the core of new
computer systems and languages.  In literal realization of an ancient
Talmudic pun, our students have become our builders.  We are lucky to
have such capable students and such accomplished builders.

   In preparing this edition, we have incorporated hundreds of
clarifications suggested by our own teaching experience and the
comments of colleagues at MIT and elsewhere.  We have redesigned most
of the major programming systems in the book, including the
generic-arithmetic system, the interpreters, the register-machine
simulator, and the compiler; and we have rewritten all the program
examples to ensure that any Scheme implementation conforming to the
IEEE Scheme standard (IEEE 1990) will be able to run the code.

   This edition emphasizes several new themes.  The most important of
these is the central role played by different approaches to dealing
with time in computational models: objects with state, concurrent
programming, functional programming, lazy evaluation, and
nondeterministic programming.  We have included new sections on
concurrency and nondeterminism, and we have tried to integrate this
theme throughout the book.

   The first edition of the book closely followed the syllabus of our
MIT one-semester subject.  With all the new material in the second
edition, it will not be possible to cover everything in a single
semester, so the instructor will have to pick and choose.  In our own
teaching, we sometimes skip the section on logic programming (section
*Note 4-4::), we have students use the register-machine simulator but
we do not cover its implementation (section *Note 5-2::), and we give
only a cursory overview of the compiler (section *Note 5-5::).  Even
so, this is still an intense course.  Some instructors may wish to
cover only the first three or four chapters, leaving the other material
for subsequent courses.

   The World-Wide-Web site `http://mitpress.mit.edu/sicp/' provides
support for users of this book.  This includes programs from the book,
sample programming assignments, supplementary materials, and
downloadable implementations of the Scheme dialect of Lisp.


File: sicp.info,  Node: Preface 1e,  Next: Acknowledgements,  Prev: Preface,  Up: Top

Preface to the First Edition
****************************

     A computer is like a violin.  You can imagine a novice trying
     first a phonograph and then a violin.  The latter, he says, sounds
     terrible.  That is the argument we have heard from our humanists
     and most of our computer scientists.  Computer programs are good,
     they say, for particular purposes, but they aren't flexible.
     Neither is a violin, or a typewriter, until you learn how to use
     it.

     --Marvin Minsky, "Why Programming Is a Good Medium for Expressing
     Poorly-Understood and Sloppily-Formulated Ideas"

   "The Structure and Interpretation of Computer Programs" is the
entry-level subject in computer science at the Massachusetts Institute
of Technology.  It is required of all students at MIT who major in
electrical engineering or in computer science, as one-fourth of the
"common core curriculum," which also includes two subjects on circuits
and linear systems and a subject on the design of digital systems.  We
have been involved in the development of this subject since 1978, and
we have taught this material in its present form since the fall of 1980
to between 600 and 700 students each year.  Most of these students have
had little or no prior formal training in computation, although many
have played with computers a bit and a few have had extensive
programming or hardware-design experience.

   Our design of this introductory computer-science subject reflects
two major concerns.  First, we want to establish the idea that a
computer language is not just a way of getting a computer to perform
operations but rather that it is a novel formal medium for expressing
ideas about methodology.  Thus, programs must be written for people to
read, and only incidentally for machines to execute.  Second, we
believe that the essential material to be addressed by a subject at
this level is not the syntax of particular programming-language
constructs, nor clever algorithms for computing particular functions
efficiently, nor even the mathematical analysis of algorithms and the
foundations of computing, but rather the techniques used to control the
intellectual complexity of large software systems.

   Our goal is that students who complete this subject should have a
good feel for the elements of style and the aesthetics of programming.
They should have command of the major techniques for controlling
complexity in a large system. They should be capable of reading a
50-page-long program, if it is written in an exemplary style. They
should know what not to read, and what they need not understand at any
moment.  They should feel secure about modifying a program, retaining
the spirit and style of the original author.

   These skills are by no means unique to computer programming.  The
techniques we teach and draw upon are common to all of engineering
design.  We control complexity by building abstractions that hide
details when appropriate.  We control complexity by establishing
conventional interfaces that enable us to construct systems by
combining standard, well-understood pieces in a "mix and match" way.
We control complexity by establishing new languages for describing a
design, each of which emphasizes particular aspects of the design and
deemphasizes others.

   Underlying our approach to this subject is our conviction that
"computer science" is not a science and that its significance has
little to do with computers.  The computer revolution is a revolution
in the way we think and in the way we express what we think.  The
essence of this change is the emergence of what might best be called "procedural
epistemology"--the study of the structure of knowledge from an
imperative point of view, as opposed to the more declarative point of
view taken by classical mathematical subjects.  Mathematics provides a
framework for dealing precisely with notions of "what is."  Computation
provides a framework for dealing precisely with notions of "how to."

   In teaching our material we use a dialect of the programming
language Lisp.  We never formally teach the language, because we don't
have to.  We just use it, and students pick it up in a few days.  This
is one great advantage of Lisp-like languages: They have very few ways
of forming compound expressions, and almost no syntactic structure.
All of the formal properties can be covered in an hour, like the rules
of chess.  After a short time we forget about syntactic details of the
language (because there are none) and get on with the real
issues--figuring out what we want to compute, how we will decompose
problems into manageable parts, and how we will work on the parts.
Another advantage of Lisp is that it supports (but does not enforce)
more of the large-scale strategies for modular decomposition of
programs than any other language we know.  We can make procedural and
data abstractions, we can use higher-order functions to capture common
patterns of usage, we can model local state using assignment and data
mutation, we can link parts of a program with streams and delayed
evaluation, and we can easily implement embedded languages.  All of
this is embedded in an interactive environment with excellent support
for incremental program design, construction, testing, and debugging.
We thank all the generations of Lisp wizards, starting with John
McCarthy, who have fashioned a fine tool of unprecedented power and
elegance.

   Scheme, the dialect of Lisp that we use, is an attempt to bring
together the power and elegance of Lisp and Algol.  From Lisp we take
the metalinguistic power that derives from the simple syntax, the
uniform representation of programs as data objects, and the
garbage-collected heap-allocated data.  From Algol we take lexical
scoping and block structure, which are gifts from the pioneers of
programming-language design who were on the Algol committee.  We wish
to cite John Reynolds and Peter Landin for their insights into the
relationship of Church's [lambda] calculus to the structure of
programming languages.  We also recognize our debt to the
mathematicians who scouted out this territory decades before computers
appeared on the scene.  These pioneers include Alonzo Church, Barkley
Rosser, Stephen Kleene, and Haskell Curry.


File: sicp.info,  Node: Acknowledgements,  Next: Chapter 1,  Prev: Preface 1e,  Up: Top

Acknowledgements
****************

We would like to thank the many people who have helped us develop this
book and this curriculum.

   Our subject is a clear intellectual descendant of "6.231," a
wonderful subject on programming linguistics and the [lambda] calculus
taught at MIT in the late 1960s by Jack Wozencraft and Arthur Evans, Jr.

   We owe a great debt to Robert Fano, who reorganized MIT's
introductory curriculum in electrical engineering and computer science
to emphasize the principles of engineering design.  He led us in
starting out on this enterprise and wrote the first set of subject
notes from which this book evolved.

   Much of the style and aesthetics of programming that we try to teach
were developed in conjunction with Guy Lewis Steele Jr., who
collaborated with Gerald Jay Sussman in the initial development of the
Scheme language.  In addition, David Turner, Peter Henderson, Dan
Friedman, David Wise, and Will Clinger have taught us many of the
techniques of the functional programming community that appear in this
book.

   Joel Moses taught us about structuring large systems.  His
experience with the Macsyma system for symbolic computation provided
the insight that one should avoid complexities of control and
concentrate on organizing the data to reflect the real structure of the
world being modeled.

   Marvin Minsky and Seymour Papert formed many of our attitudes about
programming and its place in our intellectual lives.  To them we owe
the understanding that computation provides a means of expression for
exploring ideas that would otherwise be too complex to deal with
precisely.  They emphasize that a student's ability to write and modify
programs provides a powerful medium in which exploring becomes a
natural activity.

   We also strongly agree with Alan Perlis that programming is lots of
fun and we had better be careful to support the joy of programming.
Part of this joy derives from observing great masters at work.  We are
fortunate to have been apprentice programmers at the feet of Bill
Gosper and Richard Greenblatt.

   It is difficult to identify all the people who have contributed to
the development of our curriculum.  We thank all the lecturers,
recitation instructors, and tutors who have worked with us over the
past fifteen years and put in many extra hours on our subject,
especially Bill Siebert, Albert Meyer, Joe Stoy, Randy Davis, Louis
Braida, Eric Grimson, Rod Brooks, Lynn Stein, and Peter Szolovits.  We
would like to specially acknowledge the outstanding teaching
contributions of Franklyn Turbak, now at Wellesley; his work in
undergraduate instruction set a standard that we can all aspire to.  We
are grateful to Jerry Saltzer and Jim Miller for helping us grapple
with the mysteries of concurrency, and to Peter Szolovits and David
McAllester for their contributions to the exposition of
nondeterministic evaluation in *Note Chapter 4::.

   Many people have put in significant effort presenting this material
at other universities.  Some of the people we have worked closely with
are Jacob Katzenelson at the Technion, Hardy Mayer at the University of
California at Irvine, Joe Stoy at Oxford, Elisha Sacks at Purdue, and
Jan Komorowski at the Norwegian University of Science and Technology.
We are exceptionally proud of our colleagues who have received major
teaching awards for their adaptations of this subject at other
universities, including Kenneth Yip at Yale, Brian Harvey at the
University of California at Berkeley, and Dan Huttenlocher at Cornell.

   Al Moye' arranged for us to teach this material to engineers at
Hewlett-Packard, and for the production of videotapes of these
lectures.  We would like to thank the talented instructors--in
particular Jim Miller, Bill Siebert, and Mike Eisenberg--who have
designed continuing education courses incorporating these tapes and
taught them at universities and industry all over the world.

   Many educators in other countries have put in significant work
translating the first edition.  Michel Briand, Pierre Chamard, and
Andre' Pic produced a French edition; Susanne Daniels-Herold produced a
German edition; and Fumio Motoyoshi produced a Japanese edition.  We do
not know who produced the Chinese edition, but we consider it an honor
to have been selected as the subject of an "unauthorized" translation.

   It is hard to enumerate all the people who have made technical
contributions to the development of the Scheme systems we use for
instructional purposes.  In addition to Guy Steele, principal wizards
have included Chris Hanson, Joe Bowbeer, Jim Miller, Guillermo Rozas,
and Stephen Adams.  Others who have put in significant time are Richard
Stallman, Alan Bawden, Kent Pitman, Jon Taft, Neil Mayle, John Lamping,
Gwyn Osnos, Tracy Larrabee, George Carrette, Soma Chaudhuri, Bill
Chiarchiaro, Steven Kirsch, Leigh Klotz, Wayne Noss, Todd Cass, Patrick
O'Donnell, Kevin Theobald, Daniel Weise, Kenneth Sinclair, Anthony
Courtemanche, Henry M. Wu, Andrew Berlin, and Ruth Shyu.

   Beyond the MIT implementation, we would like to thank the many people
who worked on the IEEE Scheme standard, including William Clinger and
Jonathan Rees, who edited the R^4RS, and Chris Haynes, David Bartley,
Chris Hanson, and Jim Miller, who prepared the IEEE standard.

   Dan Friedman has been a long-time leader of the Scheme community.
The community's broader work goes beyond issues of language design to
encompass significant educational innovations, such as the high-school
curriculum based on EdScheme by Schemer's Inc., and the wonderful books
by Mike Eisenberg and by Brian Harvey and Matthew Wright.

   We appreciate the work of those who contributed to making this a
real book, especially Terry Ehling, Larry Cohen, and Paul Bethge at the
MIT Press.  Ella Mazel found the wonderful cover image.  For the second
edition we are particularly grateful to Bernard and Ella Mazel for help
with the book design, and to David Jones, TeX wizard extraordinaire.
We also are indebted to those readers who made penetrating comments on
the new draft: Jacob Katzenelson, Hardy Mayer, Jim Miller, and
especially Brian Harvey, who did unto this book as Julie did unto his
book `Simply Scheme'.

   Finally, we would like to acknowledge the support of the
organizations that have encouraged this work over the years, including
suppport from Hewlett-Packard, made possible by Ira Goldstein and Joel
Birnbaum, and support from DARPA, made possible by Bob Kahn.


File: sicp.info,  Node: Chapter 1,  Next: Chapter 2,  Prev: Acknowledgements,  Up: Top

1 Building Abstractions with Procedures
***************************************

     The acts of the mind, wherein it exerts its power over simple
     ideas, are chiefly these three: 1. Combining several simple ideas
     into one compound one, and thus all complex ideas are made.  2.
     The second is bringing two ideas, whether simple or complex,
     together, and setting them by one another so as to take a view of
     them at once, without uniting them into one, by which it gets all
     its ideas of relations.  3.  The third is separating them from all
     other ideas that accompany them in their real existence: this is
     called abstraction, and thus all its general ideas are made.

     --John Locke, _An Essay Concerning Human Understanding_ (1690)

   We are about to study the idea of a "computational process".
Computational processes are abstract beings that inhabit computers.  As
they evolve, processes manipulate other abstract things called "data".
The evolution of a process is directed by a pattern of rules called a "program".
People create programs to direct processes.  In effect, we conjure the
spirits of the computer with our spells.

   A computational process is indeed much like a sorcerer's idea of a
spirit.  It cannot be seen or touched.  It is not composed of matter at
all.  However, it is very real.  It can perform intellectual work.  It
can answer questions.  It can affect the world by disbursing money at a
bank or by controlling a robot arm in a factory.  The programs we use
to conjure processes are like a sorcerer's spells.  They are carefully
composed from symbolic expressions in arcane and esoteric "programming
languages" that prescribe the tasks we want our processes to perform.

   A computational process, in a correctly working computer, executes
programs precisely and accurately.  Thus, like the sorcerer's
apprentice, novice programmers must learn to understand and to
anticipate the consequences of their conjuring.  Even small errors
(usually called "bugs" or "glitches") in programs can have complex and
unanticipated consequences.

   Fortunately, learning to program is considerably less dangerous than
learning sorcery, because the spirits we deal with are conveniently
contained in a secure way.  Real-world programming, however, requires
care, expertise, and wisdom.  A small bug in a computer-aided design
program, for example, can lead to the catastrophic collapse of an
airplane or a dam or the self-destruction of an industrial robot.

   Master software engineers have the ability to organize programs so
that they can be reasonably sure that the resulting processes will
perform the tasks intended.  They can visualize the behavior of their
systems in advance.  They know how to structure programs so that
unanticipated problems do not lead to catastrophic consequences, and
when problems do arise, they can "debug" their programs.  Well-designed
computational systems, like well-designed automobiles or nuclear
reactors, are designed in a modular manner, so that the parts can be
constructed, replaced, and debugged separately.

Programming in Lisp
...................

We need an appropriate language for describing processes, and we will
use for this purpose the programming language Lisp.  Just as our
everyday thoughts are usually expressed in our natural language (such
as English, French, or Japanese), and descriptions of quantitative
phenomena are expressed with mathematical notations, our procedural
thoughts will be expressed in Lisp.  Lisp was invented in the late
1950s as a formalism for reasoning about the use of certain kinds of
logical expressions, called "recursion equations", as a model for
computation.  The language was conceived by John McCarthy and is based
on his paper "Recursive Functions of Symbolic Expressions and Their
Computation by Machine" (McCarthy 1960).

   Despite its inception as a mathematical formalism, Lisp is a
practical programming language.  A Lisp "interpreter" is a machine that
carries out processes described in the Lisp language.  The first Lisp
interpreter was implemented by McCarthy with the help of colleagues and
students in the Artificial Intelligence Group of the MIT Research
Laboratory of Electronics and in the MIT Computation Center.(1)  Lisp,
whose name is an acronym for LISt Processing, was designed to provide
symbol-manipulating capabilities for attacking programming problems
such as the symbolic differentiation and integration of algebraic
expressions.  It included for this purpose new data objects known as
atoms and lists, which most strikingly set it apart from all other
languages of the period.

   Lisp was not the product of a concerted design effort.  Instead, it
evolved informally in an experimental manner in response to users'
needs and to pragmatic implementation considerations.  Lisp's informal
evolution has continued through the years, and the community of Lisp
users has traditionally resisted attempts to promulgate any "official"
definition of the language.  This evolution, together with the
flexibility and elegance of the initial conception, has enabled Lisp,
which is the second oldest language in widespread use today (only
Fortran is older), to continually adapt to encompass the most modern
ideas about program design.  Thus, Lisp is by now a family of dialects,
which, while sharing most of the original features, may differ from one
another in significant ways.  The dialect of Lisp used in this book is
called Scheme.(2)

   Because of its experimental character and its emphasis on symbol
manipulation, Lisp was at first very inefficient for numerical
computations, at least in comparison with Fortran.  Over the years,
however, Lisp compilers have been developed that translate programs
into machine code that can perform numerical computations reasonably
efficiently.  And for special applications, Lisp has been used with
great effectiveness.(3)  Although Lisp has not yet overcome its old
reputation as hopelessly inefficient, Lisp is now used in many
applications where efficiency is not the central concern.  For example,
Lisp has become a language of choice for operating-system shell
languages and for extension languages for editors and computer-aided
design systems.

   If Lisp is not a mainstream language, why are we using it as the
framework for our discussion of programming?  Because the language
possesses unique features that make it an excellent medium for studying
important programming constructs and data structures and for relating
them to the linguistic features that support them.  The most
significant of these features is the fact that Lisp descriptions of
processes, called "procedures", can themselves be represented and
manipulated as Lisp data.  The importance of this is that there are
powerful program-design techniques that rely on the ability to blur the
traditional distinction between "passive" data and "active" processes.
As we shall discover, Lisp's flexibility in handling procedures as data
makes it one of the most convenient languages in existence for
exploring these techniques.  The ability to represent procedures as
data also makes Lisp an excellent language for writing programs that
must manipulate other programs as data, such as the interpreters and
compilers that support computer languages.  Above and beyond these
considerations, programming in Lisp is great fun.

* Menu:

* 1-1::              The Elements of Programming
* 1-2::              Procedures and the Processes They Generate
* 1-3::              Formulating Abstractions with Higher-Order Procedures

   ---------- Footnotes ----------

   (1) The `Lisp 1 Programmer's Manual' appeared in 1960, and the `Lisp
1.5 Programmer's Manual' (McCarthy 1965) was published in 1962.  The
early history of Lisp is described in McCarthy 1978.

   (2) The two dialects in which most major Lisp programs of the 1970s
were written are MacLisp (Moon 1978; Pitman 1983), developed at the MIT
Project MAC, and Interlisp (Teitelman 1974), developed at Bolt Beranek
and Newman Inc. and the Xerox Palo Alto Research Center.  Portable
Standard Lisp (Hearn 1969; Griss 1981) was a Lisp dialect designed to
be easily portable between different machines.  MacLisp spawned a
number of subdialects, such as Franz Lisp, which was developed at the
University of California at Berkeley, and Zetalisp (Moon 1981), which
was based on a special-purpose processor designed at the MIT Artificial
Intelligence Laboratory to run Lisp very efficiently.  The Lisp dialect
used in this book, called Scheme (Steele 1975), was invented in 1975 by
Guy Lewis Steele Jr. and Gerald Jay Sussman of the MIT Artificial
Intelligence Laboratory and later reimplemented for instructional use
at MIT.  Scheme became an IEEE standard in 1990 (IEEE 1990).  The
Common Lisp dialect (Steele 1982, Steele 1990) was developed by the
Lisp community to combine features from the earlier Lisp dialects to
make an industrial standard for Lisp.  Common Lisp became an ANSI
standard in 1994 (ANSI 1994).

   (3) One such special application was a breakthrough computation of
scientific importance--an integration of the motion of the Solar System
that extended previous results by nearly two orders of magnitude, and
demonstrated that the dynamics of the Solar System is chaotic.  This
computation was made possible by new integration algorithms, a
special-purpose compiler, and a special-purpose computer all
implemented with the aid of software tools written in Lisp (Abelson et
al. 1992; Sussman and Wisdom 1992).


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1.1 The Elements of Programming
===============================

A powerful programming language is more than just a means for
instructing a computer to perform tasks.  The language also serves as a
framework within which we organize our ideas about processes.  Thus,
when we describe a language, we should pay particular attention to the
means that the language provides for combining simple ideas to form
more complex ideas.  Every powerful language has three mechanisms for
accomplishing this:

"primitive expressions"
     which represent the simplest entities the language is concerned
     with,

"means of combination"
     by which compound elements are built from simpler ones, and

"means of abstraction"
     by which compound elements can be named and manipulated as units.


   In programming, we deal with two kinds of elements: procedures and
data. (Later we will discover that they are really not so distinct.)
Informally, data is "stuff" that we want to manipulate, and procedures
are descriptions of the rules for manipulating the data.  Thus, any
powerful programming language should be able to describe primitive data
and primitive procedures and should have methods for combining and
abstracting procedures and data.

   In this chapter we will deal only with simple numerical data so that
we can focus on the rules for building procedures.(1) In later chapters
we will see that these same rules allow us to build procedures to
manipulate compound data as well.

* Menu:

* 1-1-1::            Expressions
* 1-1-2::            Naming and the Environment
* 1-1-3::            Evaluating Combinations
* 1-1-4::            Compound Procedures
* 1-1-5::            The Substitution Model for Procedure Application
* 1-1-6::            Conditional Expressions and Predicates
* 1-1-7::            Example: Square Roots by Newton's Method
* 1-1-8::            Procedures as Black-Box Abstractions

   ---------- Footnotes ----------

   (1) The characterization of numbers as "simple data" is a barefaced
bluff.  In fact, the treatment of numbers is one of the trickiest and
most confusing aspects of any programming language.  Some typical
issues involved are these: Some computer systems distinguish "integers",
such as 2, from "real numbers", such as 2.71.  Is the real number 2.00
different from the integer 2?  Are the arithmetic operations used for
integers the same as the operations used for real numbers?  Does 6
divided by 2 produce 3, or 3.0?  How large a number can we represent?
How many decimal places of accuracy can we represent?  Is the range of
integers the same as the range of real numbers?  Above and beyond these
questions, of course, lies a collection of issues concerning roundoff
and truncation errors - the entire science of numerical analysis.
Since our focus in this book is on large-scale program design rather
than on numerical techniques, we are going to ignore these problems.
The numerical examples in this chapter will exhibit the usual roundoff
behavior that one observes when using arithmetic operations that
preserve a limited number of decimal places of accuracy in noninteger
operations.


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1.1.1 Expressions
-----------------

One easy way to get started at programming is to examine some typical
interactions with an interpreter for the Scheme dialect of Lisp.
Imagine that you are sitting at a computer terminal.  You type an "expression",
and the interpreter responds by displaying the result of its "evaluating"
that expression.

   One kind of primitive expression you might type is a number.  (More
precisely, the expression that you type consists of the numerals that
represent the number in base 10.)  If you present Lisp with a number

     486

the interpreter will respond by printing (1)

     486

   Expressions representing numbers may be combined with an expression
representing a primitive procedure (such as `+' or `*') to form a
compound expression that represents the application of the procedure to
those numbers.  For example:

     (+ 137 349)
     486

     (- 1000 334)
     666

     (* 5 99)
     495

     (/ 10 5)
     2

     (+ 2.7 10)
     12.7

   Expressions such as these, formed by delimiting a list of
expressions within parentheses in order to denote procedure
application, are called "combinations".  The leftmost element in the
list is called the "operator", and the other elements are called "operands".
The value of a combination is obtained by applying the procedure
specified by the operator to the "arguments" that are the values of the
operands.

   The convention of placing the operator to the left of the operands
is known as "prefix notation", and it may be somewhat confusing at
first because it departs significantly from the customary mathematical
convention.  Prefix notation has several advantages, however.  One of
them is that it can accommodate procedures that may take an arbitrary
number of arguments, as in the following examples:

     (+ 21 35 12 7)
     75

     (* 25 4 12)
     1200

   No ambiguity can arise, because the operator is always the leftmost
element and the entire combination is delimited by the parentheses.

   A second advantage of prefix notation is that it extends in a
straightforward way to allow combinations to be nested, that is, to
have combinations whose elements are themselves combinations:

     (+ (* 3 5) (- 10 6))
     19

   There is no limit (in principle) to the depth of such nesting and to
the overall complexity of the expressions that the Lisp interpreter can
evaluate.  It is we humans who get confused by still relatively simple
expressions such as

     (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))

which the interpreter would readily evaluate to be 57.  We can help
ourselves by writing such an expression in the form

     (+ (* 3
           (+ (* 2 4)
              (+ 3 5)))
        (+ (- 10 7)
           6))

following a formatting convention known as "pretty-printing", in which
each long combination is written so that the operands are aligned
vertically.  The resulting indentations display clearly the structure
of the expression.(2)

   Even with complex expressions, the interpreter always operates in
the same basic cycle: It reads an expression from the terminal,
evaluates the expression, and prints the result.  This mode of
operation is often expressed by saying that the interpreter runs in a "read-eval-print
loop".  Observe in particular that it is not necessary to explicitly
instruct the interpreter to print the value of the expression.(3)

   ---------- Footnotes ----------

   (1) Throughout this book, when we wish to emphasize the distinction
between the input typed by the user and the response printed by the
interpreter, we will show the latter in slanted characters.

   (2) Lisp systems typically provide features to aid the user in
formatting expressions.  Two especially useful features are one that
automatically indents to the proper pretty-print position whenever a
new line is started and one that highlights the matching left
parenthesis whenever a right parenthesis is typed.

   (3) Lisp obeys the convention that every expression has a value.
This convention, together with the old reputation of Lisp as an
inefficient language, is the source of the quip by Alan Perlis
(paraphrasing Oscar Wilde) that "Lisp programmers know the value of
everything but the cost of nothing."


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1.1.2 Naming and the Environment
--------------------------------

A critical aspect of a programming language is the means it provides
for using names to refer to computational objects.  We say that the
name identifies a "variable" whose "value" is the object.

   In the Scheme dialect of Lisp, we name things with `define'.  Typing

     (define size 2)

causes the interpreter to associate the value 2 with the name
`size'.(1) Once the name `size' has been associated with the number 2,
we can refer to the value 2 by name:

     size
     2

     (* 5 size)
     10

   Here are further examples of the use of `define':

     (define pi 3.14159)

     (define radius 10)

     (* pi (* radius radius))
     314.159

     (define circumference (* 2 pi radius))

     circumference
     62.8318

   `Define' is our language's simplest means of abstraction, for it
allows us to use simple names to refer to the results of compound
operations, such as the `circumference' computed above.  In general,
computational objects may have very complex structures, and it would be
extremely inconvenient to have to remember and repeat their details
each time we want to use them.  Indeed, complex programs are
constructed by building, step by step, computational objects of
increasing complexity. The interpreter makes this step-by-step program
construction particularly convenient because name-object associations
can be created incrementally in successive interactions.  This feature
encourages the incremental development and testing of programs and is
largely responsible for the fact that a Lisp program usually consists
of a large number of relatively simple procedures.

   It should be clear that the possibility of associating values with
symbols and later retrieving them means that the interpreter must
maintain some sort of memory that keeps track of the name-object pairs.
This memory is called the "environment" (more precisely the "global
environment", since we will see later that a computation may involve a
number of different environments).(2)

   ---------- Footnotes ----------

   (1) In this book, we do not show the interpreter's response to
evaluating definitions, since this is highly implementation-dependent.

   (2) *Note Chapter 3:: will show that this notion of environment is
crucial, both for understanding how the interpreter works and for
implementing interpreters.


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1.1.3 Evaluating Combinations
-----------------------------

One of our goals in this chapter is to isolate issues about thinking
procedurally.  As a case in point, let us consider that, in evaluating
combinations, the interpreter is itself following a procedure.

     To evaluate a combination, do the following:

       1. Evaluate the subexpressions of the combination.

       2. Apply the procedure that is the value of the leftmost
          subexpression (the operator) to the arguments that are the
          values of the other subexpressions (the operands).


   Even this simple rule illustrates some important points about
processes in general.  First, observe that the first step dictates that
in order to accomplish the evaluation process for a combination we must
first perform the evaluation process on each element of the
combination.  Thus, the evaluation rule is "recursive" in nature; that
is, it includes, as one of its steps, the need to invoke the rule
itself.(1)

   Notice how succinctly the idea of recursion can be used to express
what, in the case of a deeply nested combination, would otherwise be
viewed as a rather complicated process.  For example, evaluating

     (* (+ 2 (* 4 6))
        (+ 3 5 7))

requires that the evaluation rule be applied to four different
combinations.  We can obtain a picture of this process by representing
the combination in the form of a tree, as shown in *Note Figure 1-1::.
Each combination is represented by a node with branches corresponding
to the operator and the operands of the combination stemming from it.
The terminal nodes (that is, nodes with no branches stemming from them)
represent either operators or numbers.  Viewing evaluation in terms of
the tree, we can imagine that the values of the operands percolate
upward, starting from the terminal nodes and then combining at higher
and higher levels.  In general, we shall see that recursion is a very
powerful technique for dealing with hierarchical, treelike objects.  In
fact, the "percolate values upward" form of the evaluation rule is an
example of a general kind of process known as "tree accumulation".

     *Figure 1.1:* Tree representation, showing the value of each
     subcombination.

             390
             /|\____________
            / |             \
           *  26            15
              /|\            |
             / | \         // \\
            +  2  24      / |  | \
                  /|\    +  3  5  7
                 / | \
                *  4  6

   Next, observe that the repeated application of the first step brings
us to the point where we need to evaluate, not combinations, but
primitive expressions such as numerals, built-in operators, or other
names.  We take care of the primitive cases by stipulating that

   * the values of numerals are the numbers that they name,

   * the values of built-in operators are the machine instruction
     sequences that carry out the corresponding operations, and

   * the values of other names are the objects associated with those
     names in the environment.


   We may regard the second rule as a special case of the third one by
stipulating that symbols such as `+' and `*' are also included in the
global environment, and are associated with the sequences of machine
instructions that are their "values."  The key point to notice is the
role of the environment in determining the meaning of the symbols in
expressions.  In an interactive language such as Lisp, it is
meaningless to speak of the value of an expression such as `(+ x 1)'
without specifying any information about the environment that would
provide a meaning for the symbol `x' (or even for the symbol `+').  As
we shall see in *Note Chapter 3::, the general notion of the
environment as providing a context in which evaluation takes place will
play an important role in our understanding of program execution.

   Notice that the evaluation rule given above does not handle
definitions.  For instance, evaluating `(define x 3)' does not apply
`define' to two arguments, one of which is the value of the symbol `x'
and the other of which is 3, since the purpose of the `define' is
precisely to associate `x' with a value.  (That is, `(define x 3)' is
not a combination.)

   Such exceptions to the general evaluation rule are called forms
"special forms".  `Define' is the only example of a special form that
we have seen so far, but we will meet others shortly.  Each special
form has its own evaluation rule. The various kinds of expressions
(each with its associated evaluation rule) constitute the syntax of the
programming language.  In comparison with most other programming
languages, Lisp has a very simple syntax; that is, the evaluation rule
for expressions can be described by a simple general rule together with
specialized rules for a small number of special forms.(2)

   ---------- Footnotes ----------

   (1) It may seem strange that the evaluation rule says, as part of
the first step, that we should evaluate the leftmost element of a
combination, since at this point that can only be an operator such as
`+' or `*' representing a built-in primitive procedure such as addition
or multiplication.  We will see later that it is useful to be able to
work with combinations whose operators are themselves compound
expressions.

   (2) Special syntactic forms that are simply convenient alternative
surface structures for things that can be written in more uniform ways
are sometimes called "syntactic sugar", to use a phrase coined by Peter
Landin.  In comparison with users of other languages, Lisp programmers,
as a rule, are less concerned with matters of syntax.  (By contrast,
examine any Pascal manual and notice how much of it is devoted to
descriptions of syntax.)  This disdain for syntax is due partly to the
flexibility of Lisp, which makes it easy to change surface syntax, and
partly to the observation that many "convenient" syntactic constructs,
which make the language less uniform, end up causing more trouble than
they are worth when programs become large and complex.  In the words of
Alan Perlis, "Syntactic sugar causes cancer of the semicolon."


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1.1.4 Compound Procedures
-------------------------

We have identified in Lisp some of the elements that must appear in any
powerful programming language:

   * Numbers and arithmetic operations are primitive data and
     procedures.

   * Nesting of combinations provides a means of combining operations.

   * Definitions that associate names with values provide a limited
     means of abstraction.


   Now we will learn about "procedure definitions", a much more powerful
abstraction technique by which a compound operation can be given a name
and then referred to as a unit.

   We begin by examining how to express the idea of "squaring."  We
might say, "To square something, multiply it by itself."  This is
expressed in our language as

     (define (square x) (* x x))

   We can understand this in the following way:

     (define (square    x)         (*      x         x))
       |        |       |           |      |         |
      To     square  something,  multiply  it  by  itself.

   We have here a "compound procedure", which has been given the name
`square'.  The procedure represents the operation of multiplying
something by itself.  The thing to be multiplied is given a local name,
`x', which plays the same role that a pronoun plays in natural
language.  Evaluating the definition creates this compound procedure
and associates it with the name `square'.(1)

   The general form of a procedure definition is

     (define (<NAME> <FORMAL PARAMETERS>) <BODY>)

   The <NAME> is a symbol to be associated with the procedure
definition in the environment.(2) The <FORMAL PARAMETERS> are the names
used within the body of the procedure to refer to the corresponding
arguments of the procedure.  The <BODY> is an expression that will
yield the value of the procedure application when the formal parameters
are replaced by the actual arguments to which the procedure is
applied.(3)  The <NAME> and the <FORMAL PARAMETERS> are grouped within
parentheses, just as they would be in an actual call to the procedure
being defined.

   Having defined `square', we can now use it:

     (square 21)
     441

     (square (+ 2 5))
     49

     (square (square 3))
     81

   We can also use `square' as a building block in defining other
procedures.  For example, x^2 + y^2 can be expressed as

     (+ (square x) (square y))

   We can easily define a procedure `sum-of-squares' that, given any two
numbers as arguments, produces the sum of their squares:

     (define (sum-of-squares x y)
       (+ (square x) (square y)))
     (sum-of-squares 3 4)
     25

   Now we can use `sum-of-squares' as a building block in constructing
further procedures:

     (define (f a)
       (sum-of-squares (+ a 1) (* a 2)))

     (f 5)
     136

   Compound procedures are used in exactly the same way as primitive
procedures.  Indeed, one could not tell by looking at the definition of
`sum-of-squares' given above whether `square' was built into the
interpreter, like `+' and `*', or defined as a compound procedure.

   ---------- Footnotes ----------

   (1) Observe that there are two different operations being combined
here: we are creating the procedure, and we are giving it the name
`square'.  It is possible, indeed important, to be able to separate
these two notions--to create procedures without naming them, and to
give names to procedures that have already been created.  We will see
how to do this in section *Note 1-3-2::.

   (2) Throughout this book, we will describe the general syntax of
expressions by using italic symbols delimited by angle brackets--e.g.,
<NAME>--to denote the "slots" in the expression to be filled in when
such an expression is actually used.

   (3) More generally, the body of the procedure can be a sequence of
expressions.  In this case, the interpreter evaluates each expression
in the sequence in turn and returns the value of the final expression
as the value of the procedure application.


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1.1.5 The Substitution Model for Procedure Application
------------------------------------------------------

To evaluate a combination whose operator names a compound procedure, the
interpreter follows much the same process as for combinations whose
operators name primitive procedures, which we described in section
*Note 1-1-3::.  That is, the interpreter evaluates the elements of the
combination and applies the procedure (which is the value of the
operator of the combination) to the arguments (which are the values of
the operands of the combination).

   We can assume that the mechanism for applying primitive procedures
to arguments is built into the interpreter.  For compound procedures,
the application process is as follows:

     To apply a compound procedure to arguments, evaluate the body of
     the procedure with each formal parameter replaced by the
     corresponding argument.

   To illustrate this process, let's evaluate the combination

     (f 5)

where `f' is the procedure defined in section *Note 1-1-4::.  We begin
by retrieving the body of `f':

     (sum-of-squares (+ a 1) (* a 2))

   Then we replace the formal parameter `a' by the argument 5:

     (sum-of-squares (+ 5 1) (* 5 2))

   Thus the problem reduces to the evaluation of a combination with two
operands and an operator `sum-of-squares'.  Evaluating this combination
involves three subproblems.  We must evaluate the operator to get the
procedure to be applied, and we must evaluate the operands to get the
arguments.  Now `(+ 5 1)' produces 6 and `(* 5 2)' produces 10, so we
must apply the `sum-of-squares' procedure to 6 and 10.  These values
are substituted for the formal parameters `x' and `y' in the body of
`sum-of-squares', reducing the expression to

     (+ (square 6) (square 10))

   If we use the definition of `square', this reduces to

     (+ (* 6 6) (* 10 10))

which reduces by multiplication to

     (+ 36 100)

and finally to

     136

   The process we have just described is called the "substitution model"
for procedure application.  It can be taken as a model that determines
the "meaning" of procedure application, insofar as the procedures in
this chapter are concerned.  However, there are two points that should
be stressed:

   * The purpose of the substitution is to help us think about procedure
     application, not to provide a description of how the interpreter
     really works.  Typical interpreters do not evaluate procedure
     applications by manipulating the text of a procedure to substitute
     values for the formal parameters.  In practice, the "substitution"
     is accomplished by using a local environment for the formal
     parameters.  We will discuss this more fully in *Note Chapter 3::
     and *Note Chapter 4:: when we examine the implementation of an
     interpreter in detail.

   * Over the course of this book, we will present a sequence of
     increasingly elaborate models of how interpreters work,
     culminating with a complete implementation of an interpreter and
     compiler in *Note Chapter 5::.  The substitution model is only the
     first of these models--a way to get started thinking formally
     about the evaluation process.  In general, when modeling phenomena
     in science and engineering, we begin with simplified, incomplete
     models.  As we examine things in greater detail, these simple
     models become inadequate and must be replaced by more refined
     models.  The substitution model is no exception.  In particular,
     when we address in *Note Chapter 3:: the use of procedures with
     "mutable data," we will see that the substitution model breaks
     down and must be replaced by a more complicated model of procedure
     application.(1)


Applicative order versus normal order
.....................................

According to the description of evaluation given in section *Note
1-1-3::, the interpreter first evaluates the operator and operands and
then applies the resulting procedure to the resulting arguments.  This
is not the only way to perform evaluation.  An alternative evaluation
model would not evaluate the operands until their values were needed.
Instead it would first substitute operand expressions for parameters
until it obtained an expression involving only primitive operators, and
would then perform the evaluation.  If we used this method, the
evaluation of `(f 5)' would proceed according to the sequence of
expansions

     (sum-of-squares (+ 5 1) (* 5 2))

     (+    (square (+ 5 1))      (square (* 5 2))  )

     (+    (* (+ 5 1) (+ 5 1))   (* (* 5 2) (* 5 2)))

followed by the reductions

     (+         (* 6 6)             (* 10 10))

     (+           36                   100)

                         136

   This gives the same answer as our previous evaluation model, but the
process is different.  In particular, the evaluations of `(+ 5 1)' and
`(* 5 2)' are each performed twice here, corresponding to the reduction
of the expression `(* x x)' with `x' replaced respectively by `(+ 5 1)'
and `(* 5 2)'.

   This alternative "fully expand and then reduce" evaluation method is
known as "normal-order evaluation", in contrast to the "evaluate the
arguments and then apply" method that the interpreter actually uses,
which is called "applicative-order evaluation".  It can be shown that,
for procedure applications that can be modeled using substitution
(including all the procedures in the first two chapters of this book)
and that yield legitimate values, normal-order and applicative-order
evaluation produce the same value.  (See *Note Exercise 1-5:: for an
instance of an "illegitimate" value where normal-order and
applicative-order evaluation do not give the same result.)

   Lisp uses applicative-order evaluation, partly because of the
additional efficiency obtained from avoiding multiple evaluations of
expressions such as those illustrated with `(+ 5 1)' and `(* 5 2)'
above and, more significantly, because normal-order evaluation becomes
much more complicated to deal with when we leave the realm of
procedures that can be modeled by substitution.  On the other hand,
normal-order evaluation can be an extremely valuable tool, and we will
investigate some of its implications in *Note Chapter 3:: and *Note
Chapter 4::.(2)

   ---------- Footnotes ----------

   (1) Despite the simplicity of the substitution idea, it turns out to
be surprisingly complicated to give a rigorous mathematical definition
of the substitution process.  The problem arises from the possibility of
confusion between the names used for the formal parameters of a
procedure and the (possibly identical) names used in the expressions to
which the procedure may be applied.  Indeed, there is a long history of
erroneous definitions of "substitution" in the literature of logic and
programming semantics.  See Stoy 1977 for a careful discussion of
substitution.

   (2) In *Note Chapter 3:: we will introduce "stream processing",
which is a way of handling apparently "infinite" data structures by
incorporating a limited form of normal-order evaluation.  In section
*Note 4-2:: we will modify the Scheme interpreter to produce a
normal-order variant of Scheme.


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1.1.6 Conditional Expressions and Predicates
--------------------------------------------

The expressive power of the class of procedures that we can define at
this point is very limited, because we have no way to make tests and to
perform different operations depending on the result of a test.  For
instance, we cannot define a procedure that computes the absolute value
of a number by testing whether the number is positive, negative, or
zero and taking different actions in the different cases according to
the rule

           /
           |   x  if x > 0
     |x| = <   0  if x = 0
           |  -x  if x < 0
           \

   This construct is called a "case analysis", and there is a special
form in Lisp for notating such a case analysis.  It is called `cond'
(which stands for "conditional"), and it is used as follows:

     (define (abs x)
       (cond ((> x 0) x)
             ((= x 0) 0)
             ((< x 0) (- x))))

   The general form of a conditional expression is

     (cond (<P1> <E1>)
           (<P2> <E2>)
           ...
           (<PN> <EN>))

consisting of the symbol `cond' followed by parenthesized pairs of
expressions

     (<P> <E>)

called "clauses". The first expression in each pair is a "predicate"--that
is, an expression whose value is interpreted as either true or false.(1)

   Conditional expressions are evaluated as follows.  The predicate
<P1> is evaluated first.  If its value is false, then <P2> is
evaluated.  If <P2>'s value is also false, then <P3> is evaluated.
This process continues until a predicate is found whose value is true,
in which case the interpreter returns the value of the corresponding expression
"consequent expression" <E> of the clause as the value of the
conditional expression.  If none of the <P>'s is found to be true, the
value of the `cond' is undefined.

   The word "predicate" is used for procedures that return true or
false, as well as for expressions that evaluate to true or false.  The
absolute-value procedure `abs' makes use of the primitive predicates
`>', `<', and `='.(2) These take two numbers as arguments and test
whether the first number is, respectively, greater than, less than, or
equal to the second number, returning true or false accordingly.

   Another way to write the absolute-value procedure is

     (define (abs x)
       (cond ((< x 0) (- x))
             (else x)))

which could be expressed in English as "If x is less than zero return -
x; otherwise return x."  `Else' is a special symbol that can be used in
place of the <P> in the final clause of a `cond'.  This causes the
`cond' to return as its value the value of the corresponding <E>
whenever all previous clauses have been bypassed.  In fact, any
expression that always evaluates to a true value could be used as the
<P> here.

   Here is yet another way to write the absolute-value procedure:

     (define (abs x)
       (if (< x 0)
           (- x)
           x))

   This uses the special form `if', a restricted type of conditional
that can be used when there are precisely two cases in the case
analysis.  The general form of an `if' expression is

     (if <PREDICATE> <CONSEQUENT> <ALTERNATIVE>)

   To evaluate an `if' expression, the interpreter starts by evaluating
the <PREDICATE> part of the expression.  If the <PREDICATE> evaluates
to a true value, the interpreter then evaluates the <CONSEQUENT> and
returns its value.  Otherwise it evaluates the <ALTERNATIVE> and returns
its value.(3)

   In addition to primitive predicates such as `<', `=', and `>', there
are logical composition operations, which enable us to construct
compound predicates.  The three most frequently used are these:

   * `(and <E1> ... <EN>)'

     The interpreter evaluates the expressions <E> one at a time, in
     left-to-right order.  If any <E> evaluates to false, the value of
     the `and' expression is false, and the rest of the <E>'s are not
     evaluated.  If all <E>'s evaluate to true values, the value of the
     `and' expression is the value of the last one.

   * `(or <E1> ... <EN>)'

     The interpreter evaluates the expressions <E> one at a time, in
     left-to-right order.  If any <E> evaluates to a true value, that
     value is returned as the value of the `or' expression, and the
     rest of the <E>'s are not evaluated.  If all <E>'s evaluate to
     false, the value of the `or' expression is false.

   * `(not <E>)'

     The value of a `not' expression is true when the expression <E>
     evaluates to false, and false otherwise.


   Notice that `and' and `or' are special forms, not procedures, because
the subexpressions are not necessarily all evaluated.  `Not' is an
ordinary procedure.

   As an example of how these are used, the condition that a number x
be in the range 5 < x < 10 may be expressed as

     (and (> x 5) (< x 10))

   As another example, we can define a predicate to test whether one
number is greater than or equal to another as

     (define (>= x y)
       (or (> x y) (= x y)))

or alternatively as

     (define (>= x y)
       (not (< x y)))

     *Exercise 1.1:* Below is a sequence of expressions.  What is the
     result printed by the interpreter in response to each expression?
     Assume that the sequence is to be evaluated in the order in which
     it is presented.

          10

          (+ 5 3 4)

          (- 9 1)

          (/ 6 2)

          (+ (* 2 4) (- 4 6))

          (define a 3)

          (define b (+ a 1))

          (+ a b (* a b))

          (= a b)

          (if (and (> b a) (< b (* a b)))
              b
              a)

          (cond ((= a 4) 6)
                ((= b 4) (+ 6 7 a))
                (else 25))

          (+ 2 (if (> b a) b a))

          (* (cond ((> a b) a)
                   ((< a b) b)
                   (else -1))
             (+ a 1))

     *Exercise 1.2:* Translate the following expression into prefix
     form.

          5 + 4 + (2 - (3 - (6 + 4/5)))
          -----------------------------
                 3(6 - 2)(2 - 7)

     *Exercise 1.3:* Define a procedure that takes three numbers as
     arguments and returns the sum of the squares of the two larger
     numbers.

     *Exercise 1.4:* Observe that our model of evaluation allows for
     combinations whose operators are compound expressions.  Use this
     observation to describe the behavior of the following procedure:

          (define (a-plus-abs-b a b)
            ((if (> b 0) + -) a b))

     *Exercise 1.5:* Ben Bitdiddle has invented a test to determine
     whether the interpreter he is faced with is using
     applicative-order evaluation or normal-order evaluation.  He
     defines the following two procedures:

          (define (p) (p))

          (define (test x y)
            (if (= x 0)
                0
                y))

     Then he evaluates the expression

          (test 0 (p))

     What behavior will Ben observe with an interpreter that uses
     applicative-order evaluation?  What behavior will he observe with
     an interpreter that uses normal-order evaluation?  Explain your
     answer.  (Assume that the evaluation rule for the special form
     `if' is the same whether the interpreter is using normal or
     applicative order: The predicate expression is evaluated first,
     and the result determines whether to evaluate the consequent or
     the alternative expression.)

   ---------- Footnotes ----------

   (1) "Interpreted as either true or false" means this: In Scheme,
there are two distinguished values that are denoted by the constants
`#t' and `#f'.  When the interpreter checks a predicate's value, it
interprets `#f' as false.  Any other value is treated as true.  (Thus,
providing `#t' is logically unnecessary, but it is convenient.)  In
this book we will use names `true' and `false', which are associated
with the values `#t' and `#f' respectively.

   (2) `Abs' also uses the "minus" operator `-', which, when used with
a single operand, as in `(- x)', indicates negation.

   (3) A minor difference between `if' and `cond' is that the <E> part
of each `cond' clause may be a sequence of expressions.  If the
corresponding <P> is found to be true, the expressions <E> are
evaluated in sequence and the value of the final expression in the
sequence is returned as the value of the `cond'.  In an `if'
expression, however, the <CONSEQUENT> and <ALTERNATIVE> must be single
expressions.


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1.1.7 Example: Square Roots by Newton's Method
----------------------------------------------

Procedures, as introduced above, are much like ordinary mathematical
functions.  They specify a value that is determined by one or more
parameters.  But there is an important difference between mathematical
functions and computer procedures.  Procedures must be effective.

   As a case in point, consider the problem of computing square roots.
We can define the square-root function as

     sqrt(x) = the y such that y >= 0 and y^2 = x

   This describes a perfectly legitimate mathematical function.  We
could use it to recognize whether one number is the square root of
another, or to derive facts about square roots in general.  On the
other hand, the definition does not describe a procedure.  Indeed, it
tells us almost nothing about how to actually find the square root of a
given number.  It will not help matters to rephrase this definition in
pseudo-Lisp:

     (define (sqrt x)
       (the y (and (>= y 0)
                   (= (square y) x))))

   This only begs the question.

   The contrast between function and procedure is a reflection of the
general distinction between describing properties of things and
describing how to do things, or, as it is sometimes referred to, the
distinction between declarative knowledge and imperative knowledge.  In
mathematics we are usually concerned with declarative (what is)
descriptions, whereas in computer science we are usually concerned with
imperative (how to) descriptions.(1)

   How does one compute square roots?  The most common way is to use
Newton's method of successive approximations, which says that whenever
we have a guess y for the value of the square root of a number x, we
can perform a simple manipulation to get a better guess (one closer to
the actual square root) by averaging y with x/y.(2) For example, we can
compute the square root of 2 as follows.  Suppose our initial guess is
1:

     Guess  Quotient             Average
     1      (2/1) = 2            ((2 + 1)/2) = 1.5
     1.5    (2/1.5) = 1.3333     ((1.3333 + 1.5)/2) = 1.4167
     1.4167 (2/1.4167) = 1.4118  ((1.4167 + 1.4118)/2) = 1.4142
     1.4142 ...                  ...

Continuing this process, we obtain better and better approximations to
the square root.

   Now let's formalize the process in terms of procedures.  We start
with a value for the radicand (the number whose square root we are
trying to compute) and a value for the guess.  If the guess is good
enough for our purposes, we are done; if not, we must repeat the
process with an improved guess.  We write this basic strategy as a
procedure:

     (define (sqrt-iter guess x)
       (if (good-enough? guess x)
           guess
           (sqrt-iter (improve guess x)
                      x)))

   A guess is improved by averaging it with the quotient of the
radicand and the old guess:

     (define (improve guess x)
       (average guess (/ x guess)))

where

     (define (average x y)
       (/ (+ x y) 2))

   We also have to say what we mean by "good enough."  The following
will do for illustration, but it is not really a very good test.  (See
exercise *Note Exercise 1-7::.)  The idea is to improve the answer
until it is close enough so that its square differs from the radicand
by less than a predetermined tolerance (here 0.001):(3)

     (define (good-enough? guess x)
       (< (abs (- (square guess) x)) 0.001))

   Finally, we need a way to get started.  For instance, we can always
guess that the square root of any number is 1:(4)

     (define (sqrt x)
       (sqrt-iter 1.0 x))

   If we type these definitions to the interpreter, we can use `sqrt'
just as we can use any procedure:

     (sqrt 9)
     3.00009155413138

     (sqrt (+ 100 37))
     11.704699917758145

     (sqrt (+ (sqrt 2) (sqrt 3)))
     1.7739279023207892

     (square (sqrt 1000))
     1000.000369924366

   The `sqrt' program also illustrates that the simple procedural
language we have introduced so far is sufficient for writing any purely
numerical program that one could write in, say, C or Pascal.  This
might seem surprising, since we have not included in our language any
iterative (looping) constructs that direct the computer to do something
over and over again.  `Sqrt-iter', on the other hand, demonstrates how
iteration can be accomplished using no special construct other than the
ordinary ability to call a procedure.(5)

     *Exercise 1.6:* Alyssa P. Hacker doesn't see why `if' needs to be
     provided as a special form.  "Why can't I just define it as an
     ordinary procedure in terms of `cond'?" she asks.  Alyssa's friend
     Eva Lu Ator claims this can indeed be done, and she defines a new
     version of `if':

          (define (new-if predicate then-clause else-clause)
            (cond (predicate then-clause)
                  (else else-clause)))

     Eva demonstrates the program for Alyssa:

          (new-if (= 2 3) 0 5)
          5

          (new-if (= 1 1) 0 5)
          0

     Delighted, Alyssa uses `new-if' to rewrite the square-root program:

          (define (sqrt-iter guess x)
            (new-if (good-enough? guess x)
                    guess
                    (sqrt-iter (improve guess x)
                               x)))

     What happens when Alyssa attempts to use this to compute square
     roots?  Explain.

     *Exercise 1.7:* The `good-enough?' test used in computing square
     roots will not be very effective for finding the square roots of
     very small numbers.  Also, in real computers, arithmetic operations
     are almost always performed with limited precision.  This makes
     our test inadequate for very large numbers.  Explain these
     statements, with examples showing how the test fails for small and
     large numbers.  An alternative strategy for implementing
     `good-enough?' is to watch how `guess' changes from one iteration
     to the next and to stop when the change is a very small fraction
     of the guess.  Design a square-root procedure that uses this kind
     of end test.  Does this work better for small and large numbers?

     *Exercise 1.8:* Newton's method for cube roots is based on the
     fact that if y is an approximation to the cube root of x, then a
     better approximation is given by the value

          x/y^2 + 2y
          ----------
              3

     Use this formula to implement a cube-root procedure analogous to
     the square-root procedure.  (In section *Note 1-3-4:: we will see
     how to implement Newton's method in general as an abstraction of
     these square-root and cube-root procedures.)

   ---------- Footnotes ----------

   (1) Declarative and imperative descriptions are intimately related,
as indeed are mathematics and computer science.  For instance, to say
that the answer produced by a program is "correct" is to make a
declarative statement about the program.  There is a large amount of
research aimed at establishing techniques for proving that programs are
correct, and much of the technical difficulty of this subject has to do
with negotiating the transition between imperative statements (from
which programs are constructed) and declarative statements (which can be
used to deduce things).  In a related vein, an important current area in
programming-language design is the exploration of so-called very
high-level languages, in which one actually programs in terms of
declarative statements.  The idea is to make interpreters sophisticated
enough so that, given "what is" knowledge specified by the programmer,
they can generate "how to" knowledge automatically.  This cannot be
done in general, but there are important areas where progress has been
made.  We shall revisit this idea in *Note Chapter 4::.

   (2) This square-root algorithm is actually a special case of
Newton's method, which is a general technique for finding roots of
equations.  The square-root algorithm itself was developed by Heron of
Alexandria in the first century A.D.  We will see how to express the
general Newton's method as a Lisp procedure in section *Note 1-3-4::.

   (3) We will usually give predicates names ending with question
marks, to help us remember that they are predicates.  This is just a
stylistic convention.  As far as the interpreter is concerned, the
question mark is just an ordinary character.

   (4) Observe that we express our initial guess as 1.0 rather than 1.
This would not make any difference in many Lisp implementations.  MIT
Scheme, however, distinguishes between exact integers and decimal
values, and dividing two integers produces a rational number rather
than a decimal.  For example, dividing 10 by 6 yields 5/3, while
dividing 10.0 by 6.0 yields 1.6666666666666667.  (We will learn how to
implement arithmetic on rational numbers in section *Note 2-1-1::.)  If
we start with an initial guess of 1 in our square-root program, and x
is an exact integer, all subsequent values produced in the square-root
computation will be rational numbers rather than decimals.  Mixed
operations on rational numbers and decimals always yield decimals, so
starting with an initial guess of 1.0 forces all subsequent values to
be decimals.

   (5) Readers who are worried about the efficiency issues involved in
using procedure calls to implement iteration should note the remarks on
"tail recursion" in section *Note 1-2-1::.


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1.1.8 Procedures as Black-Box Abstractions
------------------------------------------

`Sqrt' is our first example of a process defined by a set of mutually
defined procedures.  Notice that the definition of `sqrt-iter' is "recursive";
that is, the procedure is defined in terms of itself.  The idea of
being able to define a procedure in terms of itself may be disturbing;
it may seem unclear how such a "circular" definition could make sense
at all, much less specify a well-defined process to be carried out by a
computer.  This will be addressed more carefully in section *Note
1-2::.  But first let's consider some other important points
illustrated by the `sqrt' example.

   Observe that the problem of computing square roots breaks up
naturally into a number of subproblems: how to tell whether a guess is
good enough, how to improve a guess, and so on.  Each of these tasks is
accomplished by a separate procedure.  The entire `sqrt' program can be
viewed as a cluster of procedures (shown in *Note Figure 1-2::) that
mirrors the decomposition of the problem into subproblems.

     *Figure 1.2:* Procedural decomposition of the `sqrt' program.

                      sqrt
                       |
                   sqrt-iter
                   /       \
           good-enough    improve
             /    \          |
          square  abs     average

   The importance of this decomposition strategy is not simply that one
is dividing the program into parts.  After all, we could take any large
program and divide it into parts--the first ten lines, the next ten
lines, the next ten lines, and so on.  Rather, it is crucial that each
procedure accomplishes an identifiable task that can be used as a
module in defining other procedures.  For example, when we define the
`good-enough?' procedure in terms of `square', we are able to regard
the `square' procedure as a "black box."  We are not at that moment
concerned with _how_ the procedure computes its result, only with the
fact that it computes the square.  The details of how the square is
computed can be suppressed, to be considered at a later time.  Indeed,
as far as the `good-enough?' procedure is concerned, `square' is not
quite a procedure but rather an abstraction of a procedure, a so-called "procedural
abstraction".  At this level of abstraction, any procedure that
computes the square is equally good.

   Thus, considering only the values they return, the following two
procedures for squaring a number should be indistinguishable.  Each
takes a numerical argument and produces the square of that number as
the value.(1)

     (define (square x) (* x x))

     (define (square x)
       (exp (double (log x))))

     (define (double x) (+ x x))

   So a procedure definition should be able to suppress detail.  The
users of the procedure may not have written the procedure themselves,
but may have obtained it from another programmer as a black box.  A
user should not need to know how the procedure is implemented in order
to use it.

Local names
...........

One detail of a procedure's implementation that should not matter to
the user of the procedure is the implementer's choice of names for the
procedure's formal parameters.  Thus, the following procedures should
not be distinguishable:

     (define (square x) (* x x))

     (define (square y) (* y y))

   This principle--that the meaning of a procedure should be
independent of the parameter names used by its author--seems on the
surface to be self-evident, but its consequences are profound.  The
simplest consequence is that the parameter names of a procedure must be
local to the body of the procedure.  For example, we used `square' in
the definition of `good-enough?' in our square-root procedure:

     (define (good-enough? guess x)
       (< (abs (- (square guess) x)) 0.001))

   The intention of the author of `good-enough?' is to determine if the
square of the first argument is within a given tolerance of the second
argument.  We see that the author of `good-enough?' used the name
`guess' to refer to the first argument and `x' to refer to the second
argument.  The argument of `square' is `guess'.  If the author of
`square' used `x' (as above) to refer to that argument, we see that the
`x' in `good-enough?' must be a different `x' than the one in `square'.
Running the procedure `square' must not affect the value of `x' that
is used by `good-enough?', because that value of `x' may be needed by
`good-enough?' after `square' is done computing.

   If the parameters were not local to the bodies of their respective
procedures, then the parameter `x' in `square' could be confused with
the parameter `x' in `good-enough?', and the behavior of `good-enough?'
would depend upon which version of `square' we used.  Thus, `square'
would not be the black box we desired.

   A formal parameter of a procedure has a very special role in the
procedure definition, in that it doesn't matter what name the formal
parameter has.  Such a name is called a "bound variable", and we say
that the procedure definition "binds" its formal parameters.  The
meaning of a procedure definition is unchanged if a bound variable is
consistently renamed throughout the definition.(2)  If a variable is
not bound, we say that it is "free".  The set of expressions for which
a binding defines a name is called the "scope" of that name.  In a
procedure definition, the bound variables declared as the formal
parameters of the procedure have the body of the procedure as their
scope.

   In the definition of `good-enough?' above, `guess' and `x' are bound
variables but `<', `-', `abs', and `square' are free.  The meaning of
`good-enough?' should be independent of the names we choose for `guess'
and `x' so long as they are distinct and different from `<', `-',
`abs', and `square'.  (If we renamed `guess' to `abs' we would have
introduced a bug by "capturing" the variable `abs'.  It would have
changed from free to bound.)  The meaning of `good-enough?' is not
independent of the names of its free variables, however.  It surely
depends upon the fact (external to this definition) that the symbol
`abs' names a procedure for computing the absolute value of a number.
`Good-enough?' will compute a different function if we substitute `cos'
for `abs' in its definition.

Internal definitions and block structure
........................................

We have one kind of name isolation available to us so far: The formal
parameters of a procedure are local to the body of the procedure.  The
square-root program illustrates another way in which we would like to
control the use of names.  The existing program consists of separate
procedures:

     (define (sqrt x)
       (sqrt-iter 1.0 x))

     (define (sqrt-iter guess x)
       (if (good-enough? guess x)
           guess
           (sqrt-iter (improve guess x) x)))

     (define (good-enough? guess x)
       (< (abs (- (square guess) x)) 0.001))

     (define (improve guess x)
       (average guess (/ x guess)))

   The problem with this program is that the only procedure that is
important to users of `sqrt' is `sqrt'.  The other procedures
(`sqrt-iter', `good-enough?', and `improve') only clutter up their
minds.  They may not define any other procedure called `good-enough?'
as part of another program to work together with the square-root
program, because `sqrt' needs it.  The problem is especially severe in
the construction of large systems by many separate programmers.  For
example, in the construction of a large library of numerical
procedures, many numerical functions are computed as successive
approximations and thus might have procedures named `good-enough?' and
`improve' as auxiliary procedures.  We would like to localize the
subprocedures, hiding them inside `sqrt' so that `sqrt' could coexist
with other successive approximations, each having its own private
`good-enough?' procedure.  To make this possible, we allow a procedure
to have internal definitions that are local to that procedure.  For
example, in the square-root problem we can write

     (define (sqrt x)
       (define (good-enough? guess x)
         (< (abs (- (square guess) x)) 0.001))
       (define (improve guess x)
         (average guess (/ x guess)))
       (define (sqrt-iter guess x)
         (if (good-enough? guess x)
             guess
             (sqrt-iter (improve guess x) x)))
       (sqrt-iter 1.0 x))

   Such nesting of definitions, called "block structure", is basically
the right solution to the simplest name-packaging problem.  But there
is a better idea lurking here.  In addition to internalizing the
definitions of the auxiliary procedures, we can simplify them.  Since
`x' is bound in the definition of `sqrt', the procedures
`good-enough?', `improve', and `sqrt-iter', which are defined
internally to `sqrt', are in the scope of `x'.  Thus, it is not
necessary to pass `x' explicitly to each of these procedures.  Instead,
we allow `x' to be a free variable in the internal definitions, as
shown below. Then `x' gets its value from the argument with which the
enclosing procedure `sqrt' is called.  This discipline is called "lexical
scoping".(3)

     (define (sqrt x)
       (define (good-enough? guess)
         (< (abs (- (square guess) x)) 0.001))
       (define (improve guess)
         (average guess (/ x guess)))
       (define (sqrt-iter guess)
         (if (good-enough? guess)
             guess
             (sqrt-iter (improve guess))))
       (sqrt-iter 1.0))

   We will use block structure extensively to help us break up large
programs into tractable pieces.(4) The idea of block structure
originated with the programming language Algol 60. It appears in most
advanced programming languages and is an important tool for helping to
organize the construction of large programs.

   ---------- Footnotes ----------

   (1) It is not even clear which of these procedures is a more
efficient implementation.  This depends upon the hardware available.
There are machines for which the "obvious" implementation is the less
efficient one.  Consider a machine that has extensive tables of
logarithms and antilogarithms stored in a very efficient manner.

   (2) The concept of consistent renaming is actually subtle and
difficult to define formally.  Famous logicians have made embarrassing
errors here.

   (3) [Footnote 28] Lexical scoping dictates that free variables in a
procedure are taken to refer to bindings made by enclosing procedure
definitions; that is, they are looked up in the environment in which the
procedure was defined. We will see how this works in detail in chapter
3 when we study environments and the detailed behavior of the
interpreter.

   (4) Embedded definitions must come first in a procedure body. The
management is not responsible for the consequences of running programs
that intertwine definition and use.


File: sicp.info,  Node: 1-2,  Next: 1-3,  Prev: 1-1,  Up: Chapter 1

1.2 Procedures and the Processes They Generate
==============================================

We have now considered the elements of programming: We have used
primitive arithmetic operations, we have combined these operations, and
we have abstracted these composite operations by defining them as
compound procedures.  But that is not enough to enable us to say that
we know how to program.  Our situation is analogous to that of someone
who has learned the rules for how the pieces move in chess but knows
nothing of typical openings, tactics, or strategy.  Like the novice
chess player, we don't yet know the common patterns of usage in the
domain.  We lack the knowledge of which moves are worth making (which
procedures are worth defining).  We lack the experience to predict the
consequences of making a move (executing a procedure).

   The ability to visualize the consequences of the actions under
consideration is crucial to becoming an expert programmer, just as it
is in any synthetic, creative activity.  In becoming an expert
photographer, for example, one must learn how to look at a scene and
know how dark each region will appear on a print for each possible
choice of exposure and development conditions.  Only then can one
reason backward, planning framing, lighting, exposure, and development
to obtain the desired effects.  So it is with programming, where we are
planning the course of action to be taken by a process and where we
control the process by means of a program.  To become experts, we must
learn to visualize the processes generated by various types of
procedures.  Only after we have developed such a skill can we learn to
reliably construct programs that exhibit the desired behavior.

   A procedure is a pattern for the "local evolution" of a computational
process.  It specifies how each stage of the process is built upon the
previous stage.  We would like to be able to make statements about the
overall, or "global", behavior of a process whose local evolution has
been specified by a procedure.  This is very difficult to do in
general, but we can at least try to describe some typical patterns of
process evolution.

   In this section we will examine some common "shapes" for processes
generated by simple procedures.  We will also investigate the rates at
which these processes consume the important computational resources of
time and space.  The procedures we will consider are very simple.
Their role is like that played by test patterns in photography: as
oversimplified prototypical patterns, rather than practical examples in
their own right.

* Menu:

* 1-2-1::            Linear Recursion and Iteration
* 1-2-2::            Tree Recursion
* 1-2-3::            Orders of Growth
* 1-2-4::            Exponentiation
* 1-2-5::            Greatest Common Divisors
* 1-2-6::            Example: Testing for Primality


File: sicp.info,  Node: 1-2-1,  Next: 1-2-2,  Prev: 1-2,  Up: 1-2

1.2.1 Linear Recursion and Iteration
------------------------------------

     *Figure 1.3:* A linear recursive process for computing 6!.

          (factorial 6)        ------------------------.
          (* 6 (factorial 5))                          |
          (* 6 (* 5 (factorial 4)))                    |
          (* 6 (* 5 (* 4 (factorial 3))))              |
          (* 6 (* 5 (* 4 (* 3 (factorial 2)))))        |
          (* 6 (* 5 (* 4 (* 3 (* 2 (factorial 1))))))  |
          (* 6 (* 5 (* 4 (* 3 (* 2 1)))))              |
          (* 6 (* 5 (* 4 (* 3 2))))                    |
          (* 6 (* 5 (* 4 6)))                          |
          (* 6 (* 5 24))                               |
          (* 6 120)                                    |
          720          <-------------------------------'

We begin by considering the factorial function, defined by

     n! = n * (n - 1) * (n - 2) ... 3 * 2 * 1

   There are many ways to compute factorials.  One way is to make use
of the observation that n! is equal to n times (n - 1)! for any positive
integer n:

     n! = n * [(n - 1) * (n - 2) ... 3 * 2 * 1] = n * (n - 1)!

   Thus, we can compute n! by computing (n - 1)! and multiplying the
result by n.  If we add the stipulation that 1! is equal to 1, this
observation translates directly into a procedure:

     (define (factorial n)
       (if (= n 1)
           1
           (* n (factorial (- n 1)))))

   We can use the substitution model of section *Note 1-1-5:: to watch
this procedure in action computing 6!, as shown in *Note Figure 1-3::.

   Now let's take a different perspective on computing factorials.  We
could describe a rule for computing n! by specifying that we first
multiply 1 by 2, then multiply the result by 3, then by 4, and so on
until we reach n.  More formally, we maintain a running product,
together with a counter that counts from 1 up to n.  We can describe
the computation by saying that the counter and the product
simultaneously change from one step to the next according to the rule

     product  <-  counter  ...  product

     counter  <-  counter  +  1

and stipulating that n! is the value of the product when the counter
exceeds n.

     *Figure 1.4:* A linear iterative process for computing 6!.

          (factorial 6)   -----.
          (fact-iter   1 1 6)  |
          (fact-iter   1 2 6)  |
          (fact-iter   2 3 6)  |
          (fact-iter   6 4 6)  |
          (fact-iter  24 5 6)  |
          (fact-iter 120 6 6)  |
          (fact-iter 720 7 6)  V
          720

   Once again, we can recast our description as a procedure for
computing factorials:(1)

     (define (factorial n)
       (fact-iter 1 1 n))

     (define (fact-iter product counter max-count)
       (if (> counter max-count)
           product
           (fact-iter (* counter product)
                      (+ counter 1)
                      max-count)))

   As before, we can use the substitution model to visualize the
process of computing 6!, as shown in *Note Figure 1-4::.

   Compare the two processes.  From one point of view, they seem hardly
different at all.  Both compute the same mathematical function on the
same domain, and each requires a number of steps proportional to n to
compute n!.  Indeed, both processes even carry out the same sequence of
multiplications, obtaining the same sequence of partial products.  On
the other hand, when we consider the "shapes" of the two processes, we
find that they evolve quite differently.

   Consider the first process.  The substitution model reveals a shape
of expansion followed by contraction, indicated by the arrow in *Note
Figure 1-3::.  The expansion occurs as the process builds up a chain of operations
"deferred operations" (in this case, a chain of multiplications).  The
contraction occurs as the operations are actually performed.  This type
of process, characterized by a chain of deferred operations, is called
a "recursive process".  Carrying out this process requires that the
interpreter keep track of the operations to be performed later on.  In
the computation of n!, the length of the chain of deferred
multiplications, and hence the amount of information needed to keep
track of it, grows linearly with n (is proportional to n), just like
the number of steps.  Such a process is called a "linear recursive
process".

   By contrast, the second process does not grow and shrink.  At each
step, all we need to keep track of, for any n, are the current values
of the variables `product', `counter', and `max-count'.  We call this an "iterative
process".  In general, an iterative process is one whose state can be
summarized by a fixed number of "state variables", together with a
fixed rule that describes how the state variables should be updated as
the process moves from state to state and an (optional) end test that
specifies conditions under which the process should terminate.  In
computing n!, the number of steps required grows linearly with n.  Such
a process is called a "linear iterative process".

   The contrast between the two processes can be seen in another way.
In the iterative case, the program variables provide a complete
description of the state of the process at any point.  If we stopped
the computation between steps, all we would need to do to resume the
computation is to supply the interpreter with the values of the three
program variables.  Not so with the recursive process.  In this case
there is some additional "hidden" information, maintained by the
interpreter and not contained in the program variables, which indicates
"where the process is" in negotiating the chain of deferred operations.
The longer the chain, the more information must be maintained.(2)

   In contrasting iteration and recursion, we must be careful not to
confuse the notion of a recursive "process" with the notion of a
recursive "procedure".  When we describe a procedure as recursive, we
are referring to the syntactic fact that the procedure definition
refers (either directly or indirectly) to the procedure itself.  But
when we describe a process as following a pattern that is, say,
linearly recursive, we are speaking about how the process evolves, not
about the syntax of how a procedure is written.  It may seem disturbing
that we refer to a recursive procedure such as `fact-iter' as
generating an iterative process.  However, the process really is
iterative: Its state is captured completely by its three state
variables, and an interpreter need keep track of only three variables
in order to execute the process.

   One reason that the distinction between process and procedure may be
confusing is that most implementations of common languages (including
Ada, Pascal, and C) are designed in such a way that the interpretation
of any recursive procedure consumes an amount of memory that grows with
the number of procedure calls, even when the process described is, in
principle, iterative.  As a consequence, these languages can describe
iterative processes only by resorting to special-purpose "looping
constructs" such as `do', `repeat', `until', `for', and `while'.  The
implementation of Scheme we shall consider in *Note Chapter 5:: does
not share this defect.  It will execute an iterative process in
constant space, even if the iterative process is described by a
recursive procedure.  An implementation with this property is called "tail-recursive".
With a tail-recursive implementation, iteration can be expressed using
the ordinary procedure call mechanism, so that special iteration
constructs are useful only as syntactic sugar.(3)

     *Exercise 1.9:* Each of the following two procedures defines a
     method for adding two positive integers in terms of the procedures
     `inc', which increments its argument by 1, and `dec', which
     decrements its argument by 1.

          (define (+ a b)
            (if (= a 0)
                b
                (inc (+ (dec a) b))))

          (define (+ a b)
            (if (= a 0)
                b
                (+ (dec a) (inc b))))

     Using the substitution model, illustrate the process generated by
     each procedure in evaluating `(+ 4 5)'.  Are these processes
     iterative or recursive?

     *Exercise 1.10:* The following procedure computes a mathematical
     function called Ackermann's function.

          (define (A x y)
            (cond ((= y 0) 0)
                  ((= x 0) (* 2 y))
                  ((= y 1) 2)
                  (else (A (- x 1)
                           (A x (- y 1))))))

     What are the values of the following expressions?

          (A 1 10)

          (A 2 4)

          (A 3 3)

     Consider the following procedures, where `A' is the procedure
     defined above:

          (define (f n) (A 0 n))

          (define (g n) (A 1 n))

          (define (h n) (A 2 n))

          (define (k n) (* 5 n n))

     Give concise mathematical definitions for the functions computed
     by the procedures `f', `g', and `h' for positive integer values of
     n.  For example, `(k n)' computes 5n^2.

   ---------- Footnotes ----------

   (1) In a real program we would probably use the block structure
introduced in the last section to hide the definition of `fact-iter':

     (define (factorial n)
             (define (iter product counter)
                     (if (> counter n)
                         product
                         (iter (* counter product)
                               (+ counter 1))))
             (iter 1 1))

   We avoided doing this here so as to minimize the number of things to
think about at once.

   (2) When we discuss the implementation of procedures on register
machines in *Note Chapter 5::, we will see that any iterative process
can be realized "in hardware" as a machine that has a fixed set of
registers and no auxiliary memory.  In contrast, realizing a recursive
process requires a machine that uses an auxiliary data structure known
as a "stack".

   (3) Tail recursion has long been known as a compiler optimization
trick.  A coherent semantic basis for tail recursion was provided by
Carl Hewitt (1977), who explained it in terms of the "message-passing"
model of computation that we shall discuss in *Note Chapter 3::.
Inspired by this, Gerald Jay Sussman and Guy Lewis Steele Jr. (see
Steele 1975) constructed a tail-recursive interpreter for Scheme.
Steele later showed how tail recursion is a consequence of the natural
way to compile procedure calls (Steele 1977).  The IEEE standard for
Scheme requires that Scheme implementations be tail-recursive.


File: sicp.info,  Node: 1-2-2,  Next: 1-2-3,  Prev: 1-2-1,  Up: 1-2

1.2.2 Tree Recursion
--------------------

Another common pattern of computation is called "tree recursion".  As
an example, consider computing the sequence of Fibonacci numbers, in
which each number is the sum of the preceding two:

   0, 1, 1, 2, 3, 4, 8, 13, 21, ...

   In general, the Fibonacci numbers can be defined by the rule

              /
              |  0                        if n = 0
     Fib(n) = <  1                        if n = 1
              |  Fib(n - 1) + Fib(n - 2)  otherwise
              \

   We can immediately translate this definition into a recursive
procedure for computing Fibonacci numbers:

     (define (fib n)
       (cond ((= n 0) 0)
             ((= n 1) 1)
             (else (+ (fib (- n 1))
                      (fib (- n 2))))))

     *Figure 1.5:* The tree-recursive process generated in computing
     `(fib 5)'.

                             ..<............ fib5   <..........
                          ...     ___________/  \___________   .
                       ...       /       . .....            \    .
                     ..       fib4     .        . . . .     fib3  .
                   ..     ____/. \____  ..             .  __/  \__  .
                 ..      /  . .  ..   \    .        ..   /  . .   \   .
               ..     fib3 .       .  fib2 .        . fib2 .   .  fib1 .
             ..      / . \  .     .   /  \  .      .  /  \ ...  .  |  .
           ..       / . . \   .  .   /  . \   .  .   / .  \   .  . 1 .
          .      fib2 . . fib1.  .fib1 .  fib0 . .fib1. . fib0 .  .  .
          .      /  \  . . |  .  . |  .  . |   . . |   . . |   .   .>
          V     /  . \   . 1  .  . 1  .  . 0  .  . 1  .  . 0  ..
          .  fib1 .. fib0..  .   .   .   .   .   V   .   ..  .
          .   |  .  . |  . .>     .>.     . .    ..>.      .>
          .   1 .   . 0  .
           .   .     .  .
            .>.       ..

   Consider the pattern of this computation.  To compute `(fib 5)', we
compute `(fib 4)' and `(fib 3)'.  To compute `(fib 4)', we compute
`(fib 3)' and `(fib 2)'.  In general, the evolved process looks like a
tree, as shown in *Note Figure 1-5::.  Notice that the branches split
into two at each level (except at the bottom); this reflects the fact
that the `fib' procedure calls itself twice each time it is invoked.

   This procedure is instructive as a prototypical tree recursion, but
it is a terrible way to compute Fibonacci numbers because it does so
much redundant computation.  Notice in *Note Figure 1-5:: that the
entire computation of `(fib 3)'--almost half the work--is duplicated.
In fact, it is not hard to show that the number of times the procedure
will compute `(fib 1)' or `(fib 0)' (the number of leaves in the above
tree, in general) is precisely _Fib_(n + 1).  To get an idea of how bad
this is, one can show that the value of _Fib_(n) grows exponentially
with n.  More precisely (see *Note Exercise 1-13::), _Fib_(n) is the
closest integer to [phi]^n /[sqrt](5), where

     [phi] = (1 + [sqrt]5)/2 ~= 1.6180

is the "golden ratio", which satisfies the equation

     [phi]^2 = [phi] + 1

   Thus, the process uses a number of steps that grows exponentially
with the input.  On the other hand, the space required grows only
linearly with the input, because we need keep track only of which nodes
are above us in the tree at any point in the computation.  In general,
the number of steps required by a tree-recursive process will be
proportional to the number of nodes in the tree, while the space
required will be proportional to the maximum depth of the tree.

   We can also formulate an iterative process for computing the
Fibonacci numbers.  The idea is to use a pair of integers a and b,
initialized to _Fib_(1) = 1 and _Fib_(0) = 0, and to repeatedly apply
the simultaneous transformations

     a <- a + b
     b <- a

It is not hard to show that, after applying this transformation n times,
a and b will be equal, respectively, to _Fib_(n + 1) and _Fib_(n).
Thus, we can compute Fibonacci numbers iteratively using the procedure

     (define (fib n)
       (fib-iter 1 0 n))

     (define (fib-iter a b count)
       (if (= count 0)
           b
           (fib-iter (+ a b) a (- count 1))))

   This second method for computing _Fib_(n) is a linear iteration.  The
difference in number of steps required by the two methods--one linear in
n, one growing as fast as _Fib_(n) itself--is enormous, even for small
inputs.

   One should not conclude from this that tree-recursive processes are
useless.  When we consider processes that operate on hierarchically
structured data rather than numbers, we will find that tree recursion
is a natural and powerful tool.(1) But even in numerical operations,
tree-recursive processes can be useful in helping us to understand and
design programs.  For instance, although the first `fib' procedure is
much less efficient than the second one, it is more straightforward,
being little more than a translation into Lisp of the definition of the
Fibonacci sequence.  To formulate the iterative algorithm required
noticing that the computation could be recast as an iteration with
three state variables.

Example: Counting change
........................

It takes only a bit of cleverness to come up with the iterative
Fibonacci algorithm.  In contrast, consider the following problem: How
many different ways can we make change of $ 1.00, given half-dollars,
quarters, dimes, nickels, and pennies?  More generally, can we write a
procedure to compute the number of ways to change any given amount of
money?

   This problem has a simple solution as a recursive procedure.
Suppose we think of the types of coins available as arranged in some
order.  Then the following relation holds:

   The number of ways to change amount a using n kinds of coins equals

   * the number of ways to change amount a using all but the first kind
     of coin, plus

   * the number of ways to change amount a - d using all n kinds of
     coins, where d is the denomination of the first kind of coin.


   To see why this is true, observe that the ways to make change can be
divided into two groups: those that do not use any of the first kind of
coin, and those that do.  Therefore, the total number of ways to make
change for some amount is equal to the number of ways to make change
for the amount without using any of the first kind of coin, plus the
number of ways to make change assuming that we do use the first kind of
coin.  But the latter number is equal to the number of ways to make
change for the amount that remains after using a coin of the first kind.

   Thus, we can recursively reduce the problem of changing a given
amount to the problem of changing smaller amounts using fewer kinds of
coins.  Consider this reduction rule carefully, and convince yourself
that we can use it to describe an algorithm if we specify the following
degenerate cases:(2)

   * If a is exactly 0, we should count that as 1 way to make change.

   * If a is less than 0, we should count that as 0 ways to make change.

   * If n is 0, we should count that as 0 ways to make change.


   We can easily translate this description into a recursive procedure:

     (define (count-change amount)
       (cc amount 5))

     (define (cc amount kinds-of-coins)
       (cond ((= amount 0) 1)
             ((or (< amount 0) (= kinds-of-coins 0)) 0)
             (else (+ (cc amount
                          (- kinds-of-coins 1))
                      (cc (- amount
                             (first-denomination kinds-of-coins))
                          kinds-of-coins)))))

     (define (first-denomination kinds-of-coins)
       (cond ((= kinds-of-coins 1) 1)
             ((= kinds-of-coins 2) 5)
             ((= kinds-of-coins 3) 10)
             ((= kinds-of-coins 4) 25)
             ((= kinds-of-coins 5) 50)))

   (The `first-denomination' procedure takes as input the number of
kinds of coins available and returns the denomination of the first
kind.  Here we are thinking of the coins as arranged in order from
largest to smallest, but any order would do as well.)  We can now
answer our original question about changing a dollar:

     (count-change 100)
     292

   `Count-change' generates a tree-recursive process with redundancies
similar to those in our first implementation of `fib'.  (It will take
quite a while for that 292 to be computed.)  On the other hand, it is
not obvious how to design a better algorithm for computing the result,
and we leave this problem as a challenge.  The observation that a
tree-recursive process may be highly inefficient but often easy to
specify and understand has led people to propose that one could get the
best of both worlds by designing a "smart compiler" that could
transform tree-recursive procedures into more efficient procedures that
compute the same result.(3)

     *Exercise 1.11:* A function f is defined by the rule that f(n) = n
     if n<3 and f(n) = f(n - 1) + 2f(n - 2) + 3f(n - 3) if n>= 3.
     Write a procedure that computes f by means of a recursive process.
     Write a procedure that computes f by means of an iterative
     process.

     *Exercise 1.12:* The following pattern of numbers is called "Pascal's
     triangle".

                  1
                1   1
              1   2   1
            1   3   3   1
          1   4   6   4   1

     The numbers at the edge of the triangle are all 1, and each number
     inside the triangle is the sum of the two numbers above it.(4)
     Write a procedure that computes elements of Pascal's triangle by
     means of a recursive process.

     *Exercise 1.13:* Prove that _Fib_(n) is the closest integer to
     [phi]^n/[sqrt](5), where [phi] = (1 + [sqrt](5))/2.  Hint: Let
     [illegiblesymbol] = (1 - [sqrt](5))/2.  Use induction and the
     definition of the Fibonacci numbers (see section *Note 1-2-2::) to
     prove that _Fib_(n) = ([phi]^n - [illegiblesymbol]^n)/[sqrt](5).

   ---------- Footnotes ----------

   (1) An example of this was hinted at in section *Note 1-1-3::: The
interpreter itself evaluates expressions using a tree-recursive process.

   (2) For example, work through in detail how the reduction rule
applies to the problem of making change for 10 cents using pennies and
nickels.

   (3) One approach to coping with redundant computations is to arrange
matters so that we automatically construct a table of values as they
are computed.  Each time we are asked to apply the procedure to some
argument, we first look to see if the value is already stored in the
table, in which case we avoid performing the redundant computation.
This strategy, known as "tabulation" or "memoization", can be
implemented in a straightforward way.  Tabulation can sometimes be used
to transform processes that require an exponential number of steps
(such as `count-change') into processes whose space and time
requirements grow linearly with the input.  See *Note Exercise 3-27::.

   (4) The elements of Pascal's triangle are called the "binomial
coefficients", because the nth row consists of the coefficients of the
terms in the expansion of (x + y)^n.  This pattern for computing the
coefficients appeared in Blaise Pascal's 1653 seminal work on
probability theory, `Traite' du triangle arithme'tique'.  According to
Knuth (1973), the same pattern appears in the `Szu-yuen Yu"-chien'
("The Precious Mirror of the Four Elements"), published by the Chinese
mathematician Chu Shih-chieh in 1303, in the works of the
twelfth-century Persian poet and mathematician Omar Khayyam, and in the
works of the twelfth-century Hindu mathematician Bha'scara A'cha'rya.


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1.2.3 Orders of Growth
----------------------

The previous examples illustrate that processes can differ considerably
in the rates at which they consume computational resources.  One
convenient way to describe this difference is to use the notion of "order
of growth" to obtain a gross measure of the resources required by a
process as the inputs become larger.

   Let n be a parameter that measures the size of the problem, and let
R(n) be the amount of resources the process requires for a problem of
size n.  In our previous examples we took n to be the number for which
a given function is to be computed, but there are other possibilities.
For instance, if our goal is to compute an approximation to the square
root of a number, we might take n to be the number of digits accuracy
required.  For matrix multiplication we might take n to be the number
of rows in the matrices.  In general there are a number of properties
of the problem with respect to which it will be desirable to analyze a
given process.  Similarly, R(n) might measure the number of internal
storage registers used, the number of elementary machine operations
performed, and so on.  In computers that do only a fixed number of
operations at a time, the time required will be proportional to the
number of elementary machine operations performed.

   We say that R(n) has order of growth [theta](f(n)), written R(n) =
[theta](f(n)) (pronounced "theta of f(n)"), if there are positive
constants k_1 and k_2 independent of n such that

     k_1 f(n) <= R(n) <= k_2 f(n)

for any sufficiently large value of n.  (In other words, for large n,
the value R(n) is sandwiched between k_1f(n) and k_2f(n).)

   For instance, with the linear recursive process for computing
factorial described in section *Note 1-2-1:: the number of steps grows
proportionally to the input n.  Thus, the steps required for this
process grows as [theta](n).  We also saw that the space required grows
as [theta](n).  For the iterative factorial, the number of steps is
still [theta](n) but the space is [theta](1)--that is, constant.(1) The
tree-recursive Fibonacci computation requires [theta]([phi]^n) steps
and space [theta](n), where [phi] is the golden ratio described in
section *Note 1-2-2::.

   Orders of growth provide only a crude description of the behavior of
a process.  For example, a process requiring n^2 steps and a process
requiring 1000n^2 steps and a process requiring 3n^2 + 10n + 17 steps
all have [theta](n^2) order of growth.  On the other hand, order of
growth provides a useful indication of how we may expect the behavior
of the process to change as we change the size of the problem.  For a
[theta](n) (linear) process, doubling the size will roughly double the
amount of resources used.  For an exponential process, each increment
in problem size will multiply the resource utilization by a constant
factor.  In the remainder of section *Note 1-2:: we will examine two
algorithms whose order of growth is logarithmic, so that doubling the
problem size increases the resource requirement by a constant amount.

     *Exercise 1.14:* Draw the tree illustrating the process generated
     by the `count-change' procedure of section *Note 1-2-2:: in making
     change for 11 cents.  What are the orders of growth of the space
     and number of steps used by this process as the amount to be
     changed increases?

     *Exercise 1.15:* The sine of an angle (specified in radians) can
     be computed by making use of the approximation `sin' xapprox x if
     x is sufficiently small, and the trigonometric identity

                         x             x
          sin x = 3 sin --- - 4 sin^3 ---
                         3             3

     to reduce the size of the argument of `sin'.  (For purposes of this
     exercise an angle is considered "sufficiently small" if its
     magnitude is not greater than 0.1 radians.) These ideas are
     incorporated in the following procedures:

          (define (cube x) (* x x x))

          (define (p x) (- (* 3 x) (* 4 (cube x))))

          (define (sine angle)
             (if (not (> (abs angle) 0.1))
                 angle
                 (p (sine (/ angle 3.0)))))

       a. How many times is the procedure `p' applied when `(sine
          12.15)' is evaluated?

       b. What is the order of growth in space and number of steps (as
          a function of a) used by the process generated by the `sine'
          procedure when `(sine a)' is evaluated?


   ---------- Footnotes ----------

   (1) These statements mask a great deal of oversimplification.  For
instance, if we count process steps as "machine operations" we are
making the assumption that the number of machine operations needed to
perform, say, a multiplication is independent of the size of the
numbers to be multiplied, which is false if the numbers are
sufficiently large.  Similar remarks hold for the estimates of space.
Like the design and description of a process, the analysis of a process
can be carried out at various levels of abstraction.


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1.2.4 Exponentiation
--------------------

Consider the problem of computing the exponential of a given number.
We would like a procedure that takes as arguments a base b and a
positive integer exponent n and computes b^n.  One way to do this is
via the recursive definition

     b^n = b * b^(n - 1)
     b^0 = 1

which translates readily into the procedure

     (define (expt b n)
       (if (= n 0)
           1
           (* b (expt b (- n 1)))))

   This is a linear recursive process, which requires [theta](n) steps
and [theta](n) space.  Just as with factorial, we can readily formulate
an equivalent linear iteration:

     (define (expt b n)
       (expt-iter b n 1))

     (define (expt-iter b counter product)
       (if (= counter 0)
           product
           (expt-iter b
                     (- counter 1)
                     (* b product))))

   This version requires [theta](n) steps and [theta](1) space.

   We can compute exponentials in fewer steps by using successive
squaring.  For instance, rather than computing b^8 as

     b * (b * (b * (b * (b * (b * (b * b))))))

we can compute it using three multiplications:

     b^2 = b * b
     b^4 = b^2 * b^2
     b^8 = b^4 * b^4

   This method works fine for exponents that are powers of 2.  We can
also take advantage of successive squaring in computing exponentials in
general if we use the rule

     b^n = (b^(b/2))^2    if n is even
     b^n = b * b^(n - 1)  if n is odd

   We can express this method as a procedure:

     (define (fast-expt b n)
       (cond ((= n 0) 1)
             ((even? n) (square (fast-expt b (/ n 2))))
             (else (* b (fast-expt b (- n 1))))))

where the predicate to test whether an integer is even is defined in
terms of the primitive procedure `remainder' by

     (define (even? n)
       (= (remainder n 2) 0))

   The process evolved by `fast-expt' grows logarithmically with n in
both space and number of steps.  To see this, observe that computing
b^(2n) using `fast-expt' requires only one more multiplication than
computing b^n.  The size of the exponent we can compute therefore
doubles (approximately) with every new multiplication we are allowed.
Thus, the number of multiplications required for an exponent of n grows
about as fast as the logarithm of n to the base 2.  The process has
[theta](`log' n) growth.(1)

   The difference between [theta](`log' n) growth and [theta](n) growth
becomes striking as n becomes large.  For example, `fast-expt' for n =
1000 requires only 14 multiplications.(2) It is also possible to use
the idea of successive squaring to devise an iterative algorithm that
computes exponentials with a logarithmic number of steps (see *Note
Exercise 1-16::), although, as is often the case with iterative
algorithms, this is not written down so straightforwardly as the
recursive algorithm.(3)

     *Exercise 1.16:* Design a procedure that evolves an iterative
     exponentiation process that uses successive squaring and uses a
     logarithmic number of steps, as does `fast-expt'.  (Hint: Using the
     observation that (b^(n/2))^2 = (b^2)^(n/2), keep, along with the
     exponent n and the base b, an additional state variable a, and
     define the state transformation in such a way that the product a
     b^n is unchanged from state to state.  At the beginning of the
     process a is taken to be 1, and the answer is given by the value
     of a at the end of the process.  In general, the technique of
     defining an "invariant quantity" that remains unchanged from state
     to state is a powerful way to think about the design of iterative
     algorithms.)

     *Exercise 1.17:* The exponentiation algorithms in this section are
     based on performing exponentiation by means of repeated
     multiplication.  In a similar way, one can perform integer
     multiplication by means of repeated addition.  The following
     multiplication procedure (in which it is assumed that our language
     can only add, not multiply) is analogous to the `expt' procedure:

          (define (* a b)
            (if (= b 0)
                0
                (+ a (* a (- b 1)))))

     This algorithm takes a number of steps that is linear in `b'.  Now
     suppose we include, together with addition, operations `double',
     which doubles an integer, and `halve', which divides an (even)
     integer by 2.  Using these, design a multiplication procedure
     analogous to `fast-expt' that uses a logarithmic number of steps.

     *Exercise 1.18:* Using the results of *Note Exercise 1-16:: and
     *Note Exercise 1-17::, devise a procedure that generates an
     iterative process for multiplying two integers in terms of adding,
     doubling, and halving and uses a logarithmic number of steps.(4)

     *Exercise 1.19:* There is a clever algorithm for computing the
     Fibonacci numbers in a logarithmic number of steps.  Recall the
     transformation of the state variables a and b in the `fib-iter'
     process of section *Note 1-2-2::: a <- a + b and b <- a.  Call
     this transformation T, and observe that applying T over and over
     again n times, starting with 1 and 0, produces the pair _Fib_(n +
     1) and _Fib_(n).  In other words, the Fibonacci numbers are
     produced by applying T^n, the nth power of the transformation T,
     starting with the pair (1,0).  Now consider T to be the special
     case of p = 0 and q = 1 in a family of transformations T_(pq),
     where T_(pq) transforms the pair (a,b) according to a <- bq + aq +
     ap and b <- bp + aq.  Show that if we apply such a transformation
     T_(pq) twice, the effect is the same as using a single
     transformation T_(p'q') of the same form, and compute p' and q' in
     terms of p and q.  This gives us an explicit way to square these
     transformations, and thus we can compute T^n using successive
     squaring, as in the `fast-expt' procedure.  Put this all together
     to complete the following procedure, which runs in a logarithmic
     number of steps:(5)

          (define (fib n)
            (fib-iter 1 0 0 1 n))

          (define (fib-iter a b p q count)
            (cond ((= count 0) b)
                  ((even? count)
                   (fib-iter a
                             b
                             <??>      ; compute p'
                             <??>      ; compute q'
                             (/ count 2)))
                  (else (fib-iter (+ (* b q) (* a q) (* a p))
                                  (+ (* b p) (* a q))
                                  p
                                  q
                                  (- count 1)))))

   ---------- Footnotes ----------

   (1) More precisely, the number of multiplications required is equal
to 1 less than the log base 2 of n plus the number of ones in the
binary representation of n.  This total is always less than twice the
log base 2 of n.  The arbitrary constants k_1 and k_2 in the definition
of order notation imply that, for a logarithmic process, the base to
which logarithms are taken does not matter, so all such processes are
described as [theta](`log' n).

   (2) You may wonder why anyone would care about raising numbers to
the 1000th power.  See section *Note 1-2-6::.

   (3) This iterative algorithm is ancient.  It appears in the
`Chandah-sutra' by A'cha'rya Pingala, written before 200 B.C. See Knuth
1981, section 4.6.3, for a full discussion and analysis of this and
other methods of exponentiation.

   (4) This algorithm, which is sometimes known as the "Russian peasant
method" of multiplication, is ancient.  Examples of its use are found
in the Rhind Papyrus, one of the two oldest mathematical documents in
existence, written about 1700 B.C.  (and copied from an even older
document) by an Egyptian scribe named A'h-mose.

   (5) This exercise was suggested to us by Joe Stoy, based on an
example in Kaldewaij 1990.


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1.2.5 Greatest Common Divisors
------------------------------

The greatest common divisor (GCD) of two integers a and b is defined to
be the largest integer that divides both a and b with no remainder.
For example, the GCD of 16 and 28 is 4.  In *Note Chapter 2::, when we
investigate how to implement rational-number arithmetic, we will need
to be able to compute GCDs in order to reduce rational numbers to
lowest terms.  (To reduce a rational number to lowest terms, we must
divide both the numerator and the denominator by their GCD.  For
example, 16/28 reduces to 4/7.)  One way to find the GCD of two
integers is to factor them and search for common factors, but there is
a famous algorithm that is much more efficient.

   The idea of the algorithm is based on the observation that, if r is
the remainder when a is divided by b, then the common divisors of a and
b are precisely the same as the common divisors of b and r.  Thus, we
can use the equation

     GCD(a,b) = GCD(b,r)

to successively reduce the problem of computing a GCD to the problem of
computing the GCD of smaller and smaller pairs of integers.  For
example,

     GCD(206,40) = GCD(40,6)
                 = GCD(6,4)
                 = GCD(4,2)
                 = GCD(2,0)
                 = 2

reduces GCD(206,40) to GCD(2,0), which is 2.  It is possible to show
that starting with any two positive integers and performing repeated
reductions will always eventually produce a pair where the second
number is 0.  Then the GCD is the other number in the pair.  This
method for computing the GCD is known as Algorithm "Euclid's
Algorithm".(1)

   It is easy to express Euclid's Algorithm as a procedure:

     (define (gcd a b)
       (if (= b 0)
           a
           (gcd b (remainder a b))))

   This generates an iterative process, whose number of steps grows as
the logarithm of the numbers involved.

   The fact that the number of steps required by Euclid's Algorithm has
logarithmic growth bears an interesting relation to the Fibonacci
numbers:

     *Lame''s Theorem:* If Euclid's Algorithm requires k steps to
     compute the GCD of some pair, then the smaller number in the pair
     must be greater than or equal to the kth Fibonacci number.(2)

   We can use this theorem to get an order-of-growth estimate for
Euclid's Algorithm.  Let n be the smaller of the two inputs to the
procedure.  If the process takes k steps, then we must have n>= _Fib_(k)
approx [phi]^k/[sqrt](5).  Therefore the number of steps k grows as the
logarithm (to the base [phi]) of n.  Hence, the order of growth is
[theta](`log' n).

     *Exercise 1.20:* The process that a procedure generates is of
     course dependent on the rules used by the interpreter.  As an
     example, consider the iterative `gcd' procedure given above.
     Suppose we were to interpret this procedure using normal-order
     evaluation, as discussed in section *Note 1-1-5::.  (The
     normal-order-evaluation rule for `if' is described in *Note
     Exercise 1-5::.)  Using the substitution method (for normal
     order), illustrate the process generated in evaluating `(gcd 206
     40)' and indicate the `remainder' operations that are actually
     performed.  How many `remainder' operations are actually performed
     in the normal-order evaluation of `(gcd 206 40)'?  In the
     applicative-order evaluation?

   ---------- Footnotes ----------

   (1) Euclid's Algorithm is so called because it appears in Euclid's
`Elements' (Book 7, ca. 300 B.C.).  According to Knuth (1973), it can
be considered the oldest known nontrivial algorithm.  The ancient
Egyptian method of multiplication (*Note Exercise 1-18::) is surely
older, but, as Knuth explains, Euclid's algorithm is the oldest known
to have been presented as a general algorithm, rather than as a set of
illustrative examples.

   (2) This theorem was proved in 1845 by Gabriel Lame', a French
mathematician and engineer known chiefly for his contributions to
mathematical physics.  To prove the theorem, we consider pairs (a_k
,b_k), where a_k>= b_k, for which Euclid's Algorithm terminates in k
steps.  The proof is based on the claim that, if (a_(k+1), b_(k+1)) ->
(a_k, b_k) -> (a_(k-1), b_(k-1)) are three successive pairs in the
reduction process, then we must have b_(k+1)>= b_k + b_(k-1).  To
verify the claim, consider that a reduction step is defined by applying
the transformation a_(k-1) = b_k, b_(k-1) = remainder of a_k divided by
b_k.  The second equation means that a_k = qb_k + b_(k-1) for some
positive integer q.  And since q must be at least 1 we have a_k = qb_k
+ b_(k-1) >= b_k + b_(k-1).  But in the previous reduction step we have
b_(k+1) = a_k.  Therefore, b_(k+1) = a_k>= b_k + b_(k-1).  This verifies
the claim.  Now we can prove the theorem by induction on k, the number
of steps that the algorithm requires to terminate.  The result is true
for k = 1, since this merely requires that b be at least as large as
_Fib_(1) = 1.  Now, assume that the result is true for all integers
less than or equal to k and establish the result for k + 1.  Let
(a_(k+1), b_(k+1)) -> (a_k, b_k) -> (a_k-1), b_(k-1)) be successive
pairs in the reduction process.  By our induction hypotheses, we have
b_(k-1) >= _Fib_(k - 1) and b_k >= _Fib_(k).  Thus, applying the claim
we just proved together with the definition of the Fibonacci numbers
gives b_(k+1) >= b_k + b_(k-1) >= _Fib_(k) + _Fib_(k - 1) = _Fib_(k +
1), which completes the proof of Lame''s Theorem.


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1.2.6 Example: Testing for Primality
------------------------------------

This section describes two methods for checking the primality of an
integer n, one with order of growth [theta](_[sqrt]_(n)), and a
"probabilistic" algorithm with order of growth [theta](`log' n).  The
exercises at the end of this section suggest programming projects based
on these algorithms.

Searching for divisors
......................

Since ancient times, mathematicians have been fascinated by problems
concerning prime numbers, and many people have worked on the problem of
determining ways to test if numbers are prime.  One way to test if a
number is prime is to find the number's divisors.  The following
program finds the smallest integral divisor (greater than 1) of a given
number n.  It does this in a straightforward way, by testing n for
divisibility by successive integers starting with 2.

     (define (smallest-divisor n)
       (find-divisor n 2))

     (define (find-divisor n test-divisor)
       (cond ((> (square test-divisor) n) n)
             ((divides? test-divisor n) test-divisor)
             (else (find-divisor n (+ test-divisor 1)))))

     (define (divides? a b)
       (= (remainder b a) 0))

   We can test whether a number is prime as follows: n is prime if and
only if n is its own smallest divisor.

     (define (prime? n)
       (= n (smallest-divisor n)))

   The end test for `find-divisor' is based on the fact that if n is not
prime it must have a divisor less than or equal to _[sqrt]_(n).(1)
This means that the algorithm need only test divisors between 1 and
_[sqrt]_(n).  Consequently, the number of steps required to identify n
as prime will have order of growth [theta](_[sqrt]_(n)).

The Fermat test
...............

The [theta](`log' n) primality test is based on a result from number
theory known as Fermat's Little Theorem.(2)

     *Fermat's Little Theorem:* If n is a prime number and a is any
     positive integer less than n, then a raised to the nth power is
     congruent to a modulo n.

   (Two numbers are said to be "congruent modulo" n if they both have
the same remainder when divided by n.  The remainder of a number a when
divided by n is also referred to as the "remainder of" a "modulo" n, or
simply as a "modulo" n.)

   If n is not prime, then, in general, most of the numbers a< n will
not satisfy the above relation.  This leads to the following algorithm
for testing primality: Given a number n, pick a random number a < n and
compute the remainder of a^n modulo n.  If the result is not equal to
a, then n is certainly not prime.  If it is a, then chances are good
that n is prime.  Now pick another random number a and test it with the
same method.  If it also satisfies the equation, then we can be even
more confident that n is prime.  By trying more and more values of a,
we can increase our confidence in the result.  This algorithm is known
as the Fermat test.

   To implement the Fermat test, we need a procedure that computes the
exponential of a number modulo another number:

     (define (expmod base exp m)
       (cond ((= exp 0) 1)
             ((even? exp)
              (remainder (square (expmod base (/ exp 2) m))
                         m))
             (else
              (remainder (* base (expmod base (- exp 1) m))
                         m))))

   This is very similar to the `fast-expt' procedure of section *Note
1-2-4::.  It uses successive squaring, so that the number of steps
grows logarithmically with the exponent.(3)

   The Fermat test is performed by choosing at random a number a
between 1 and n - 1 inclusive and checking whether the remainder modulo
n of the nth power of a is equal to a.  The random number a is chosen
using the procedure `random', which we assume is included as a primitive
in Scheme. `Random' returns a nonnegative integer less than its integer
input.  Hence, to obtain a random number between 1 and n - 1, we call
`random' with an input of n - 1 and add 1 to the result:

     (define (fermat-test n)
       (define (try-it a)
         (= (expmod a n n) a))
       (try-it (+ 1 (random (- n 1)))))

   The following procedure runs the test a given number of times, as
specified by a parameter.  Its value is true if the test succeeds every
time, and false otherwise.

     (define (fast-prime? n times)
       (cond ((= times 0) true)
             ((fermat-test n) (fast-prime? n (- times 1)))
             (else false)))

Probabilistic methods
.....................

The Fermat test differs in character from most familiar algorithms, in
which one computes an answer that is guaranteed to be correct.  Here,
the answer obtained is only probably correct.  More precisely, if n
ever fails the Fermat test, we can be certain that n is not prime.  But
the fact that n passes the test, while an extremely strong indication,
is still not a guarantee that n is prime.  What we would like to say is
that for any number n, if we perform the test enough times and find
that n always passes the test, then the probability of error in our
primality test can be made as small as we like.

   Unfortunately, this assertion is not quite correct.  There do exist
numbers that fool the Fermat test: numbers n that are not prime and yet
have the property that a^n is congruent to a modulo n for all integers
a < n.  Such numbers are extremely rare, so the Fermat test is quite
reliable in practice.(4)

   There are variations of the Fermat test that cannot be fooled.  In
these tests, as with the Fermat method, one tests the primality of an
integer n by choosing a random integer a<n and checking some condition
that depends upon n and a.  (See *Note Exercise 1-28:: for an example
of such a test.)  On the other hand, in contrast to the Fermat test,
one can prove that, for any n, the condition does not hold for most of
the integers a<n unless n is prime.  Thus, if n passes the test for
some random choice of a, the chances are better than even that n is
prime.  If n passes the test for two random choices of a, the chances
are better than 3 out of 4 that n is prime. By running the test with
more and more randomly chosen values of a we can make the probability
of error as small as we like.

   The existence of tests for which one can prove that the chance of
error becomes arbitrarily small has sparked interest in algorithms of
this type, which have come to be known as "probabilistic algorithms".
There is a great deal of research activity in this area, and
probabilistic algorithms have been fruitfully applied to many fields.(5)

     *Exercise 1.21:* Use the `smallest-divisor' procedure to find the
     smallest divisor of each of the following numbers: 199, 1999,
     19999.

     *Exercise 1.22:* Most Lisp implementations include a primitive
     called `runtime' that returns an integer that specifies the amount
     of time the system has been running (measured, for example, in
     microseconds).  The following `timed-prime-test' procedure, when
     called with an integer n, prints n and checks to see if n is
     prime.  If n is prime, the procedure prints three asterisks
     followed by the amount of time used in performing the test.

          (define (timed-prime-test n)
            (newline)
            (display n)
            (start-prime-test n (runtime)))

          (define (start-prime-test n start-time)
            (if (prime? n)
                (report-prime (- (runtime) start-time))))

          (define (report-prime elapsed-time)
            (display " *** ")
            (display elapsed-time))

     Using this procedure, write a procedure `search-for-primes' that
     checks the primality of consecutive odd integers in a specified
     range.  Use your procedure to find the three smallest primes
     larger than 1000; larger than 10,000; larger than 100,000; larger
     than 1,000,000.  Note the time needed to test each prime.  Since
     the testing algorithm has order of growth of [theta](_[sqrt]_(n)),
     you should expect that testing for primes around 10,000 should
     take about _[sqrt]_(10) times as long as testing for primes around
     1000.  Do your timing data bear this out?  How well do the data
     for 100,000 and 1,000,000 support the _[sqrt]_(n) prediction?  Is
     your result compatible with the notion that programs on your
     machine run in time proportional to the number of steps required
     for the computation?

     *Exercise 1.23:* The `smallest-divisor' procedure shown at the
     start of this section does lots of needless testing: After it
     checks to see if the number is divisible by 2 there is no point in
     checking to see if it is divisible by any larger even numbers.
     This suggests that the values used for `test-divisor' should not
     be 2, 3, 4, 5, 6, ..., but rather 2, 3, 5, 7, 9, ....  To
     implement this change, define a procedure `next' that returns 3 if
     its input is equal to 2 and otherwise returns its input plus 2.
     Modify the `smallest-divisor' procedure to use `(next
     test-divisor)' instead of `(+ test-divisor 1)'.  With
     `timed-prime-test' incorporating this modified version of
     `smallest-divisor', run the test for each of the 12 primes found in
     *Note Exercise 1-22::.  Since this modification halves the number
     of test steps, you should expect it to run about twice as fast.
     Is this expectation confirmed?  If not, what is the observed ratio
     of the speeds of the two algorithms, and how do you explain the
     fact that it is different from 2?

     *Exercise 1.24:* Modify the `timed-prime-test' procedure of *Note
     Exercise 1-22:: to use `fast-prime?' (the Fermat method), and test
     each of the 12 primes you found in that exercise.  Since the
     Fermat test has [theta](`log' n) growth, how would you expect the
     time to test primes near 1,000,000 to compare with the time needed
     to test primes near 1000?  Do your data bear this out?  Can you
     explain any discrepancy you find?

     *Exercise 1.25:* Alyssa P. Hacker complains that we went to a lot
     of extra work in writing `expmod'.  After all, she says, since we
     already know how to compute exponentials, we could have simply
     written

          (define (expmod base exp m)
            (remainder (fast-expt base exp) m))

     Is she correct?  Would this procedure serve as well for our fast
     prime tester?  Explain.

     *Exercise 1.26:* Louis Reasoner is having great difficulty doing
     *Note Exercise 1-24::.  His `fast-prime?' test seems to run more
     slowly than his `prime?' test.  Louis calls his friend Eva Lu Ator
     over to help.  When they examine Louis's code, they find that he
     has rewritten the `expmod' procedure to use an explicit
     multiplication, rather than calling `square':

          (define (expmod base exp m)
            (cond ((= exp 0) 1)
                  ((even? exp)
                   (remainder (* (expmod base (/ exp 2) m)
                                 (expmod base (/ exp 2) m))
                              m))
                  (else
                   (remainder (* base (expmod base (- exp 1) m))
                              m))))

     "I don't see what difference that could make," says Louis.  "I
     do."  says Eva.  "By writing the procedure like that, you have
     transformed the [theta](`log' n) process into a [theta](n)
     process."  Explain.

     *Exercise 1.27:* Demonstrate that the Carmichael numbers listed in
     *Note Footnote 1-47:: really do fool the Fermat test.  That is,
     write a procedure that takes an integer n and tests whether a^n is
     congruent to a modulo n for every a<n, and try your procedure on
     the given Carmichael numbers.

     *Exercise 1.28:* One variant of the Fermat test that cannot be
     fooled is called the "Miller-Rabin test" (Miller 1976; Rabin
     1980).  This starts from an alternate form of Fermat's Little
     Theorem, which states that if n is a prime number and a is any
     positive integer less than n, then a raised to the (n - 1)st power
     is congruent to 1 modulo n.  To test the primality of a number n
     by the Miller-Rabin test, we pick a random number a<n and raise a
     to the (n - 1)st power modulo n using the `expmod' procedure.
     However, whenever we perform the squaring step in `expmod', we
     check to see if we have discovered a "nontrivial square root of 1
     modulo n," that is, a number not equal to 1 or n - 1 whose square
     is equal to 1 modulo n.  It is possible to prove that if such a
     nontrivial square root of 1 exists, then n is not prime.  It is
     also possible to prove that if n is an odd number that is not
     prime, then, for at least half the numbers a<n, computing a^(n-1)
     in this way will reveal a nontrivial square root of 1 modulo n.
     (This is why the Miller-Rabin test cannot be fooled.)  Modify the
     `expmod' procedure to signal if it discovers a nontrivial square
     root of 1, and use this to implement the Miller-Rabin test with a
     procedure analogous to `fermat-test'.  Check your procedure by
     testing various known primes and non-primes.  Hint: One convenient
     way to make `expmod' signal is to have it return 0.

   ---------- Footnotes ----------

   (1) If d is a divisor of n, then so is n/d.  But d and n/d cannot
both be greater than _[sqrt]_(n).

   (2) Pierre de Fermat (1601-1665) is considered to be the founder of
modern number theory.  He obtained many important number-theoretic
results, but he usually announced just the results, without providing
his proofs.  Fermat's Little Theorem was stated in a letter he wrote in
1640.  The first published proof was given by Euler in 1736 (and an
earlier, identical proof was discovered in the unpublished manuscripts
of Leibniz).  The most famous of Fermat's results--known as Fermat's
Last Theorem--was jotted down in 1637 in his copy of the book
`Arithmetic' (by the third-century Greek mathematician Diophantus) with
the remark "I have discovered a truly remarkable proof, but this margin
is too small to contain it."  Finding a proof of Fermat's Last Theorem
became one of the most famous challenges in number theory.  A complete
solution was finally given in 1995 by Andrew Wiles of Princeton
University.

   (3) The reduction steps in the cases where the exponent e is greater
than 1 are based on the fact that, for any integers x, y, and m, we can
find the remainder of x times y modulo m by computing separately the
remainders of x modulo m and y modulo m, multiplying these, and then
taking the remainder of the result modulo m.  For instance, in the case
where e is even, we compute the remainder of b^(e/2) modulo m, square
this, and take the remainder modulo m.  This technique is useful
because it means we can perform our computation without ever having to
deal with numbers much larger than m.  (Compare *Note Exercise 1-25::.)

   (4) [Footnote 1.47] Numbers that fool the Fermat test are called "Carmichael
numbers", and little is known about them other than that they are
extremely rare.  There are 255 Carmichael numbers below 100,000,000.
The smallest few are 561, 1105, 1729, 2465, 2821, and 6601.  In testing
primality of very large numbers chosen at random, the chance of
stumbling upon a value that fools the Fermat test is less than the
chance that cosmic radiation will cause the computer to make an error
in carrying out a "correct" algorithm.  Considering an algorithm to be
inadequate for the first reason but not for the second illustrates the
difference between mathematics and engineering.

   (5) One of the most striking applications of probabilistic prime
testing has been to the field of cryptography.  Although it is now
computationally infeasible to factor an arbitrary 200-digit number, the
primality of such a number can be checked in a few seconds with the
Fermat test.  This fact forms the basis of a technique for constructing
"unbreakable codes" suggested by Rivest, Shamir, and Adleman (1977).
The resulting "RSA algorithm" has become a widely used technique for
enhancing the security of electronic communications.  Because of this
and related developments, the study of prime numbers, once considered
the epitome of a topic in "pure" mathematics to be studied only for its
own sake, now turns out to have important practical applications to
cryptography, electronic funds transfer, and information retrieval.


File: sicp.info,  Node: 1-3,  Prev: 1-2,  Up: Chapter 1

1.3 Formulating Abstractions with Higher-Order Procedures
=========================================================

We have seen that procedures are, in effect, abstractions that describe
compound operations on numbers independent of the particular numbers.
For example, when we

     (define (cube x) (* x x x))

we are not talking about the cube of a particular number, but rather
about a method for obtaining the cube of any number.  Of course we
could get along without ever defining this procedure, by always writing
expressions such as

     (* 3 3 3)
     (* x x x)
     (* y y y)

and never mentioning `cube' explicitly.  This would place us at a
serious disadvantage, forcing us to work always at the level of the
particular operations that happen to be primitives in the language
(multiplication, in this case) rather than in terms of higher-level
operations.  Our programs would be able to compute cubes, but our
language would lack the ability to express the concept of cubing.  One
of the things we should demand from a powerful programming language is
the ability to build abstractions by assigning names to common patterns
and then to work in terms of the abstractions directly.  Procedures
provide this ability.  This is why all but the most primitive
programming languages include mechanisms for defining procedures.

   Yet even in numerical processing we will be severely limited in our
ability to create abstractions if we are restricted to procedures whose
parameters must be numbers.  Often the same programming pattern will be
used with a number of different procedures.  To express such patterns
as concepts, we will need to construct procedures that can accept
procedures as arguments or return procedures as values.  Procedures
that manipulate procedures are called "higher-order procedures".  This
section shows how higher-order procedures can serve as powerful
abstraction mechanisms, vastly increasing the expressive power of our
language.

* Menu:

* 1-3-1::            Procedures as Arguments
* 1-3-2::            Constructing Procedures Using `Lambda'
* 1-3-3::            Procedures as General Methods
* 1-3-4::            Procedures as Returned Values


File: sicp.info,  Node: 1-3-1,  Next: 1-3-2,  Prev: 1-3,  Up: 1-3

1.3.1 Procedures as Arguments
-----------------------------

Consider the following three procedures.  The first computes the sum of
the integers from `a' through `b':

     (define (sum-integers a b)
       (if (> a b)
           0
           (+ a (sum-integers (+ a 1) b))))

   The second computes the sum of the cubes of the integers in the
given range:

     (define (sum-cubes a b)
       (if (> a b)
           0
           (+ (cube a) (sum-cubes (+ a 1) b))))

   The third computes the sum of a sequence of terms in the series

       1       1       1
     ----- + ----- + ------ + ...
     1 * 3   5 * 7   9 * 11

which converges to [pi]/8 (very slowly):(1)

     (define (pi-sum a b)
       (if (> a b)
           0
           (+ (/ 1.0 (* a (+ a 2))) (pi-sum (+ a 4) b))))

   These three procedures clearly share a common underlying pattern.
They are for the most part identical, differing only in the name of the
procedure, the function of `a' used to compute the term to be added,
and the function that provides the next value of `a'.  We could
generate each of the procedures by filling in slots in the same
template:

     (define (<NAME> a b)
       (if (> a b)
           0
           (+ (<TERM> a)
              (<NAME> (<NEXT> a) b))))

   The presence of such a common pattern is strong evidence that there
is a useful abstraction waiting to be brought to the surface.  Indeed,
mathematicians long ago identified the abstraction of "summation of a
series" and invented "sigma notation," for example

       b
      ---
      >    f(n) = f(a) + ... + f(b)
      ---
      n=a

to express this concept.  The power of sigma notation is that it allows
mathematicians to deal with the concept of summation itself rather than
only with particular sums--for example, to formulate general results
about sums that are independent of the particular series being summed.

   Similarly, as program designers, we would like our language to be
powerful enough so that we can write a procedure that expresses the
concept of summation itself rather than only procedures that compute
particular sums.  We can do so readily in our procedural language by
taking the common template shown above and transforming the "slots"
into formal parameters:

     (define (sum term a next b)
       (if (> a b)
           0
           (+ (term a)
              (sum term (next a) next b))))

   Notice that `sum' takes as its arguments the lower and upper bounds
`a' and `b' together with the procedures `term' and `next'.  We can use
`sum' just as we would any procedure.  For example, we can use it
(along with a procedure `inc' that increments its argument by 1) to
define `sum-cubes':

     (define (inc n) (+ n 1))

     (define (sum-cubes a b)
       (sum cube a inc b))

   Using this, we can compute the sum of the cubes of the integers from
1 to 10:

     (sum-cubes 1 10)
     3025

   With the aid of an identity procedure to compute the term, we can
define `sum-integers' in terms of `sum':

     (define (identity x) x)

     (define (sum-integers a b)
       (sum identity a inc b))

   Then we can add up the integers from 1 to 10:

     (sum-integers 1 10)
     55

   We can also define `pi-sum' in the same way:(2)

     (define (pi-sum a b)
       (define (pi-term x)
         (/ 1.0 (* x (+ x 2))))
       (define (pi-next x)
         (+ x 4))
       (sum pi-term a pi-next b))

   Using these procedures, we can compute an approximation to [pi]:

     (* 8 (pi-sum 1 1000))
     3.139592655589783

   Once we have `sum', we can use it as a building block in formulating
further concepts.  For instance, the definite integral of a function f
between the limits a and b can be approximated numerically using the
formula

     /b     /  /     dx \    /          dx \    /           dx \      \
     |  f = | f| a + -- | + f| a + dx + -- | + f| a + 2dx + -- | + ...| dx
     /a     \  \     2  /    \          2  /    \           2  /      /

for small values of dx.  We can express this directly as a procedure:

     (define (integral f a b dx)
       (define (add-dx x) (+ x dx))
       (* (sum f (+ a (/ dx 2.0)) add-dx b)
          dx))

     (integral cube 0 1 0.01)
     .24998750000000042

     (integral cube 0 1 0.001)
     .249999875000001

(The exact value of the integral of `cube' between 0 and 1 is 1/4.)

     *Exercise 1.29:* Simpson's Rule is a more accurate method of
     numerical integration than the method illustrated above.  Using
     Simpson's Rule, the integral of a function f between a and b is
     approximated as

          h
          - (y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + ... + 2y_(n-2) + 4y_(n-1) + y_n)
          3

     where h = (b - a)/n, for some even integer n, and y_k = f(a + kh).
     (Increasing n increases the accuracy of the approximation.)
     Define a procedure that takes as arguments f, a, b, and n and
     returns the value of the integral, computed using Simpson's Rule.
     Use your procedure to integrate `cube' between 0 and 1 (with n =
     100 and n = 1000), and compare the results to those of the
     `integral' procedure shown above.

     *Exercise 1.30:* The `sum' procedure above generates a linear
     recursion.  The procedure can be rewritten so that the sum is
     performed iteratively.  Show how to do this by filling in the
     missing expressions in the following definition:

          (define (sum term a next b)
            (define (iter a result)
              (if <??>
                  <??>
                  (iter <??> <??>)))
            (iter <??> <??>))

     *Exercise 1.31:*
       a. The `sum' procedure is only the simplest of a vast number of
          similar abstractions that can be captured as higher-order
          procedures.(3)  Write an analogous procedure called `product'
          that returns the product of the values of a function at
          points over a given range.  Show how to define `factorial' in
          terms of `product'.  Also use `product' to compute
          approximations to [pi] using the formula(4)

               pi   2 * 4 * 4 * 6 * 6 * 8 ...
               -- = -------------------------
                4   3 * 3 * 5 * 5 * 7 * 7 ...

       b. If your `product' procedure generates a recursive process,
          write one that generates an iterative process.  If it
          generates an iterative process, write one that generates a
          recursive process.


     *Exercise 1.32:*
       a. Show that `sum' and `product' (*Note Exercise 1-31::) are
          both special cases of a still more general notion called
          `accumulate' that combines a collection of terms, using some
          general accumulation function:

               (accumulate combiner null-value term a next b)

          `Accumulate' takes as arguments the same term and range
          specifications as `sum' and `product', together with a
          `combiner' procedure (of two arguments) that specifies how
          the current term is to be combined with the accumulation of
          the preceding terms and a `null-value' that specifies what
          base value to use when the terms run out.  Write `accumulate'
          and show how `sum' and `product' can both be defined as
          simple calls to `accumulate'.

       b. If your `accumulate' procedure generates a recursive process,
          write one that generates an iterative process.  If it
          generates an iterative process, write one that generates a
          recursive process.


     *Exercise 1.33:* You can obtain an even more general version of
     `accumulate' (*Note Exercise 1-32::) by introducing the notion of
     a "filter" on the terms to be combined.  That is, combine only
     those terms derived from values in the range that satisfy a
     specified condition.  The resulting `filtered-accumulate'
     abstraction takes the same arguments as accumulate, together with
     an additional predicate of one argument that specifies the filter.
     Write `filtered-accumulate' as a procedure.  Show how to express
     the following using `filtered-accumulate':

       a. the sum of the squares of the prime numbers in the interval a
          to b (assuming that you have a `prime?' predicate already
          written)

       b. the product of all the positive integers less than n that are
          relatively prime to n (i.e., all positive integers i < n such
          that GCD(i,n) = 1).


   ---------- Footnotes ----------

   (1) This series, usually written in the equivalent form ([pi]/4) = 1
- (1/3) + (1/5) - (1/7) + ..., is due to Leibniz.  We'll see how to use
this as the basis for some fancy numerical tricks in section *Note
3-5-3::.

   (2) Notice that we have used block structure (section *Note 1-1-8::)
to embed the definitions of `pi-next' and `pi-term' within `pi-sum',
since these procedures are unlikely to be useful for any other purpose.
We will see how to get rid of them altogether in section *Note 1-3-2::.

   (3) The intent of *Note Exercise 1-31:: through *Note Exercise
1-33:: is to demonstrate the expressive power that is attained by using
an appropriate abstraction to consolidate many seemingly disparate
operations.  However, though accumulation and filtering are elegant
ideas, our hands are somewhat tied in using them at this point since we
do not yet have data structures to provide suitable means of
combination for these abstractions.  We will return to these ideas in
section *Note 2-2-3:: when we show how to use "sequences" as interfaces
for combining filters and accumulators to build even more powerful
abstractions.  We will see there how these methods really come into
their own as a powerful and elegant approach to designing programs.

   (4) This formula was discovered by the seventeenth-century English
mathematician John Wallis.


File: sicp.info,  Node: 1-3-2,  Next: 1-3-3,  Prev: 1-3-1,  Up: 1-3

1.3.2 Constructing Procedures Using `Lambda'
--------------------------------------------

In using `sum' as in section *Note 1-3-1::, it seems terribly awkward to
have to define trivial procedures such as `pi-term' and `pi-next' just
so we can use them as arguments to our higher-order procedure.  Rather
than define `pi-next' and `pi-term', it would be more convenient to
have a way to directly specify "the procedure that returns its input
incremented by 4" and "the procedure that returns the reciprocal of its
input times its input plus 2."  We can do this by introducing the
special form `lambda', which creates procedures.  Using `lambda' we can
describe what we want as

     (lambda (x) (+ x 4))

and

     (lambda (x) (/ 1.0 (* x (+ x 2))))

   Then our `pi-sum' procedure can be expressed without defining any
auxiliary procedures as

     (define (pi-sum a b)
       (sum (lambda (x) (/ 1.0 (* x (+ x 2))))
            a
            (lambda (x) (+ x 4))
            b))

   Again using `lambda', we can write the `integral' procedure without
having to define the auxiliary procedure `add-dx':

     (define (integral f a b dx)
       (* (sum f
               (+ a (/ dx 2.0))
               (lambda (x) (+ x dx))
               b)
          dx))

   In general, `lambda' is used to create procedures in the same way as
`define', except that no name is specified for the procedure:

     (lambda (<FORMAL-PARAMETERS>) <BODY>)

   The resulting procedure is just as much a procedure as one that is
created using `define'.  The only difference is that it has not been
associated with any name in the environment.  In fact,

     (define (plus4 x) (+ x 4))

is equivalent to

     (define plus4 (lambda (x) (+ x 4)))

   We can read a `lambda' expression as follows:

     (lambda                     (x)     (+   x     4))
         |                        |       |   |     |
     the procedure of an argument x that adds x and 4

   Like any expression that has a procedure as its value, a `lambda'
expression can be used as the operator in a combination such as

     ((lambda (x y z) (+ x y (square z))) 1 2 3)
     12

or, more generally, in any context where we would normally use a
procedure name.(1)

Using `let' to create local variables
.....................................

Another use of `lambda' is in creating local variables.  We often need
local variables in our procedures other than those that have been bound
as formal parameters.  For example, suppose we wish to compute the
function

     f(x,y) = x(1 + xy)^2 + y(1 - y) + (1 + xy)(1 - y)

which we could also express as

          a = 1 + xy
          b = 1 - y
     f(x,y) = xa^2 + yb + ab

   In writing a procedure to compute f, we would like to include as
local variables not only x and y but also the names of intermediate
quantities like a and b.  One way to accomplish this is to use an
auxiliary procedure to bind the local variables:

     (define (f x y)
       (define (f-helper a b)
         (+ (* x (square a))
            (* y b)
            (* a b)))
       (f-helper (+ 1 (* x y))
                 (- 1 y)))

   Of course, we could use a `lambda' expression to specify an anonymous
procedure for binding our local variables.  The body of `f' then
becomes a single call to that procedure:

     (define (f x y)
       ((lambda (a b)
          (+ (* x (square a))
             (* y b)
             (* a b)))
        (+ 1 (* x y))
        (- 1 y)))

   This construct is so useful that there is a special form called
`let' to make its use more convenient.  Using `let', the `f' procedure
could be written as

     (define (f x y)
       (let ((a (+ 1 (* x y)))
             (b (- 1 y)))
         (+ (* x (square a))
            (* y b)
            (* a b))))

   The general form of a `let' expression is

     (let ((<VAR1> <EXP1>)
           (<VAR2> <EXP2>)
           ...
           (<VARN> <VARN>))
        <BODY>)

which can be thought of as saying

     let <VAR_1> have the value <EXP_1> and
         <VAR_2> have the value <EXP_2> and
         ...
         <VAR_N> have the value <EXP_N>
     in  <BODY>

   The first part of the `let' expression is a list of name-expression
pairs.  When the `let' is evaluated, each name is associated with the
value of the corresponding expression.  The body of the `let' is
evaluated with these names bound as local variables.  The way this
happens is that the `let' expression is interpreted as an alternate
syntax for

     ((lambda (<VAR_1> ... <VAR_N>)
         <BODY>)
      <EXP_1>
      ...
      <EXP_N>)

   No new mechanism is required in the interpreter in order to provide
local variables.  A `let' expression is simply syntactic sugar for the
underlying `lambda' application.

   We can see from this equivalence that the scope of a variable
specified by a `let' expression is the body of the `let'.  This implies
that:

   * `Let' allows one to bind variables as locally as possible to where
     they are to be used.  For example, if the value of `x' is 5, the
     value of the expression

          (+ (let ((x 3))
               (+ x (* x 10)))
             x)

     is 38.  Here, the `x' in the body of the `let' is 3, so the value
     of the `let' expression is 33.  On the other hand, the `x' that is
     the second argument to the outermost `+' is still 5.

   * The variables' values are computed outside the `let'.  This
     matters when the expressions that provide the values for the local
     variables depend upon variables having the same names as the local
     variables themselves.  For example, if the value of `x' is 2, the
     expression

          (let ((x 3)
                (y (+ x 2)))
            (* x y))

     will have the value 12 because, inside the body of the `let', `x'
     will be 3 and `y' will be 4 (which is the outer `x' plus 2).


   Sometimes we can use internal definitions to get the same effect as
with `let'.  For example, we could have defined the procedure `f' above
as

     (define (f x y)
       (define a (+ 1 (* x y)))
       (define b (- 1 y))
       (+ (* x (square a))
          (* y b)
          (* a b)))

   We prefer, however, to use `let' in situations like this and to use
internal `define' only for internal procedures.(2)

     *Exercise 1.34:* Suppose we define the procedure

          (define (f g)
            (g 2))

     Then we have

          (f square)
          4

          (f (lambda (z) (* z (+ z 1))))
          6

     What happens if we (perversely) ask the interpreter to evaluate
     the combination `(f f)'?  Explain.

   ---------- Footnotes ----------

   (1) It would be clearer and less intimidating to people learning
Lisp if a name more obvious than `lambda', such as `make-procedure',
were used.  But the convention is firmly entrenched.  The notation is
adopted from the [lambda] calculus, a mathematical formalism introduced
by the mathematical logician Alonzo Church (1941).  Church developed
the [lambda] calculus to provide a rigorous foundation for studying the
notions of function and function application.  The [lambda] calculus
has become a basic tool for mathematical investigations of the
semantics of programming languages.

   (2) Understanding internal definitions well enough to be sure a
program means what we intend it to mean requires a more elaborate model
of the evaluation process than we have presented in this chapter.  The
subtleties do not arise with internal definitions of procedures,
however.  We will return to this issue in section *Note 4-1-6::, after
we learn more about evaluation.


File: sicp.info,  Node: 1-3-3,  Next: 1-3-4,  Prev: 1-3-2,  Up: 1-3

1.3.3 Procedures as General Methods
-----------------------------------

We introduced compound procedures in section *Note 1-1-4:: as a
mechanism for abstracting patterns of numerical operations so as to
make them independent of the particular numbers involved.  With
higher-order procedures, such as the `integral' procedure of section
*Note 1-3-1::, we began to see a more powerful kind of abstraction:
procedures used to express general methods of computation, independent
of the particular functions involved.  In this section we discuss two
more elaborate examples--general methods for finding zeros and fixed
points of functions--and show how these methods can be expressed
directly as procedures.

Finding roots of equations by the half-interval method
......................................................

The "half-interval method" is a simple but powerful technique for
finding roots of an equation f(x) = 0, where f is a continuous
function.  The idea is that, if we are given points a and b such that
f(a) < 0 < f(b), then f must have at least one zero between a and b.
To locate a zero, let x be the average of a and b and compute f(x).  If
f(x) > 0, then f must have a zero between a and x.  If f(x) < 0, then f
must have a zero between x and b.  Continuing in this way, we can
identify smaller and smaller intervals on which f must have a zero.
When we reach a point where the interval is small enough, the process
stops.  Since the interval of uncertainty is reduced by half at each
step of the process, the number of steps required grows as
[theta](`log'( L/T)), where L is the length of the original interval
and T is the error tolerance (that is, the size of the interval we will
consider "small enough").  Here is a procedure that implements this
strategy:

     (define (search f neg-point pos-point)
       (let ((midpoint (average neg-point pos-point)))
         (if (close-enough? neg-point pos-point)
             midpoint
             (let ((test-value (f midpoint)))
               (cond ((positive? test-value)
                      (search f neg-point midpoint))
                     ((negative? test-value)
                      (search f midpoint pos-point))
                     (else midpoint))))))

   We assume that we are initially given the function f together with
points at which its values are negative and positive.  We first compute
the midpoint of the two given points.  Next we check to see if the
given interval is small enough, and if so we simply return the midpoint
as our answer.  Otherwise, we compute as a test value the value of f at
the midpoint.  If the test value is positive, then we continue the
process with a new interval running from the original negative point to
the midpoint.  If the test value is negative, we continue with the
interval from the midpoint to the positive point.  Finally, there is
the possibility that the test value is 0, in which case the midpoint is
itself the root we are searching for.

   To test whether the endpoints are "close enough" we can use a
procedure similar to the one used in section *Note 1-1-7:: for
computing square roots:(1)

     (define (close-enough? x y)
       (< (abs (- x y)) 0.001))

   `Search' is awkward to use directly, because we can accidentally
give it points at which f's values do not have the required sign, in
which case we get a wrong answer.  Instead we will use `search' via the
following procedure, which checks to see which of the endpoints has a
negative function value and which has a positive value, and calls the
`search' procedure accordingly.  If the function has the same sign on
the two given points, the half-interval method cannot be used, in which
case the procedure signals an error.(2)

     (define (half-interval-method f a b)
       (let ((a-value (f a))
             (b-value (f b)))
         (cond ((and (negative? a-value) (positive? b-value))
                (search f a b))
               ((and (negative? b-value) (positive? a-value))
                (search f b a))
               (else
                (error "Values are not of opposite sign" a b)))))

   The following example uses the half-interval method to approximate
[pi] as the root between 2 and 4 of `sin' x = 0:

     (half-interval-method sin 2.0 4.0)
     3.14111328125

   Here is another example, using the half-interval method to search
for a root of the equation x^3 - 2x - 3 = 0 between 1 and 2:

     (half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))
                           1.0
                           2.0)
     1.89306640625

Finding fixed points of functions
.................................

A number x is called a "fixed point" of a function f if x satisfies the
equation f(x) = x.  For some functions f we can locate a fixed point by
beginning with an initial guess and applying f repeatedly,

     f(x), f(f(x), (f(f(f(x))))

until the value does not change very much.  Using this idea, we can
devise a procedure `fixed-point' that takes as inputs a function and an
initial guess and produces an approximation to a fixed point of the
function.  We apply the function repeatedly until we find two
successive values whose difference is less than some prescribed
tolerance:

     (define tolerance 0.00001)

     (define (fixed-point f first-guess)
       (define (close-enough? v1 v2)
         (< (abs (- v1 v2)) tolerance))
       (define (try guess)
         (let ((next (f guess)))
           (if (close-enough? guess next)
               next
               (try next))))
       (try first-guess))

   For example, we can use this method to approximate the fixed point
of the cosine function, starting with 1 as an initial approximation:(3)

     (fixed-point cos 1.0)
     .7390822985224023

   Similarly, we can find a solution to the equation y = `sin' y +
`cos' y:

     (fixed-point (lambda (y) (+ (sin y) (cos y)))
                  1.0)
     1.2587315962971173

   The fixed-point process is reminiscent of the process we used for
finding square roots in section *Note 1-1-7::.  Both are based on the
idea of repeatedly improving a guess until the result satisfies some
criterion.  In fact, we can readily formulate the square-root
computation as a fixed-point search.  Computing the square root of some
number x requires finding a y such that y^2 = x.  Putting this equation
into the equivalent form y = x/y, we recognize that we are looking for
a fixed point of the function(4) y |-> x/y, and we can therefore try to
compute square roots as

     (define (sqrt x)
       (fixed-point (lambda (y) (/ x y))
                    1.0))

   Unfortunately, this fixed-point search does not converge.  Consider
an initial guess y_1.  The next guess is y_2 = x/y_1 and the next guess
is y_3 = x/y_2 = x/(x/y_1) = y_1.  This results in an infinite loop in
which the two guesses y_1 and y_2 repeat over and over, oscillating
about the answer.

   One way to control such oscillations is to prevent the guesses from
changing so much.  Since the answer is always between our guess y and
x/y, we can make a new guess that is not as far from y as x/y by
averaging y with x/y, so that the next guess after y is (1/2)(y + x/y)
instead of x/y.  The process of making such a sequence of guesses is
simply the process of looking for a fixed point of y |-> (1/2)(y + x/y):

     (define (sqrt x)
       (fixed-point (lambda (y) (average y (/ x y)))
                    1.0))

   (Note that y = (1/2)(y + x/y) is a simple transformation of the
equation y = x/y; to derive it, add y to both sides of the equation and
divide by 2.)

   With this modification, the square-root procedure works.  In fact,
if we unravel the definitions, we can see that the sequence of
approximations to the square root generated here is precisely the same
as the one generated by our original square-root procedure of section
*Note 1-1-7::.  This approach of averaging successive approximations to
a solution, a technique we that we call "average damping", often aids
the convergence of fixed-point searches.

     *Exercise 1.35:* Show that the golden ratio [phi] (section *Note
     1-2-2::) is a fixed point of the transformation x |-> 1 + 1/x, and
     use this fact to compute [phi] by means of the `fixed-point'
     procedure.

     *Exercise 1.36:* Modify `fixed-point' so that it prints the
     sequence of approximations it generates, using the `newline' and
     `display' primitives shown in *Note Exercise 1-22::.  Then find a
     solution to x^x = 1000 by finding a fixed point of x |->
     `log'(1000)/`log'(x).  (Use Scheme's primitive `log' procedure,
     which computes natural logarithms.)  Compare the number of steps
     this takes with and without average damping.  (Note that you
     cannot start `fixed-point' with a guess of 1, as this would cause
     division by `log'(1) = 0.)

     *Exercise 1.37:*
       a. An infinite "continued fraction" is an expression of the form

                          N_1
               f = ---------------------
                              N_2
                   D_1 + ---------------
                                  N_3
                         D_2 + ---------
                               D_3 + ...

          As an example, one can show that the infinite continued
          fraction expansion with the n_i and the D_i all equal to 1
          produces 1/[phi], where [phi] is the golden ratio (described
          in section *Note 1-2-2::).  One way to approximate an
          infinite continued fraction is to truncate the expansion
          after a given number of terms.  Such a truncation--a
          so-called finite continued fraction "k-term finite continued
          fraction"--has the form

                      N_1
               -----------------
                         N_2
               D_1 + -----------
                     ...    N_K
                         + -----
                            D_K

          Suppose that `n' and `d' are procedures of one argument (the
          term index i) that return the n_i and D_i of the terms of the
          continued fraction.  Define a procedure `cont-frac' such that
          evaluating `(cont-frac n d k)' computes the value of the
          k-term finite continued fraction.  Check your procedure by
          approximating 1/[phi] using

               (cont-frac (lambda (i) 1.0)
                          (lambda (i) 1.0)
                          k)

          for successive values of `k'.  How large must you make `k' in
          order to get an approximation that is accurate to 4 decimal
          places?

       b. If your `cont-frac' procedure generates a recursive process,
          write one that generates an iterative process.  If it
          generates an iterative process, write one that generates a
          recursive process.


     *Exercise 1.38:* In 1737, the Swiss mathematician Leonhard Euler
     published a memoir `De Fractionibus Continuis', which included a
     continued fraction expansion for e - 2, where e is the base of the
     natural logarithms.  In this fraction, the n_i are all 1, and the
     D_i are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ....  Write
     a program that uses your `cont-frac' procedure from *Note Exercise
     1-37:: to approximate e, based on Euler's expansion.

     *Exercise 1.39:* A continued fraction representation of the
     tangent function was published in 1770 by the German mathematician
     J.H. Lambert:

                        x
          tan x = ---------------
                          x^2
                  1 - -----------
                            x^2
                      3 - -------
                          5 - ...

     where x is in radians.  Define a procedure `(tan-cf x k)' that
     computes an approximation to the tangent function based on
     Lambert's formula.  `K' specifies the number of terms to compute,
     as in *Note Exercise 1-37::.

   ---------- Footnotes ----------

   (1) We have used 0.001 as a representative "small" number to
indicate a tolerance for the acceptable error in a calculation.  The
appropriate tolerance for a real calculation depends upon the problem
to be solved and the limitations of the computer and the algorithm.
This is often a very subtle consideration, requiring help from a
numerical analyst or some other kind of magician.

   (2) This can be accomplished using `error', which takes as arguments
a number of items that are printed as error messages.

   (3) Try this during a boring lecture: Set your calculator to radians
mode and then repeatedly press the `cos' button until you obtain the
fixed point.

   (4) |-> (pronounced "maps to") is the mathematician's way of writing
`lambda'.  y |-> x/y means `(lambda(y) (/ x y))', that is, the function
whose value at y is x/y.


File: sicp.info,  Node: 1-3-4,  Prev: 1-3-3,  Up: 1-3

1.3.4 Procedures as Returned Values
-----------------------------------

The above examples demonstrate how the ability to pass procedures as
arguments significantly enhances the expressive power of our
programming language.  We can achieve even more expressive power by
creating procedures whose returned values are themselves procedures.

   We can illustrate this idea by looking again at the fixed-point
example described at the end of section *Note 1-3-3::.  We formulated a
new version of the square-root procedure as a fixed-point search,
starting with the observation that [sqrt]x is a fixed-point of the
function y |-> x/y.  Then we used average damping to make the
approximations converge.  Average damping is a useful general technique
in itself.  Namely, given a function f, we consider the function whose
value at x is equal to the average of x and f(x).

   We can express the idea of average damping by means of the following
procedure:

     (define (average-damp f)
       (lambda (x) (average x (f x))))

   `Average-damp' is a procedure that takes as its argument a procedure
`f' and returns as its value a procedure (produced by the `lambda')
that, when applied to a number `x', produces the average of `x' and `(f
x)'.  For example, applying `average-damp' to the `square' procedure
produces a procedure whose value at some number x is the average of x
and x^2.  Applying this resulting procedure to 10 returns the average
of 10 and 100, or 55:(1)

     ((average-damp square) 10)
     55

   Using `average-damp', we can reformulate the square-root procedure as
follows:

     (define (sqrt x)
       (fixed-point (average-damp (lambda (y) (/ x y)))
                    1.0))

   Notice how this formulation makes explicit the three ideas in the
method: fixed-point search, average damping, and the function y |-> x/y.
It is instructive to compare this formulation of the square-root method
with the original version given in section *Note 1-1-7::.  Bear in mind
that these procedures express the same process, and notice how much
clearer the idea becomes when we express the process in terms of these
abstractions.  In general, there are many ways to formulate a process
as a procedure.  Experienced programmers know how to choose procedural
formulations that are particularly perspicuous, and where useful
elements of the process are exposed as separate entities that can be
reused in other applications.  As a simple example of reuse, notice
that the cube root of x is a fixed point of the function y |-> x/y^2,
so we can immediately generalize our square-root procedure to one that
extracts cube roots:(2)

     (define (cube-root x)
       (fixed-point (average-damp (lambda (y) (/ x (square y))))
                    1.0))

Newton's method
...............

When we first introduced the square-root procedure, in section *Note
1-1-7::, we mentioned that this was a special case of "Newton's
method".  If x |-> g(x) is a differentiable function, then a solution
of the equation g(x) = 0 is a fixed point of the function x |-> f(x)
where

                g(x)
     f(x) = x - -----
                Dg(x)

and Dg(x) is the derivative of g evaluated at x.  Newton's method is
the use of the fixed-point method we saw above to approximate a
solution of the equation by finding a fixed point of the function f.(3)

   For many functions g and for sufficiently good initial guesses for x,
Newton's method converges very rapidly to a solution of g(x) = 0.(4)

   In order to implement Newton's method as a procedure, we must first
express the idea of derivative.  Note that "derivative," like average
damping, is something that transforms a function into another function.
For instance, the derivative of the function x |-> x^3 is the function
x |-> 3x^2.  In general, if g is a function and dx is a small number,
then the derivative Dg of g is the function whose value at any number x
is given (in the limit of small dx) by

             g(x + dx) - g(x)
     Dg(c) = ----------------
                    dx

Thus, we can express the idea of derivative (taking dx to be, say,
0.00001) as the procedure

     (define (deriv g)
       (lambda (x)
         (/ (- (g (+ x dx)) (g x))
            dx)))

along with the definition

     (define dx 0.00001)

   Like `average-damp', `deriv' is a procedure that takes a procedure as
argument and returns a procedure as value.  For example, to approximate
the derivative of x |-> x^3 at 5 (whose exact value is 75) we can
evaluate

     (define (cube x) (* x x x))

     ((deriv cube) 5)
     75.00014999664018

   With the aid of `deriv', we can express Newton's method as a
fixed-point process:

     (define (newton-transform g)
       (lambda (x)
         (- x (/ (g x) ((deriv g) x)))))

     (define (newtons-method g guess)
       (fixed-point (newton-transform g) guess))

   The `newton-transform' procedure expresses the formula at the
beginning of this section, and `newtons-method' is readily defined in
terms of this.  It takes as arguments a procedure that computes the
function for which we want to find a zero, together with an initial
guess.  For instance, to find the square root of x, we can use Newton's
method to find a zero of the function y |-> y^2 - x starting with an
initial guess of 1.(5)

   This provides yet another form of the square-root procedure:

     (define (sqrt x)
       (newtons-method (lambda (y) (- (square y) x))
                       1.0))

Abstractions and first-class procedures
.......................................

We've seen two ways to express the square-root computation as an
instance of a more general method, once as a fixed-point search and
once using Newton's method.  Since Newton's method was itself expressed
as a fixed-point process, we actually saw two ways to compute square
roots as fixed points.  Each method begins with a function and finds a
fixed point of some transformation of the function.  We can express
this general idea itself as a procedure:

     (define (fixed-point-of-transform g transform guess)
       (fixed-point (transform g) guess))

   This very general procedure takes as its arguments a procedure `g'
that computes some function, a procedure that transforms `g', and an
initial guess.  The returned result is a fixed point of the transformed
function.

   Using this abstraction, we can recast the first square-root
computation from this section (where we look for a fixed point of the
average-damped version of y |-> x/y) as an instance of this general
method:

     (define (sqrt x)
       (fixed-point-of-transform (lambda (y) (/ x y))
                                 average-damp
                                 1.0))

   Similarly, we can express the second square-root computation from
this section (an instance of Newton's method that finds a fixed point
of the Newton transform of y |-> y^2 - x) as

     (define (sqrt x)
       (fixed-point-of-transform (lambda (y) (- (square y) x))
                                 newton-transform
                                 1.0))

   We began section *Note 1-3:: with the observation that compound
procedures are a crucial abstraction mechanism, because they permit us
to express general methods of computing as explicit elements in our
programming language.  Now we've seen how higher-order procedures
permit us to manipulate these general methods to create further
abstractions.

   As programmers, we should be alert to opportunities to identify the
underlying abstractions in our programs and to build upon them and
generalize them to create more powerful abstractions.  This is not to
say that one should always write programs in the most abstract way
possible; expert programmers know how to choose the level of
abstraction appropriate to their task.  But it is important to be able
to think in terms of these abstractions, so that we can be ready to
apply them in new contexts.  The significance of higher-order
procedures is that they enable us to represent these abstractions
explicitly as elements in our programming language, so that they can be
handled just like other computational elements.

   In general, programming languages impose restrictions on the ways in
which computational elements can be manipulated.  Elements with the
fewest restrictions are said to have "first-class" status.  Some of the
"rights and privileges" of first-class elements are:(6)

   * They may be named by variables.

   * They may be passed as arguments to procedures.

   * They may be returned as the results of procedures.

   * They may be included in data structures.(7)


   Lisp, unlike other common programming languages, awards procedures
full first-class status.  This poses challenges for efficient
implementation, but the resulting gain in expressive power is
enormous.(8)

     *Exercise 1.40:* Define a procedure `cubic' that can be used
     together with the `newtons-method' procedure in expressions of the
     form

          (newtons-method (cubic a b c) 1)

     to approximate zeros of the cubic x^3 + ax^2 + bx + c.

     *Exercise 1.41:* Define a procedure `double' that takes a
     procedure of one argument as argument and returns a procedure that
     applies the original procedure twice.  For example, if `inc' is a
     procedure that adds 1 to its argument, then `(double inc)' should
     be a procedure that adds 2.  What value is returned by

          (((double (double double)) inc) 5)

     *Exercise 1.42:* Let f and g be two one-argument functions.  The "composition"
     f after g is defined to be the function x |-> f(g(x)).  Define a
     procedure `compose' that implements composition.  For example, if
     `inc' is a procedure that adds 1 to its argument,

          ((compose square inc) 6)
          49

     *Exercise 1.43:* If f is a numerical function and n is a positive
     integer, then we can form the nth repeated application of f, which
     is defined to be the function whose value at x is
     f(f(...(f(x))...)).  For example, if f is the function x |-> x +
     1, then the nth repeated application of f is the function x |-> x
     + n.  If f is the operation of squaring a number, then the nth
     repeated application of f is the function that raises its argument
     to the 2^nth power.  Write a procedure that takes as inputs a
     procedure that computes f and a positive integer n and returns the
     procedure that computes the nth repeated application of f.  Your
     procedure should be able to be used as follows:

          ((repeated square 2) 5)
          625

     Hint: You may find it convenient to use `compose' from *Note
     Exercise 1-42::.

     *Exercise 1.44:* The idea of "smoothing" a function is an
     important concept in signal processing.  If f is a function and dx
     is some small number, then the smoothed version of f is the
     function whose value at a point x is the average of f(x - dx),
     f(x), and f(x + dx).  Write a procedure `smooth' that takes as
     input a procedure that computes f and returns a procedure that
     computes the smoothed f.  It is sometimes valuable to repeatedly
     smooth a function (that is, smooth the smoothed function, and so
     on) to obtained the "n-fold smoothed function".  Show how to
     generate the n-fold smoothed function of any given function using
     `smooth' and `repeated' from *Note Exercise 1-43::.

     *Exercise 1.45:* We saw in section *Note 1-3-3:: that attempting
     to compute square roots by naively finding a fixed point of y |->
     x/y does not converge, and that this can be fixed by average
     damping.  The same method works for finding cube roots as fixed
     points of the average-damped y |-> x/y^2.  Unfortunately, the
     process does not work for fourth roots--a single average damp is
     not enough to make a fixed-point search for y |-> x/y^3 converge.
     On the other hand, if we average damp twice (i.e., use the average
     damp of the average damp of y |-> x/y^3) the fixed-point search
     does converge.  Do some experiments to determine how many average
     damps are required to compute nth roots as a fixed-point search
     based upon repeated average damping of y |-> x/y^(n-1).  Use this
     to implement a simple procedure for computing nth roots using
     `fixed-point', `average-damp', and the `repeated' procedure of
     *Note Exercise 1-43::.  Assume that any arithmetic operations you
     need are available as primitives.

     *Exercise 1.46:* Several of the numerical methods described in
     this chapter are instances of an extremely general computational
     strategy known as "iterative improvement".  Iterative improvement
     says that, to compute something, we start with an initial guess
     for the answer, test if the guess is good enough, and otherwise
     improve the guess and continue the process using the improved
     guess as the new guess.  Write a procedure `iterative-improve'
     that takes two procedures as arguments: a method for telling
     whether a guess is good enough and a method for improving a guess.
     `Iterative-improve' should return as its value a procedure that
     takes a guess as argument and keeps improving the guess until it
     is good enough.  Rewrite the `sqrt' procedure of section *Note
     1-1-7:: and the `fixed-point' procedure of section *Note 1-3-3::
     in terms of `iterative-improve'.

   ---------- Footnotes ----------

   (1) Observe that this is a combination whose operator is itself a
combination.  *Note Exercise 1-4:: already demonstrated the ability to
form such combinations, but that was only a toy example.  Here we begin
to see the real need for such combinations--when applying a procedure
that is obtained as the value returned by a higher-order procedure.

   (2) See *Note Exercise 1-45:: for a further generalization.

   (3) Elementary calculus books usually describe Newton's method in
terms of the sequence of approximations x_(n+1) = x_n - g(x_n)/Dg(x_n).
Having language for talking about processes and using the idea of
fixed points simplifies the description of the method.

   (4) Newton's method does not always converge to an answer, but it can
be shown that in favorable cases each iteration doubles the
number-of-digits accuracy of the approximation to the solution.  In
such cases, Newton's method will converge much more rapidly than the
half-interval method.

   (5) For finding square roots, Newton's method converges rapidly to
the correct solution from any starting point.

   (6) The notion of first-class status of programming-language
elements is due to the British computer scientist Christopher Strachey
(1916-1975).

   (7) We'll see examples of this after we introduce data structures in
*Note Chapter 2::.

   (8) The major implementation cost of first-class procedures is that
allowing procedures to be returned as values requires reserving storage
for a procedure's free variables even while the procedure is not
executing.  In the Scheme implementation we will study in section *Note
4-1::, these variables are stored in the procedure's environment.


File: sicp.info,  Node: Chapter 2,  Next: Chapter 3,  Prev: Chapter 1,  Up: Top

2 Building Abstractions with Data
*********************************

     We now come to the decisive step of mathematical abstraction: we
     forget about what the symbols stand for. ...[The mathematician]
     need not be idle; there are many operations which he may carry out
     with these symbols, without ever having to look at the things they
     stand for.

     --Hermann Weyl, `The Mathematical Way of Thinking'

   We concentrated in *Note Chapter 1:: on computational processes and
on the role of procedures in program design.  We saw how to use
primitive data (numbers) and primitive operations (arithmetic
operations), how to combine procedures to form compound procedures
through composition, conditionals, and the use of parameters, and how
to abstract procedures by using `define'.  We saw that a procedure can
be regarded as a pattern for the local evolution of a process, and we
classified, reasoned about, and performed simple algorithmic analyses of
some common patterns for processes as embodied in procedures.  We also
saw that higher-order procedures enhance the power of our language by
enabling us to manipulate, and thereby to reason in terms of, general
methods of computation.  This is much of the essence of programming.

   In this chapter we are going to look at more complex data.  All the
procedures in *Note Chapter 1:: operate on simple numerical data, and
simple data are not sufficient for many of the problems we wish to
address using computation.  Programs are typically designed to model
complex phenomena, and more often than not one must construct
computational objects that have several parts in order to model
real-world phenomena that have several aspects.  Thus, whereas our
focus in *Note Chapter 1:: was on building abstractions by combining
procedures to form compound procedures, we turn in this chapter to
another key aspect of any programming language: the means it provides
for building abstractions by combining data objects to form "compound
data".

   Why do we want compound data in a programming language?  For the
same reasons that we want compound procedures: to elevate the
conceptual level at which we can design our programs, to increase the
modularity of our designs, and to enhance the expressive power of our
language.  Just as the ability to define procedures enables us to deal
with processes at a higher conceptual level than that of the primitive
operations of the language, the ability to construct compound data
objects enables us to deal with data at a higher conceptual level than
that of the primitive data objects of the language.

   Consider the task of designing a system to perform arithmetic with
rational numbers.  We could imagine an operation `add-rat' that takes
two rational numbers and produces their sum.  In terms of simple data,
a rational number can be thought of as two integers: a numerator and a
denominator.  Thus, we could design a program in which each rational
number would be represented by two integers (a numerator and a
denominator) and where `add-rat' would be implemented by two procedures
(one producing the numerator of the sum and one producing the
denominator).  But this would be awkward, because we would then need to
explicitly keep track of which numerators corresponded to which
denominators.  In a system intended to perform many operations on many
rational numbers, such bookkeeping details would clutter the programs
substantially, to say nothing of what they would do to our minds.  It
would be much better if we could "glue together" a numerator and
denominator to form a pair--a "compound data object"--that our programs
could manipulate in a way that would be consistent with regarding a
rational number as a single conceptual unit.

   The use of compound data also enables us to increase the modularity
of our programs.  If we can manipulate rational numbers directly as
objects in their own right, then we can separate the part of our
program that deals with rational numbers per se from the details of how
rational numbers may be represented as pairs of integers.  The general
technique of isolating the parts of a program that deal with how data
objects are represented from the parts of a program that deal with how
data objects are used is a powerful design methodology called "data
abstraction".  We will see how data abstraction makes programs much
easier to design, maintain, and modify.

   The use of compound data leads to a real increase in the expressive
power of our programming language.  Consider the idea of forming a
"linear combination" ax + by.  We might like to write a procedure that
would accept a, b, x, and y as arguments and return the value of ax +
by.  This presents no difficulty if the arguments are to be numbers,
because we can readily define the procedure

     (define (linear-combination a b x y)
       (+ (* a x) (* b y)))

   But suppose we are not concerned only with numbers.  Suppose we
would like to express, in procedural terms, the idea that one can form
linear combinations whenever addition and multiplication are
defined--for rational numbers, complex numbers, polynomials, or
whatever.  We could express this as a procedure of the form

     (define (linear-combination a b x y)
       (add (mul a x) (mul b y)))

where `add' and `mul' are not the primitive procedures `+' and `*' but
rather more complex things that will perform the appropriate operations
for whatever kinds of data we pass in as the arguments `a', `b', `x',
and `y'. The key point is that the only thing `linear-combination'
should need to know about `a', `b', `x', and `y' is that the procedures
`add' and `mul' will perform the appropriate manipulations.  From the
perspective of the procedure `linear-combination', it is irrelevant
what `a', `b', `x', and `y' are and even more irrelevant how they might
happen to be represented in terms of more primitive data.  This same
example shows why it is important that our programming language provide
the ability to manipulate compound objects directly: Without this,
there is no way for a procedure such as `linear-combination' to pass
its arguments along to `add' and `mul' without having to know their
detailed structure.(1)

   We begin this chapter by implementing the rational-number arithmetic
system mentioned above.  This will form the background for our
discussion of compound data and data abstraction.  As with compound
procedures, the main issue to be addressed is that of abstraction as a
technique for coping with complexity, and we will see how data
abstraction enables us to erect suitable "abstraction barriers" between
different parts of a program.

   We will see that the key to forming compound data is that a
programming language should provide some kind of "glue" so that data
objects can be combined to form more complex data objects.  There are
many possible kinds of glue.  Indeed, we will discover how to form
compound data using no special "data" operations at all, only
procedures.  This will further blur the distinction between "procedure"
and "data," which was already becoming tenuous toward the end of *Note
Chapter 1::.  We will also explore some conventional techniques for
representing sequences and trees.  One key idea in dealing with
compound data is the notion of "closure"--that the glue we use for
combining data objects should allow us to combine not only primitive
data objects, but compound data objects as well.  Another key idea is
that compound data objects can serve as "conventional interfaces" for
combining program modules in mix-and-match ways.  We illustrate some of
these ideas by presenting a simple graphics language that exploits
closure.

   We will then augment the representational power of our language by
introducing "symbolic expressions"--data whose elementary parts can be
arbitrary symbols rather than only numbers.  We explore various
alternatives for representing sets of objects.  We will find that, just
as a given numerical function can be computed by many different
computational processes, there are many ways in which a given data
structure can be represented in terms of simpler objects, and the
choice of representation can have significant impact on the time and
space requirements of processes that manipulate the data.  We will
investigate these ideas in the context of symbolic differentiation, the
representation of sets, and the encoding of information.

   Next we will take up the problem of working with data that may be
represented differently by different parts of a program.  This leads to
the need to implement "generic operations", which must handle many
different types of data.  Maintaining modularity in the presence of
generic operations requires more powerful abstraction barriers than can
be erected with simple data abstraction alone.  In particular, we
introduce programming "data-directed programming" as a technique that
allows individual data representations to be designed in isolation and
then combined "additively" (i.e., without modification).  To illustrate
the power of this approach to system design, we close the chapter by
applying what we have learned to the implementation of a package for
performing symbolic arithmetic on polynomials, in which the
coefficients of the polynomials can be integers, rational numbers,
complex numbers, and even other polynomials.

* Menu:

* 2-1::              Introduction to Data Abstraction
* 2-2::              Hierarchical Data and the Closure Property
* 2-3::              Symbolic Data
* 2-4::              Multiple Representations for Abstract Data
* 2-5::              Systems with Generic Operations

   ---------- Footnotes ----------

   (1) The ability to directly manipulate procedures provides an
analogous increase in the expressive power of a programming language.
For example, in section *Note 1-3-1:: we introduced the `sum'
procedure, which takes a procedure `term' as an argument and computes
the sum of the values of `term' over some specified interval.  In order
to define `sum', it is crucial that we be able to speak of a procedure
such as `term' as an entity in its own right, without regard for how
`term' might be expressed with more primitive operations.  Indeed, if
we did not have the notion of "a procedure," it is doubtful that we
would ever even think of the possibility of defining an operation such
as `sum'.  Moreover, insofar as performing the summation is concerned,
the details of how `term' may be constructed from more primitive
operations are irrelevant.


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2.1 Introduction to Data Abstraction
====================================

In section *Note 1-1-8::, we noted that a procedure used as an element
in creating a more complex procedure could be regarded not only as a
collection of particular operations but also as a procedural
abstraction.  That is, the details of how the procedure was implemented
could be suppressed, and the particular procedure itself could be
replaced by any other procedure with the same overall behavior.  In
other words, we could make an abstraction that would separate the way
the procedure would be used from the details of how the procedure would
be implemented in terms of more primitive procedures.  The analogous
notion for compound data is called "data abstraction".  Data
abstraction is a methodology that enables us to isolate how a compound
data object is used from the details of how it is constructed from more
primitive data objects.

   The basic idea of data abstraction is to structure the programs that
are to use compound data objects so that they operate on "abstract
data." That is, our programs should use data in such a way as to make
no assumptions about the data that are not strictly necessary for
performing the task at hand.  At the same time, a "concrete" data
representation is defined independent of the programs that use the
data.  The interface between these two parts of our system will be a
set of procedures, called "selectors" and "constructors", that
implement the abstract data in terms of the concrete representation.  To
illustrate this technique, we will consider how to design a set of
procedures for manipulating rational numbers.

* Menu:

* 2-1-1::            Example: Arithmetic Operations for Rational Numbers
* 2-1-2::            Abstraction Barriers
* 2-1-3::            What Is Meant by Data?
* 2-1-4::            Extended Exercise: Interval Arithmetic


File: sicp.info,  Node: 2-1-1,  Next: 2-1-2,  Prev: 2-1,  Up: 2-1

2.1.1 Example: Arithmetic Operations for Rational Numbers
---------------------------------------------------------

Suppose we want to do arithmetic with rational numbers.  We want to be
able to add, subtract, multiply, and divide them and to test whether
two rational numbers are equal.

   Let us begin by assuming that we already have a way of constructing
a rational number from a numerator and a denominator.  We also assume
that, given a rational number, we have a way of extracting (or
selecting) its numerator and its denominator.  Let us further assume
that the constructor and selectors are available as procedures:

   * `(make-rat <N> <D>)' returns therational number whose numerator is
     the integer `<N>' and whose denominator is the integer `<D>'.

   * `(numer <X>)' returns the numerator of the rational number `<X>'.

   * `(denom <X>)' returns the denominator of the rational number `<X>'.


   We are using here a powerful strategy of synthesis: "wishful
thinking".  We haven't yet said how a rational number is represented,
or how the procedures `numer', `denom', and `make-rat' should be
implemented.  Even so, if we did have these three procedures, we could
then add, subtract, multiply, divide, and test equality by using the
following relations:

     n_1   n_2   n_1 d_2 + n_2 d_1
     --- + --- = -----------------
     d_1   d_2        d_1 d_2

     n_1   n_2   n_1 d_2 - n_2 d_1
     --- - --- = -----------------
     d_1   d_2        d_1 d_2

     n_1   n_2   n_1 n_2
     --- * --- = -------
     d_1   d_2   d_1 d_2

     n_1 / d_1   n_1 d_2
     --------- = -------
     n_2 / d_2   d_1 n_2

     n_1   n_2
     --- = ---  if and only if n_1 d_2 = n_2 d_1
     d_1   d_2

We can express these rules as procedures:

     (define (add-rat x y)
       (make-rat (+ (* (numer x) (denom y))
                    (* (numer y) (denom x)))
                 (* (denom x) (denom y))))

     (define (sub-rat x y)
       (make-rat (- (* (numer x) (denom y))
                    (* (numer y) (denom x)))
                 (* (denom x) (denom y))))

     (define (mul-rat x y)
       (make-rat (* (numer x) (numer y))
                 (* (denom x) (denom y))))

     (define (div-rat x y)
       (make-rat (* (numer x) (denom y))
                 (* (denom x) (numer y))))

     (define (equal-rat? x y)
       (= (* (numer x) (denom y))
          (* (numer y) (denom x))))

   Now we have the operations on rational numbers defined in terms of
the selector and constructor procedures `numer', `denom', and
`make-rat'.  But we haven't yet defined these.  What we need is some
way to glue together a numerator and a denominator to form a rational
number.

Pairs
.....

To enable us to implement the concrete level of our data abstraction,
our language provides a compound structure called a "pair", which can be
constructed with the primitive procedure `cons'.  This procedure takes
two arguments and returns a compound data object that contains the two
arguments as parts.  Given a pair, we can extract the parts using the
primitive procedures `car' and `cdr'.(1) Thus, we can use `cons',
`car', and `cdr' as follows:

     (define x (cons 1 2))

     (car x)
     1

     (cdr x)
     2

   Notice that a pair is a data object that can be given a name and
manipulated, just like a primitive data object.  Moreover, `cons' can
be used to form pairs whose elements are pairs, and so on:

     (define x (cons 1 2))

     (define y (cons 3 4))

     (define z (cons x y))

     (car (car z))
     1

     (car (cdr z))
     3

   In section *Note 2-2:: we will see how this ability to combine pairs
means that pairs can be used as general-purpose building blocks to
create all sorts of complex data structures.  The single compound-data
primitive "pair", implemented by the procedures `cons', `car', and
`cdr', is the only glue we need.  Data objects constructed from pairs
are called "list-structured" data.

Representing rational numbers
.............................

Pairs offer a natural way to complete the rational-number system.
Simply represent a rational number as a pair of two integers: a
numerator and a denominator.  Then `make-rat', `numer', and `denom' are
readily implemented as follows:(2)

     (define (make-rat n d) (cons n d))

     (define (numer x) (car x))

     (define (denom x) (cdr x))

   Also, in order to display the results of our computations, we can
print rational numbers by printing the numerator, a slash, and the
denominator:(3)

     (define (print-rat x)
       (newline)
       (display (numer x))
       (display "/")
       (display (denom x)))

   Now we can try our rational-number procedures:

     (define one-half (make-rat 1 2))

     (print-rat one-half)
     1/2

     (define one-third (make-rat 1 3))

     (print-rat (add-rat one-half one-third))
     5/6

     (print-rat (mul-rat one-half one-third))
     1/6

     (print-rat (add-rat one-third one-third))
     6/9

   As the final example shows, our rational-number implementation does
not reduce rational numbers to lowest terms.  We can remedy this by
changing `make-rat'. If we have a `gcd' procedure like the one in
section *Note 1-2-5:: that produces the greatest common divisor of two
integers, we can use `gcd' to reduce the numerator and the denominator
to lowest terms before constructing the pair:

     (define (make-rat n d)
       (let ((g (gcd n d)))
         (cons (/ n g) (/ d g))))

   Now we have

     (print-rat (add-rat one-third one-third))
     2/3

as desired.  This modification was accomplished by changing the
constructor `make-rat' without changing any of the procedures (such as
`add-rat' and `mul-rat') that implement the actual operations.

     *Exercise 2.1:* Define a better version of `make-rat' that handles
     both positive and negative arguments.  `Make-rat' should normalize
     the sign so that if the rational number is positive, both the
     numerator and denominator are positive, and if the rational number
     is negative, only the numerator is negative.

   ---------- Footnotes ----------

   (1) The name `cons' stands for "construct."  The names `car' and
`cdr' derive from the original implementation of Lisp on the IBM 704.
That machine had an addressing scheme that allowed one to reference the
"address" and "decrement" parts of a memory location.  `Car' stands for
"Contents of Address part of Register" and `cdr' (pronounced
"could-er") stands for "Contents of Decrement part of Register."

   (2) Another way to define the selectors and constructor is

     (define make-rat cons)
     (define numer car)
     (define denom cdr)

   The first definition associates the name `make-rat' with the value
of the expression `cons', which is the primitive procedure that
constructs pairs.  Thus `make-rat' and `cons' are names for the same
primitive constructor.

   Defining selectors and constructors in this way is efficient:
Instead of `make-rat' _calling_ `cons', `make-rat' _is_ `cons', so
there is only one procedure called, not two, when `make-rat' is called.
On the other hand, doing this defeats debugging aids that trace
procedure calls or put breakpoints on procedure calls: You may want to
watch `make-rat' being called, but you certainly don't want to watch
every call to `cons'.

   We have chosen not to use this style of definition in this book.

   (3) `Display' is the Scheme primitive for printing data.  The Scheme
primitive `newline' starts a new line for printing.  Neither of these
procedures returns a useful value, so in the uses of `print-rat' below,
we show only what `print-rat' prints, not what the interpreter prints
as the value returned by `print-rat'.


File: sicp.info,  Node: 2-1-2,  Next: 2-1-3,  Prev: 2-1-1,  Up: 2-1

2.1.2 Abstraction Barriers
--------------------------

Before continuing with more examples of compound data and data
abstraction, let us consider some of the issues raised by the
rational-number example.  We defined the rational-number operations in
terms of a constructor `make-rat' and selectors `numer' and `denom'.
In general, the underlying idea of data abstraction is to identify for
each type of data object a basic set of operations in terms of which
all manipulations of data objects of that type will be expressed, and
then to use only those operations in manipulating the data.

   We can envision the structure of the rational-number system as shown
in figure *Note Figure 2-1::.  The horizontal lines represent barriers
"abstraction barriers" that isolate different "levels" of the system.
At each level, the barrier separates the programs (above) that use the
data abstraction from the programs (below) that implement the data
abstraction.  Programs that use rational numbers manipulate them solely
in terms of the procedures supplied "for public use" by the
rational-number package: `add-rat', `sub-rat', `mul-rat', `div-rat',
and `equal-rat?'. These, in turn, are implemented solely in terms of
the constructor and selectors `make-rat', `numer', and `denom', which
themselves are implemented in terms of pairs.  The details of how pairs
are implemented are irrelevant to the rest of the rational-number
package so long as pairs can be manipulated by the use of `cons',
`car', and `cdr'.  In effect, procedures at each level are the
interfaces that define the abstraction barriers and connect the
different levels.

     *Figure 2.1:* Data-abstraction barriers in the rational-number
     package.

                  +------------------------------------+
          --------| Programs that use rational numbers |--------
                  +------------------------------------+
                    Rational numbers in promblem domain
                      +---------------------------+
          ------------|   add-rat  sub-rat  ...   |-------------
                      +---------------------------+
             Rational numbers as numerators and denominators
                        +------------------------+
          --------------| make-rat  numer  denom |--------------
                        +------------------------+
                        Rational numbers as pairs
                            +----------------+
          ------------------| cons  car  cdr |------------------
                            +----------------+
                      However pairs are implemented

   This simple idea has many advantages.  One advantage is that it
makes programs much easier to maintain and to modify.  Any complex data
structure can be represented in a variety of ways with the primitive
data structures provided by a programming language.  Of course, the
choice of representation influences the programs that operate on it;
thus, if the representation were to be changed at some later time, all
such programs might have to be modified accordingly.  This task could
be time-consuming and expensive in the case of large programs unless
the dependence on the representation were to be confined by design to a
very few program modules.

   For example, an alternate way to address the problem of reducing
rational numbers to lowest terms is to perform the reduction whenever
we access the parts of a rational number, rather than when we construct
it.  This leads to different constructor and selector procedures:

     (define (make-rat n d)
       (cons n d))

     (define (numer x)
       (let ((g (gcd (car x) (cdr x))))
         (/ (car x) g)))

     (define (denom x)
       (let ((g (gcd (car x) (cdr x))))
         (/ (cdr x) g)))

   The difference between this implementation and the previous one lies
in when we compute the `gcd'.  If in our typical use of rational
numbers we access the numerators and denominators of the same rational
numbers many times, it would be preferable to compute the `gcd' when
the rational numbers are constructed.  If not, we may be better off
waiting until access time to compute the `gcd'.  In any case, when we
change from one representation to the other, the procedures `add-rat',
`sub-rat', and so on do not have to be modified at all.

   Constraining the dependence on the representation to a few interface
procedures helps us design programs as well as modify them, because it
allows us to maintain the flexibility to consider alternate
implementations.  To continue with our simple example, suppose we are
designing a rational-number package and we can't decide initially
whether to perform the `gcd' at construction time or at selection time.
The data-abstraction methodology gives us a way to defer that decision
without losing the ability to make progress on the rest of the system.

     *Exercise 2.2:* Consider the problem of representing line segments
     in a plane.  Each segment is represented as a pair of points: a
     starting point and an ending point.  Define a constructor
     `make-segment' and selectors `start-segment' and `end-segment'
     that define the representation of segments in terms of points.
     Furthermore, a point can be represented as a pair of numbers: the
     x coordinate and the y coordinate.  Accordingly, specify a
     constructor `make-point' and selectors `x-point' and `y-point'
     that define this representation.  Finally, using your selectors
     and constructors, define a procedure `midpoint-segment' that takes
     a line segment as argument and returns its midpoint (the point
     whose coordinates are the average of the coordinates of the
     endpoints).  To try your procedures, you'll need a way to print
     points:

          (define (print-point p)
            (newline)
            (display "(")
            (display (x-point p))
            (display ",")
            (display (y-point p))
            (display ")"))

     *Exercise 2.3:* Implement a representation for rectangles in a
     plane.  (Hint: You may want to make use of *Note Exercise 2-2::.)
     In terms of your constructors and selectors, create procedures
     that compute the perimeter and the area of a given rectangle.  Now
     implement a different representation for rectangles.  Can you
     design your system with suitable abstraction barriers, so that the
     same perimeter and area procedures will work using either
     representation?


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2.1.3 What Is Meant by Data?
----------------------------

We began the rational-number implementation in section *Note 2-1-1:: by
implementing the rational-number operations `add-rat', `sub-rat', and
so on in terms of three unspecified procedures: `make-rat', `numer',
and `denom'.  At that point, we could think of the operations as being
defined in terms of data objects--numerators, denominators, and rational
numbers--whose behavior was specified by the latter three procedures.

   But exactly what is meant by "data"?  It is not enough to say
"whatever is implemented by the given selectors and constructors."
Clearly, not every arbitrary set of three procedures can serve as an
appropriate basis for the rational-number implementation.  We need to
guarantee that, if we construct a rational number `x' from a pair of
integers `n' and `d', then extracting the `numer' and the `denom' of
`x' and dividing them should yield the same result as dividing `n' by
`d'.  In other words, `make-rat', `numer', and `denom' must satisfy the
condition that, for any integer `n' and any non-zero integer `d', if
`x' is (`make-rat n d'), then

     (numer x)    n
     --------- = ---
     (denom x)    d

   In fact, this is the only condition `make-rat', `numer', and `denom'
must fulfill in order to form a suitable basis for a rational-number
representation.  In general, we can think of data as defined by some
collection of selectors and constructors, together with specified
conditions that these procedures must fulfill in order to be a valid
representation.(1)

   This point of view can serve to define not only "high-level" data
objects, such as rational numbers, but lower-level objects as well.
Consider the notion of a pair, which we used in order to define our
rational numbers.  We never actually said what a pair was, only that
the language supplied procedures `cons', `car', and `cdr' for operating
on pairs.  But the only thing we need to know about these three
operations is that if we glue two objects together using `cons' we can
retrieve the objects using `car' and `cdr'.  That is, the operations
satisfy the condition that, for any objects `x' and `y', if `z' is
`(cons x y)' then `(car z)' is `x' and `(cdr z)' is `y'.  Indeed, we
mentioned that these three procedures are included as primitives in our
language.  However, any triple of procedures that satisfies the above
condition can be used as the basis for implementing pairs.  This point
is illustrated strikingly by the fact that we could implement `cons',
`car', and `cdr' without using any data structures at all but only
using procedures.  Here are the definitions:

     (define (cons x y)
       (define (dispatch m)
         (cond ((= m 0) x)
               ((= m 1) y)
               (else (error "Argument not 0 or 1 -- CONS" m))))
       dispatch)

     (define (car z) (z 0))

     (define (cdr z) (z 1))

   This use of procedures corresponds to nothing like our intuitive
notion of what data should be.  Nevertheless, all we need to do to show
that this is a valid way to represent pairs is to verify that these
procedures satisfy the condition given above.

   The subtle point to notice is that the value returned by `(cons x
y)' is a procedure--namely the internally defined procedure `dispatch',
which takes one argument and returns either `x' or `y' depending on
whether the argument is 0 or 1.  Correspondingly, `(car z)' is defined
to apply `z' to 0.  Hence, if `z' is the procedure formed by `(cons x
y)', then `z' applied to 0 will yield `x'. Thus, we have shown that
`(car (cons x y))' yields `x', as desired.  Similarly, `(cdr (cons x
y))' applies the procedure returned by `(cons x y)' to 1, which returns
`y'.  Therefore, this procedural implementation of pairs is a valid
implementation, and if we access pairs using only `cons', `car', and
`cdr' we cannot distinguish this implementation from one that uses
"real" data structures.

   The point of exhibiting the procedural representation of pairs is
not that our language works this way (Scheme, and Lisp systems in
general, implement pairs directly, for efficiency reasons) but that it
could work this way.  The procedural representation, although obscure,
is a perfectly adequate way to represent pairs, since it fulfills the
only conditions that pairs need to fulfill.  This example also
demonstrates that the ability to manipulate procedures as objects
automatically provides the ability to represent compound data.  This
may seem a curiosity now, but procedural representations of data will
play a central role in our programming repertoire.  This style of
programming is often called "message passing", and we will be using it
as a basic tool in *Note Chapter 3:: when we address the issues of
modeling and simulation.

     *Exercise 2.4:* Here is an alternative procedural representation
     of pairs.  For this representation, verify that `(car (cons x y))'
     yields `x' for any objects `x' and `y'.

          (define (cons x y)
            (lambda (m) (m x y)))

          (define (car z)
            (z (lambda (p q) p)))

     What is the corresponding definition of `cdr'? (Hint: To verify
     that this works, make use of the substitution model of section
     *Note 1-1-5::.)

     *Exercise 2.5:* Show that we can represent pairs of nonnegative
     integers using only numbers and arithmetic operations if we
     represent the pair a and b as the integer that is the product 2^a
     3^b.  Give the corresponding definitions of the procedures `cons',
     `car', and `cdr'.

     *Exercise 2.6:* In case representing pairs as procedures wasn't
     mind-boggling enough, consider that, in a language that can
     manipulate procedures, we can get by without numbers (at least
     insofar as nonnegative integers are concerned) by implementing 0
     and the operation of adding 1 as

          (define zero (lambda (f) (lambda (x) x)))

          (define (add-1 n)
            (lambda (f) (lambda (x) (f ((n f) x)))))

     This representation is known as "Church numerals", after its
     inventor, Alonzo Church, the logician who invented the [lambda]
     calculus.

     Define `one' and `two' directly (not in terms of `zero' and
     `add-1').  (Hint: Use substitution to evaluate `(add-1 zero)').
     Give a direct definition of the addition procedure `+' (not in
     terms of repeated application of `add-1').

   ---------- Footnotes ----------

   (1) Surprisingly, this idea is very difficult to formulate
rigorously. There are two approaches to giving such a formulation.  One,
pioneered by C. A. R. Hoare (1972), is known as the method of models
"abstract models".  It formalizes the "procedures plus conditions"
specification as outlined in the rational-number example above.  Note
that the condition on the rational-number representation was stated in
terms of facts about integers (equality and division).  In general,
abstract models define new kinds of data objects in terms of previously
defined types of data objects.  Assertions about data objects can
therefore be checked by reducing them to assertions about previously
defined data objects.  Another approach, introduced by Zilles at MIT,
by Goguen, Thatcher, Wagner, and Wright at IBM (see Thatcher, Wagner,
and Wright 1978), and by Guttag at Toronto (see Guttag 1977), is called "algebraic
specification".  It regards the "procedures" as elements of an abstract
algebraic system whose behavior is specified by axioms that correspond
to our "conditions," and uses the techniques of abstract algebra to
check assertions about data objects.  Both methods are surveyed in the
paper by Liskov and Zilles (1975).


File: sicp.info,  Node: 2-1-4,  Prev: 2-1-3,  Up: 2-1

2.1.4 Extended Exercise: Interval Arithmetic
--------------------------------------------

Alyssa P. Hacker is designing a system to help people solve engineering
problems.  One feature she wants to provide in her system is the
ability to manipulate inexact quantities (such as measured parameters
of physical devices) with known precision, so that when computations
are done with such approximate quantities the results will be numbers
of known precision.

   Electrical engineers will be using Alyssa's system to compute
electrical quantities.  It is sometimes necessary for them to compute
the value of a parallel equivalent resistance R_p of two resistors R_1
and R_2 using the formula

                 1
     R_p = -------------
           1/R_1 + 1/R_2

   Resistance values are usually known only up to some tolerance
guaranteed by the manufacturer of the resistor.  For example, if you
buy a resistor labeled "6.8 ohms with 10% tolerance" you can only be
sure that the resistor has a resistance between 6.8 - 0.68 = 6.12 and
6.8 + 0.68 = 7.48 ohms.  Thus, if you have a 6.8-ohm 10% resistor in
parallel with a 4.7-ohm 5% resistor, the resistance of the combination
can range from about 2.58 ohms (if the two resistors are at the lower
bounds) to about 2.97 ohms (if the two resistors are at the upper
bounds).

   Alyssa's idea is to implement "interval arithmetic" as a set of
arithmetic operations for combining "intervals" (objects that represent
the range of possible values of an inexact quantity).  The result of
adding, subtracting, multiplying, or dividing two intervals is itself
an interval, representing the range of the result.

   Alyssa postulates the existence of an abstract object called an
"interval" that has two endpoints: a lower bound and an upper bound.
She also presumes that, given the endpoints of an interval, she can
construct the interval using the data constructor `make-interval'.
Alyssa first writes a procedure for adding two intervals.  She reasons
that the minimum value the sum could be is the sum of the two lower
bounds and the maximum value it could be is the sum of the two upper
bounds:

     (define (add-interval x y)
       (make-interval (+ (lower-bound x) (lower-bound y))
                      (+ (upper-bound x) (upper-bound y))))

   Alyssa also works out the product of two intervals by finding the
minimum and the maximum of the products of the bounds and using them as
the bounds of the resulting interval.  (`Min' and `max' are primitives
that find the minimum or maximum of any number of arguments.)

     (define (mul-interval x y)
       (let ((p1 (* (lower-bound x) (lower-bound y)))
             (p2 (* (lower-bound x) (upper-bound y)))
             (p3 (* (upper-bound x) (lower-bound y)))
             (p4 (* (upper-bound x) (upper-bound y))))
         (make-interval (min p1 p2 p3 p4)
                        (max p1 p2 p3 p4))))

   To divide two intervals, Alyssa multiplies the first by the
reciprocal of the second.  Note that the bounds of the reciprocal
interval are the reciprocal of the upper bound and the reciprocal of
the lower bound, in that order.

     (define (div-interval x y)
       (mul-interval x
                     (make-interval (/ 1.0 (upper-bound y))
                                    (/ 1.0 (lower-bound y)))))

     *Exercise 2.7:* Alyssa's program is incomplete because she has not
     specified the implementation of the interval abstraction.  Here is
     a definition of the interval constructor:

          (define (make-interval a b) (cons a b))

     Define selectors `upper-bound' and `lower-bound' to complete the
     implementation.

     *Exercise 2.8:* Using reasoning analogous to Alyssa's, describe
     how the difference of two intervals may be computed.  Define a
     corresponding subtraction procedure, called `sub-interval'.

     *Exercise 2.9:* The "width" of an interval is half of the
     difference between its upper and lower bounds.  The width is a
     measure of the uncertainty of the number specified by the
     interval.  For some arithmetic operations the width of the result
     of combining two intervals is a function only of the widths of the
     argument intervals, whereas for others the width of the
     combination is not a function of the widths of the argument
     intervals.  Show that the width of the sum (or difference) of two
     intervals is a function only of the widths of the intervals being
     added (or subtracted).  Give examples to show that this is not
     true for multiplication or division.

     *Exercise 2.10:* Ben Bitdiddle, an expert systems programmer,
     looks over Alyssa's shoulder and comments that it is not clear what
     it means to divide by an interval that spans zero.  Modify
     Alyssa's code to check for this condition and to signal an error
     if it occurs.

     *Exercise 2.11:* In passing, Ben also cryptically comments: "By
     testing the signs of the endpoints of the intervals, it is
     possible to break `mul-interval' into nine cases, only one of which
     requires more than two multiplications."  Rewrite this procedure
     using Ben's suggestion.

     After debugging her program, Alyssa shows it to a potential user,
     who complains that her program solves the wrong problem.  He wants
     a program that can deal with numbers represented as a center value
     and an additive tolerance; for example, he wants to work with
     intervals such as 3.5 +/- 0.15 rather than [3.35, 3.65].  Alyssa
     returns to her desk and fixes this problem by supplying an
     alternate constructor and alternate selectors:

          (define (make-center-width c w)
            (make-interval (- c w) (+ c w)))

          (define (center i)
            (/ (+ (lower-bound i) (upper-bound i)) 2))

          (define (width i)
            (/ (- (upper-bound i) (lower-bound i)) 2))

     Unfortunately, most of Alyssa's users are engineers.  Real
     engineering situations usually involve measurements with only a
     small uncertainty, measured as the ratio of the width of the
     interval to the midpoint of the interval.  Engineers usually
     specify percentage tolerances on the parameters of devices, as in
     the resistor specifications given earlier.

     *Exercise 2.12:* Define a constructor `make-center-percent' that
     takes a center and a percentage tolerance and produces the desired
     interval.  You must also define a selector `percent' that produces
     the percentage tolerance for a given interval.  The `center'
     selector is the same as the one shown above.

     *Exercise 2.13:* Show that under the assumption of small
     percentage tolerances there is a simple formula for the approximate
     percentage tolerance of the product of two intervals in terms of
     the tolerances of the factors.  You may simplify the problem by
     assuming that all numbers are positive.

     After considerable work, Alyssa P. Hacker delivers her finished
     system.  Several years later, after she has forgotten all about
     it, she gets a frenzied call from an irate user, Lem E. Tweakit.
     It seems that Lem has noticed that the formula for parallel
     resistors can be written in two algebraically equivalent ways:

           R_1 R_2
          ---------
          R_1 + R_2

     and

                1
          -------------
          1/R_1 + 1/R_2

     He has written the following two programs, each of which computes
     the parallel-resistors formula differently:

          (define (par1 r1 r2)
            (div-interval (mul-interval r1 r2)
                          (add-interval r1 r2)))

          (define (par2 r1 r2)
            (let ((one (make-interval 1 1)))
              (div-interval one
                            (add-interval (div-interval one r1)
                                          (div-interval one r2)))))

     Lem complains that Alyssa's program gives different answers for
     the two ways of computing. This is a serious complaint.

     *Exercise 2.14:* Demonstrate that Lem is right.  Investigate the
     behavior of the system on a variety of arithmetic expressions.
     Make some intervals A and B, and use them in computing the
     expressions A/A and A/B.  You will get the most insight by using
     intervals whose width is a small percentage of the center value.
     Examine the results of the computation in center-percent form (see
     *Note Exercise 2-12::).

     *Exercise 2.15:* Eva Lu Ator, another user, has also noticed the
     different intervals computed by different but algebraically
     equivalent expressions. She says that a formula to compute with
     intervals using Alyssa's system will produce tighter error bounds
     if it can be written in such a form that no variable that
     represents an uncertain number is repeated.  Thus, she says,
     `par2' is a "better" program for parallel resistances than `par1'.
     Is she right?  Why?

     *Exercise 2.16:* Explain, in general, why equivalent algebraic
     expressions may lead to different answers.  Can you devise an
     interval-arithmetic package that does not have this shortcoming,
     or is this task impossible?  (Warning: This problem is very
     difficult.)


File: sicp.info,  Node: 2-2,  Next: 2-3,  Prev: 2-1,  Up: Chapter 2

2.2 Hierarchical Data and the Closure Property
==============================================

As we have seen, pairs provide a primitive "glue" that we can use to
construct compound data objects.  *Note Figure 2-2:: shows a standard
way to visualize a pair--in this case, the pair formed by `(cons 1 2)'.
In this representation, which is called "box-and-pointer notation",
each object is shown as a "pointer" to a box.  The box for a primitive
object contains a representation of the object.  For example, the box
for a number contains a numeral.  The box for a pair is actually a
double box, the left part containing (a pointer to) the `car' of the
pair and the right part containing the `cdr'.

     *Figure 2.2:* Box-and-pointer representation of `(cons 1 2)'.

               +---+---+     +---+
          ---->| * | *-+---->| 2 |
               +-|-+---+     +---+
                 |
                 V
               +---+
               | 1 |
               +---+

   We have already seen that `cons' can be used to combine not only
numbers but pairs as well.  (You made use of this fact, or should have,
in doing *Note Exercise 2-2:: and *Note Exercise 2-3::.)  As a
consequence, pairs provide a universal building block from which we can
construct all sorts of data structures.  *Note Figure 2-3:: shows two
ways to use pairs to combine the numbers 1, 2, 3, and 4.

     *Figure 2.3:* Two ways to combine 1, 2, 3, and 4 using pairs.

               +---+---+     +---+---+         +---+---+     +---+
          ---->| * | *-+---->| * | * |    ---->| * | *-+---->| 4 |
               +-|-+---+     +-|-+-|-+         +-|-+---+     +---+
                 |             |   |             |
                 V             V   V             V
             +---+---+      +---+ +---+      +---+---+     +---+---+
             | * | * |      | 3 | | 4 |      | * | *-+---->| * | * |
             +-|-+-|-+      +---+ +---+      +-|-+---+     +-|-+-|-+
               |   |                           |             |   |
               V   V                           V             V   V
            +---+ +---+                      +---+        +---+ +---+
            | 1 | | 2 |                      | 1 |        | 2 | | 3 |
            +---+ +---+                      +---+        +---+ +---+

            (cons (cons 1 2)                 (cons (cons 1
                  (cons 3 4))                            (cons 2 3))
                                                   4)

   The ability to create pairs whose elements are pairs is the essence
of list structure's importance as a representational tool.  We refer to
this ability as the "closure property" of `cons'.  In general, an
operation for combining data objects satisfies the closure property if
the results of combining things with that operation can themselves be
combined using the same operation.(1) Closure is the key to power in
any means of combination because it permits us to create "hierarchical"
structures--structures made up of parts, which themselves are made up
of parts, and so on.

   From the outset of *Note Chapter 1::, we've made essential use of
closure in dealing with procedures, because all but the very simplest
programs rely on the fact that the elements of a combination can
themselves be combinations.  In this section, we take up the
consequences of closure for compound data.  We describe some
conventional techniques for using pairs to represent sequences and
trees, and we exhibit a graphics language that illustrates closure in a
vivid way.(2)

* Menu:

* 2-2-1::            Representing Sequences
* 2-2-2::            Hierarchical Structures
* 2-2-3::            Sequences as Conventional Interfaces
* 2-2-4::            Example: A Picture Language

   ---------- Footnotes ----------

   (1) The use of the word "closure" here comes from abstract algebra,
where a set of elements is said to be closed under an operation if
applying the operation to elements in the set produces an element that
is again an element of the set.  The Lisp community also
(unfortunately) uses the word "closure" to describe a totally unrelated
concept: A closure is an implementation technique for representing
procedures with free variables.  We do not use the word "closure" in
this second sense in this book.

   (2) The notion that a means of combination should satisfy closure is
a straightforward idea.  Unfortunately, the data combiners provided in
many popular programming languages do not satisfy closure, or make
closure cumbersome to exploit.  In Fortran or Basic, one typically
combines data elements by assembling them into arrays--but one cannot
form arrays whose elements are themselves arrays.  Pascal and C admit
structures whose elements are structures.  However, this requires that
the programmer manipulate pointers explicitly, and adhere to the
restriction that each field of a structure can contain only elements of
a prespecified form.  Unlike Lisp with its pairs, these languages have
no built-in general-purpose glue that makes it easy to manipulate
compound data in a uniform way.  This limitation lies behind Alan
Perlis's comment in his foreword to this book: "In Pascal the plethora
of declarable data structures induces a specialization within functions
that inhibits and penalizes casual cooperation.  It is better to have
100 functions operate on one data structure than to have 10 functions
operate on 10 data structures."


File: sicp.info,  Node: 2-2-1,  Next: 2-2-2,  Prev: 2-2,  Up: 2-2

2.2.1 Representing Sequences
----------------------------

     *Figure 2.4:* The sequence 1, 2, 3, 4 represented as a chain of
     pairs.

               +---+---+     +---+---+     +---+---+     +---+---+
          ---->| * | *-+---->| * | *-+---->| * | *-+---->| * | / |
               +-|-+---+     +-|-+---+     +-|-+---+     +-|-+---+
                 |             |             |             |
                 V             V             V             V
               +---+         +---+         +---+         +---+
               | 1 |         | 2 |         | 3 |         | 4 |
               +---+         +---+         +---+         +---+

One of the useful structures we can build with pairs is a "sequence"--an
ordered collection of data objects.  There are, of course, many ways to
represent sequences in terms of pairs.  One particularly
straightforward representation is illustrated in *Note Figure 2-4::,
where the sequence 1, 2, 3, 4 is represented as a chain of pairs.  The
`car' of each pair is the corresponding item in the chain, and the
`cdr' of the pair is the next pair in the chain.  The `cdr' of the
final pair signals the end of the sequence by pointing to a
distinguished value that is not a pair, represented in box-and-pointer
diagrams as a diagonal line and in programs as the value of the
variable `nil'.  The entire sequence is constructed by nested `cons'
operations:

     (cons 1
           (cons 2
                 (cons 3
                       (cons 4 nil))))

   Such a sequence of pairs, formed by nested `cons'es, is called a "list",
and Scheme provides a primitive called `list' to help in constructing
lists.(1)  The above sequence could be produced by `(list 1 2 3 4)'.
In general,

     (list <A_1> <A_2> ... <A_N>)

is equivalent to

     (cons <A_1>
           (cons <A_2>
                 (cons ...
                       (cons <A_N>
                             nil)
     ...)))

   Lisp systems conventionally print lists by printing the sequence of
elements, enclosed in parentheses.  Thus, the data object in *Note
Figure 2-4:: is printed as `(1 2 3 4)':

     (define one-through-four (list 1 2 3 4))

     one-through-four
     (1 2 3 4)

   Be careful not to confuse the expression `(list 1 2 3 4)' with the
list `(1 2 3 4)', which is the result obtained when the expression is
evaluated.  Attempting to evaluate the expression `(1 2 3 4)' will
signal an error when the interpreter tries to apply the procedure `1' to
arguments `2', `3', and `4'.

   We can think of `car' as selecting the first item in the list, and of
`cdr' as selecting the sublist consisting of all but the first item.
Nested applications of `car' and `cdr' can be used to extract the
second, third, and subsequent items in the list.(2) The constructor
`cons' makes a list like the original one, but with an additional item
at the beginning.

     (car one-through-four)
     1

     (cdr one-through-four)
     (2 3 4)

     (car (cdr one-through-four))
     2

     (cons 10 one-through-four)
     (10 1 2 3 4)

     (cons 5 one-through-four)
     (5 1 2 3 4)

   The value of `nil', used to terminate the chain of pairs, can be
thought of as a sequence of no elements, the "empty list".  The word "nil"
is a contraction of the Latin word _nihil_, which means "nothing."(3)

List operations
...............

The use of pairs to represent sequences of elements as lists is
accompanied by conventional programming techniques for manipulating
lists by successively "`cdr'ing down" the lists.  For example, the
procedure `list-ref' takes as arguments a list and a number n and
returns the nth item of the list.  It is customary to number the
elements of the list beginning with 0.  The method for computing
`list-ref' is the following:

   * For n = 0, `list-ref' should return the `car' of the list.

   * Otherwise, `list-ref' should return  the (n - 1)st item of the
     `cdr' of the list.


     (define (list-ref items n)
       (if (= n 0)
           (car items)
           (list-ref (cdr items) (- n 1))))

     (define squares (list 1 4 9 16 25))

     (list-ref squares 3)
     16

   Often we `cdr' down the whole list.  To aid in this, Scheme includes
a primitive predicate `null?', which tests whether its argument is the
empty list.  The procedure `length', which returns the number of items
in a list, illustrates this typical pattern of use:

     (define (length items)
       (if (null? items)
           0
           (+ 1 (length (cdr items)))))

     (define odds (list 1 3 5 7))

     (length odds)
     4

   The `length' procedure implements a simple recursive plan. The
reduction step is:

   * The `length' of any list is 1 plus the `length' of the `cdr' of
     the list.


   This is applied successively until we reach the base case:

   * The `length' of the empty list is 0.


   We could also compute `length' in an iterative style:

     (define (length items)
       (define (length-iter a count)
         (if (null? a)
             count
             (length-iter (cdr a) (+ 1 count))))
       (length-iter items 0))

   Another conventional programming technique is to "`cons' up" an
answer list while `cdr'ing down a list, as in the procedure `append',
which takes two lists as arguments and combines their elements to make
a new list:

     (append squares odds)
     (1 4 9 16 25 1 3 5 7)

     (append odds squares)
     (1 3 5 7 1 4 9 16 25)

   `Append' is also implemented using a recursive plan.  To `append'
lists `list1' and `list2', do the following:

   * If `list1' is the empty list, then the result is just `list2'.

   * Otherwise, `append' the `cdr' of `list1' and `list2', and `cons'
     the `car' of `list1' onto the result:


     (define (append list1 list2)
       (if (null? list1)
           list2
           (cons (car list1) (append (cdr list1) list2))))

     *Exercise 2.17:* Define a procedure `last-pair' that returns the
     list that contains only the last element of a given (nonempty)
     list:

          (last-pair (list 23 72 149 34))
          (34)

     *Exercise 2.18:* Define a procedure `reverse' that takes a list as
     argument and returns a list of the same elements in reverse order:

          (reverse (list 1 4 9 16 25))
          (25 16 9 4 1)

     *Exercise 2.19:* Consider the change-counting program of section
     *Note 1-2-2::.  It would be nice to be able to easily change the
     currency used by the program, so that we could compute the number
     of ways to change a British pound, for example.  As the program is
     written, the knowledge of the currency is distributed partly into
     the procedure `first-denomination' and partly into the procedure
     `count-change' (which knows that there are five kinds of U.S.
     coins).  It would be nicer to be able to supply a list of coins to
     be used for making change.

     We want to rewrite the procedure `cc' so that its second argument
     is a list of the values of the coins to use rather than an integer
     specifying which coins to use.  We could then have lists that
     defined each kind of currency:

          (define us-coins (list 50 25 10 5 1))

          (define uk-coins (list 100 50 20 10 5 2 1 0.5))

     We could then call `cc' as follows:

          (cc 100 us-coins)
          292

     To do this will require changing the program `cc' somewhat.  It
     will still have the same form, but it will access its second
     argument differently, as follows:

          (define (cc amount coin-values)
            (cond ((= amount 0) 1)
                  ((or (< amount 0) (no-more? coin-values)) 0)
                  (else
                   (+ (cc amount
                          (except-first-denomination coin-values))
                      (cc (- amount
                             (first-denomination coin-values))
                          coin-values)))))

     Define the procedures `first-denomination',
     `except-first-denomination', and `no-more?' in terms of primitive
     operations on list structures.  Does the order of the list
     `coin-values' affect the answer produced by `cc'?  Why or why not?

     *Exercise 2.20:* The procedures `+', `*', and `list' take
     arbitrary numbers of arguments. One way to define such procedures
     is to use `define' with notation "dotted-tail notation".  In a
     procedure definition, a parameter list that has a dot before the
     last parameter name indicates that, when the procedure is called,
     the initial parameters (if any) will have as values the initial
     arguments, as usual, but the final parameter's value will be a "list"
     of any remaining arguments.  For instance, given the definition

          (define (f x y . z) <BODY>)

     the procedure `f' can be called with two or more arguments.  If we
     evaluate

          (f 1 2 3 4 5 6)

     then in the body of `f', `x' will be 1, `y' will be 2, and `z'
     will be the list `(3 4 5 6)'.  Given the definition

          (define (g . w) <BODY>)

     the procedure `g' can be called with zero or more arguments.  If we
     evaluate

          (g 1 2 3 4 5 6)

     then in the body of `g', `w' will be the list `(1 2 3 4 5 6)'.(4)

     Use this notation to write a procedure `same-parity' that takes
     one or more integers and returns a list of all the arguments that
     have the same even-odd parity as the first argument.  For example,

          (same-parity 1 2 3 4 5 6 7)
          (1 3 5 7)

          (same-parity 2 3 4 5 6 7)
          (2 4 6)

Mapping over lists
..................

One extremely useful operation is to apply some transformation to each
element in a list and generate the list of results.  For instance, the
following procedure scales each number in a list by a given factor:

     (define (scale-list items factor)
       (if (null? items)
           nil
           (cons (* (car items) factor)
                 (scale-list (cdr items) factor))))

     (scale-list (list 1 2 3 4 5) 10)
     (10 20 30 40 50)

   We can abstract this general idea and capture it as a common pattern
expressed as a higher-order procedure, just as in section *Note 1-3::.
The higher-order procedure here is called `map'.  `Map' takes as
arguments a procedure of one argument and a list, and returns a list of
the results produced by applying the procedure to each element in the
list:(5)

     (define (map proc items)
       (if (null? items)
           nil
           (cons (proc (car items))
                 (map proc (cdr items)))))

     (map abs (list -10 2.5 -11.6 17))
     (10 2.5 11.6 17)

     (map (lambda (x) (* x x))
          (list 1 2 3 4))
     (1 4 9 16)

   Now we can give a new definition of `scale-list' in terms of `map':

     (define (scale-list items factor)
       (map (lambda (x) (* x factor))
            items))

   `Map' is an important construct, not only because it captures a
common pattern, but because it establishes a higher level of
abstraction in dealing with lists.  In the original definition of
`scale-list', the recursive structure of the program draws attention to
the element-by-element processing of the list.  Defining `scale-list'
in terms of `map' suppresses that level of detail and emphasizes that
scaling transforms a list of elements to a list of results.  The
difference between the two definitions is not that the computer is
performing a different process (it isn't) but that we think about the
process differently.  In effect, `map' helps establish an abstraction
barrier that isolates the implementation of procedures that transform
lists from the details of how the elements of the list are extracted
and combined.  Like the barriers shown in *Note Figure 2-1::, this
abstraction gives us the flexibility to change the low-level details of
how sequences are implemented, while preserving the conceptual
framework of operations that transform sequences to sequences.  Section
*Note 2-2-3:: expands on this use of sequences as a framework for
organizing programs.

     *Exercise 2.21:* The procedure `square-list' takes a list of
     numbers as argument and returns a list of the squares of those
     numbers.

          (square-list (list 1 2 3 4))
          (1 4 9 16)

     Here are two different definitions of `square-list'.  Complete
     both of them by filling in the missing expressions:

          (define (square-list items)
            (if (null? items)
                nil
                (cons <??> <??>)))

          (define (square-list items)
            (map <??> <??>))

     *Exercise 2.22:* Louis Reasoner tries to rewrite the first
     `square-list' procedure of *Note Exercise 2-21:: so that it
     evolves an iterative process:

          (define (square-list items)
            (define (iter things answer)
              (if (null? things)
                  answer
                  (iter (cdr things)
                        (cons (square (car things))
                              answer))))
            (iter items nil))

     Unfortunately, defining `square-list' this way produces the answer
     list in the reverse order of the one desired.  Why?

     Louis then tries to fix his bug by interchanging the arguments to
     `cons':

          (define (square-list items)
            (define (iter things answer)
              (if (null? things)
                  answer
                  (iter (cdr things)
                        (cons answer
                              (square (car things))))))
            (iter items nil))

     This doesn't work either.  Explain.

     *Exercise 2.23:* The procedure `for-each' is similar to `map'.  It
     takes as arguments a procedure and a list of elements.  However,
     rather than forming a list of the results, `for-each' just applies
     the procedure to each of the elements in turn, from left to right.
     The values returned by applying the procedure to the elements are
     not used at all--`for-each' is used with procedures that perform
     an action, such as printing.  For example,

          (for-each (lambda (x) (newline) (display x))
                    (list 57 321 88))
          57
          321
          88

     The value returned by the call to `for-each' (not illustrated
     above) can be something arbitrary, such as true.  Give an
     implementation of `for-each'.

   ---------- Footnotes ----------

   (1) In this book, we use "list" to mean a chain of pairs terminated
by the end-of-list marker.  In contrast, the term "list structure"
refers to any data structure made out of pairs, not just to lists.

   (2) Since nested applications of `car' and `cdr' are cumbersome to
write, Lisp dialects provide abbreviations for them--for instance,

     (cadr (ARG)) = (car (cdr (ARG)))

   The names of all such procedures start with `c' and end with `r'.
Each `a' between them stands for a `car' operation and each `d' for a
`cdr' operation, to be applied in the same order in which they appear
in the name.  The names `car' and `cdr' persist because simple
combinations like `cadr' are pronounceable.

   (3) It's remarkable how much energy in the standardization of Lisp
dialects has been dissipated in arguments that are literally over
nothing: Should `nil' be an ordinary name?  Should the value of `nil'
be a symbol?  Should it be a list?  Should it be a pair?  In Scheme,
`nil' is an ordinary name, which we use in this section as a variable
whose value is the end-of-list marker (just as `true' is an ordinary
variable that has a true value).  Other dialects of Lisp, including
Common Lisp, treat `nil' as a special symbol.  The authors of this
book, who have endured too many language standardization brawls, would
like to avoid the entire issue.  Once we have introduced quotation in
section *Note 2-3::, we will denote the empty list as `'()' and
dispense with the variable `nil' entirely.

   (4) To define `f' and `g' using `lambda' we would write

     (define f (lambda (x y . z) <BODY>))
     (define g (lambda w <BODY>))

   (5) [Footnote 12] Scheme standardly provides a `map' procedure that
is more general than the one described here.  This more general `map'
takes a procedure of n arguments, together with n lists, and applies
the procedure to all the first elements of the lists, all the second
elements of the lists, and so on, returning a list of the results.  For
example:

     (map + (list 1 2 3) (list 40 50 60) (list 700 800 900))
     (741 852 963)

     (map (lambda (x y) (+ x (* 2 y)))
          (list 1 2 3)
          (list 4 5 6))
     (9 12 15)


File: sicp.info,  Node: 2-2-2,  Next: 2-2-3,  Prev: 2-2-1,  Up: 2-2

2.2.2 Hierarchical Structures
-----------------------------

The representation of sequences in terms of lists generalizes naturally
to represent sequences whose elements may themselves be sequences.  For
example, we can regard the object `((1 2) 3 4)' constructed by

     (cons (list 1 2) (list 3 4))

as a list of three items, the first of which is itself a list, `(1 2)'.
Indeed, this is suggested by the form in which the result is printed by
the interpreter.  *Note Figure 2-5:: shows the representation of this
structure in terms of pairs.

     *Figure 2.5:* Structure formed by `(cons (list 1 2) (list 3 4))'.

                                                    (3 4)
                                                      |
                                                      V
          ((1 2) 3 4)  +---+---+                  +---+---+     +---+---+
                  ---->| * | *-+----------------->| * | *-+---->| * | / |
                       +-|-+---+                  +-|-+---+     +-|-+---+
                         |                          |             |
                         V                          V             V
                (1 2)  +---+---+     +---+---+    +---+         +---+
                  ---->| * | *-+---->| * | / |    | 3 |         | 4 |
                       +-|-+---+     +-|-+---+    +---+         +---+
                         |             |
                         V             V
                       +---+         +---+
                       | 1 |         | 2 |
                       +---+         +---+

   Another way to think of sequences whose elements are sequences is as "trees".
The elements of the sequence are the branches of the tree, and
elements that are themselves sequences are subtrees.  *Note Figure 2-6::
shows the structure in *Note Figure 2-5:: viewed as a tree.

     *Figure 2.6:* The list structure in *Note Figure 2-5:: viewed as a
     tree.

           ((1 2) 3 4)
               /\\
              /  | \
          (1 2)  3 4
           / \
           1 2

   Recursion is a natural tool for dealing with tree structures, since
we can often reduce operations on trees to operations on their
branches, which reduce in turn to operations on the branches of the
branches, and so on, until we reach the leaves of the tree.  As an
example, compare the `length' procedure of section *Note 2-2-1:: with
the `count-leaves' procedure, which returns the total number of leaves
of a tree:

     (define x (cons (list 1 2) (list 3 4)))

     (length x)
     3

     (count-leaves x)
     4

     (list x x)
     (((1 2) 3 4) ((1 2) 3 4))

     (length (list x x))
     2

     (count-leaves (list x x))
     8

   To implement `count-leaves', recall the recursive plan for computing
`length':

   * `Length' of a list `x' is 1 plus `length' of the `cdr' of `x'.

   * `Length' of the empty list is 0.


   `Count-leaves' is similar.  The value for the empty list is the same:

   * `Count-leaves' of the empty list is 0.


   But in the reduction step, where we strip off the `car' of the list,
we must take into account that the `car' may itself be a tree whose
leaves we need to count.  Thus, the appropriate reduction step is

   * `Count-leaves' of a tree `x' is `count-leaves' of the `car' of `x'
     plus `count-leaves' of the `cdr' of `x'.


   Finally, by taking `car's we reach actual leaves, so we need another
base case:

   * `Count-leaves' of a leaf is 1.


   To aid in writing recursive procedures on trees, Scheme provides the
primitive predicate `pair?', which tests whether its argument is a
pair.  Here is the complete procedure:(1)

     (define (count-leaves x)
       (cond ((null? x) 0)
             ((not (pair? x)) 1)
             (else (+ (count-leaves (car x))
                      (count-leaves (cdr x))))))

     *Exercise 2.24:* Suppose we evaluate the expression `(list 1 (list
     2 (list 3 4)))'.  Give the result printed by the interpreter, the
     corresponding box-and-pointer structure, and the interpretation of
     this as a tree (as in *Note Figure 2-6::).

     *Exercise 2.25:* Give combinations of `car's and `cdr's that will
     pick 7 from each of the following lists:

          (1 3 (5 7) 9)

          ((7))

          (1 (2 (3 (4 (5 (6 7))))))

     *Exercise 2.26:* Suppose we define `x' and `y' to be two lists:

          (define x (list 1 2 3))

          (define y (list 4 5 6))

     What result is printed by the interpreter in response to
     evaluating each of the following expressions:

          (append x y)

          (cons x y)

          (list x y)

     *Exercise 2.27:* Modify your `reverse' procedure of *Note Exercise
     2-18:: to produce a `deep-reverse' procedure that takes a list as
     argument and returns as its value the list with its elements
     reversed and with all sublists deep-reversed as well.  For example,

          (define x (list (list 1 2) (list 3 4)))

          x
          ((1 2) (3 4))

          (reverse x)
          ((3 4) (1 2))

          (deep-reverse x)
          ((4 3) (2 1))

     *Exercise 2.28:* Write a procedure `fringe' that takes as argument
     a tree (represented as a list) and returns a list whose elements
     are all the leaves of the tree arranged in left-to-right order.
     For example,

          (define x (list (list 1 2) (list 3 4)))

          (fringe x)
          (1 2 3 4)

          (fringe (list x x))
          (1 2 3 4 1 2 3 4)

     *Exercise 2.29:* A binary mobile consists of two branches, a left
     branch and a right branch.  Each branch is a rod of a certain
     length, from which hangs either a weight or another binary mobile.
     We can represent a binary mobile using compound data by
     constructing it from two branches (for example, using `list'):

          (define (make-mobile left right)
            (list left right))

     A branch is constructed from a `length' (which must be a number)
     together with a `structure', which may be either a number
     (representing a simple weight) or another mobile:

          (define (make-branch length structure)
            (list length structure))

       a. Write the corresponding selectors `left-branch' and
          `right-branch', which return the branches of a mobile, and
          `branch-length' and `branch-structure', which return the
          components of a branch.

       b. Using your selectors, define a procedure `total-weight' that
          returns the total weight of a mobile.

       c. A mobile is said to be "balanced" if the torque applied by
          its top-left branch is equal to that applied by its top-right
          branch (that is, if the length of the left rod multiplied by
          the weight hanging from that rod is equal to the
          corresponding product for the right side) and if each of the
          submobiles hanging off its branches is balanced. Design a
          predicate that tests whether a binary mobile is balanced.

       d. Suppose we change the representation of mobiles so that the
          constructors are

               (define (make-mobile left right)
                 (cons left right))

               (define (make-branch length structure)
                 (cons length structure))

          How much do you need to change your programs to convert to
          the new representation?


Mapping over trees
..................

Just as `map' is a powerful abstraction for dealing with sequences,
`map' together with recursion is a powerful abstraction for dealing with
trees.  For instance, the `scale-tree' procedure, analogous to
`scale-list' of section *Note 2-2-1::, takes as arguments a numeric
factor and a tree whose leaves are numbers.  It returns a tree of the
same shape, where each number is multiplied by the factor.  The
recursive plan for `scale-tree' is similar to the one for
`count-leaves':

     (define (scale-tree tree factor)
       (cond ((null? tree) nil)
             ((not (pair? tree)) (* tree factor))
             (else (cons (scale-tree (car tree) factor)
                         (scale-tree (cdr tree) factor)))))

     (scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7))
                 10)
     (10 (20 (30 40) 50) (60 70))

   Another way to implement `scale-tree' is to regard the tree as a
sequence of sub-trees and use `map'.  We map over the sequence, scaling
each sub-tree in turn, and return the list of results.  In the base
case, where the tree is a leaf, we simply multiply by the factor:

     (define (scale-tree tree factor)
       (map (lambda (sub-tree)
              (if (pair? sub-tree)
                  (scale-tree sub-tree factor)
                  (* sub-tree factor)))
            tree))

   Many tree operations can be implemented by similar combinations of
sequence operations and recursion.

     *Exercise 2.30:* Define a procedure `square-tree' analogous to the
     `square-list' procedure of *Note Exercise 2-21::.  That is,
     `square-list' should behave as follows:

          (square-tree
           (list 1
                 (list 2 (list 3 4) 5)
                 (list 6 7)))
          (1 (4 (9 16) 25) (36 49))

     Define `square-tree' both directly (i.e., without using any
     higher-order procedures) and also by using `map' and recursion.

     *Exercise 2.31:* Abstract your answer to *Note Exercise 2-30:: to
     produce a procedure `tree-map' with the property that
     `square-tree' could be defined as

          (define (square-tree tree) (tree-map square tree))

     *Exercise 2.32:* We can represent a set as a list of distinct
     elements, and we can represent the set of all subsets of the set as
     a list of lists.  For example, if the set is `(1 2 3)', then the
     set of all subsets is `(() (3) (2) (2 3) (1) (1 3) (1 2) (1 2
     3))'.  Complete the following definition of a procedure that
     generates the set of subsets of a set and give a clear explanation
     of why it works:

          (define (subsets s)
            (if (null? s)
                (list nil)
                (let ((rest (subsets (cdr s))))
                  (append rest (map <??> rest)))))

   ---------- Footnotes ----------

   (1) The order of the first two clauses in the `cond' matters, since
the empty list satisfies `null?' and also is not a pair.


File: sicp.info,  Node: 2-2-3,  Next: 2-2-4,  Prev: 2-2-2,  Up: 2-2

2.2.3 Sequences as Conventional Interfaces
------------------------------------------

In working with compound data, we've stressed how data abstraction
permits us to design programs without becoming enmeshed in the details
of data representations, and how abstraction preserves for us the
flexibility to experiment with alternative representations.  In this
section, we introduce another powerful design principle for working
with data structures--the use of "conventional interfaces".

   In section *Note 1-3:: we saw how program abstractions, implemented
as higher-order procedures, can capture common patterns in programs
that deal with numerical data.  Our ability to formulate analogous
operations for working with compound data depends crucially on the
style in which we manipulate our data structures.  Consider, for
example, the following procedure, analogous to the `count-leaves'
procedure of section *Note 2-2-2::, which takes a tree as argument and
computes the sum of the squares of the leaves that are odd:

     (define (sum-odd-squares tree)
       (cond ((null? tree) 0)
             ((not (pair? tree))
              (if (odd? tree) (square tree) 0))
             (else (+ (sum-odd-squares (car tree))
                      (sum-odd-squares (cdr tree))))))

   On the surface, this procedure is very different from the following
one, which constructs a list of all the even Fibonacci numbers
_Fib_(k), where k is less than or equal to a given integer n:

     (define (even-fibs n)
       (define (next k)
         (if (> k n)
             nil
             (let ((f (fib k)))
               (if (even? f)
                   (cons f (next (+ k 1)))
                   (next (+ k 1))))))
       (next 0))

   Despite the fact that these two procedures are structurally very
different, a more abstract description of the two computations reveals
a great deal of similarity.  The first program

   * enumerates the leaves of a tree;

   * filters them, selecting the odd ones;

   * squares each of the selected ones; and

   * accumulates the results using `+', starting with 0.


   The second program

   * enumerates the integers from 0 to n;

   * computes the Fibonacci number for each integer;

   * filters them, selecting the even ones; and

   * accumulates the results using `cons',  starting with the empty
     list.


   A signal-processing engineer would find it natural to conceptualize
these processes in terms of signals flowing through a cascade of
stages, each of which implements part of the program plan, as shown in
*Note Figure 2-7::.  In `sum-odd-squares', we begin with an "enumerator",
which generates a "signal" consisting of the leaves of a given tree.
This signal is passed through a "filter", which eliminates all but the
odd elements.  The resulting signal is in turn passed through a "map",
which is a "transducer" that applies the `square' procedure to each
element.  The output of the map is then fed to an "accumulator", which
combines the elements using `+', starting from an initial 0.  The plan
for `even-fibs' is analogous.

     *Figure 2.7:* The signal-flow plans for the procedures
     `sum-odd-squares' (top) and `even-fibs' (bottom) reveal the
     commonality between the two programs.

          +-------------+   +-------------+   +-------------+   +-------------+
          | enumerate:  |-->| filter:     |-->| map:        |-->| accumulate: |
          | tree leaves |   | odd?        |   | square      |   | +, 0        |
          +-------------+   +-------------+   +-------------+   +-------------+

          +-------------+   +-------------+   +-------------+   +-------------+
          | enumerate:  |-->| map:        |-->| filter:     |-->| accumulate: |
          | integers    |   | fib         |   | even?       |   | cons, ()    |
          +-------------+   +-------------+   +-------------+   +-------------+

   Unfortunately, the two procedure definitions above fail to exhibit
this signal-flow structure.  For instance, if we examine the
`sum-odd-squares' procedure, we find that the enumeration is
implemented partly by the `null?' and `pair?' tests and partly by the
tree-recursive structure of the procedure.  Similarly, the accumulation
is found partly in the tests and partly in the addition used in the
recursion.  In general, there are no distinct parts of either procedure
that correspond to the elements in the signal-flow description.  Our
two procedures decompose the computations in a different way, spreading
the enumeration over the program and mingling it with the map, the
filter, and the accumulation.  If we could organize our programs to
make the signal-flow structure manifest in the procedures we write, this
would increase the conceptual clarity of the resulting code.

Sequence Operations
...................

The key to organizing programs so as to more clearly reflect the
signal-flow structure is to concentrate on the "signals" that flow from
one stage in the process to the next.  If we represent these signals as
lists, then we can use list operations to implement the processing at
each of the stages.  For instance, we can implement the mapping stages
of the signal-flow diagrams using the `map' procedure from section
*Note 2-2-1:::

     (map square (list 1 2 3 4 5))
     (1 4 9 16 25)

   Filtering a sequence to select only those elements that satisfy a
given predicate is accomplished by

     (define (filter predicate sequence)
       (cond ((null? sequence) nil)
             ((predicate (car sequence))
              (cons (car sequence)
                    (filter predicate (cdr sequence))))
             (else (filter predicate (cdr sequence)))))

   For example,

     (filter odd? (list 1 2 3 4 5))
     (1 3 5)

   Accumulations can be implemented by

     (define (accumulate op initial sequence)
       (if (null? sequence)
           initial
           (op (car sequence)
               (accumulate op initial (cdr sequence)))))

     (accumulate + 0 (list 1 2 3 4 5))
     15

     (accumulate * 1 (list 1 2 3 4 5))
     120

     (accumulate cons nil (list 1 2 3 4 5))
     (1 2 3 4 5)

   All that remains to implement signal-flow diagrams is to enumerate
the sequence of elements to be processed.  For `even-fibs', we need to
generate the sequence of integers in a given range, which we can do as
follows:

     (define (enumerate-interval low high)
       (if (> low high)
           nil
           (cons low (enumerate-interval (+ low 1) high))))

     (enumerate-interval 2 7)
     (2 3 4 5 6 7)

   To enumerate the leaves of a tree, we can use(1)

     (define (enumerate-tree tree)
       (cond ((null? tree) nil)
             ((not (pair? tree)) (list tree))
             (else (append (enumerate-tree (car tree))
                           (enumerate-tree (cdr tree))))))

     (enumerate-tree (list 1 (list 2 (list 3 4)) 5))
     (1 2 3 4 5)

   Now we can reformulate `sum-odd-squares' and `even-fibs' as in the
signal-flow diagrams.  For `sum-odd-squares', we enumerate the sequence
of leaves of the tree, filter this to keep only the odd numbers in the
sequence, square each element, and sum the results:

     (define (sum-odd-squares tree)
       (accumulate +
                   0
                   (map square
                        (filter odd?
                                (enumerate-tree tree)))))

   For `even-fibs', we enumerate the integers from 0 to n, generate the
Fibonacci number for each of these integers, filter the resulting
sequence to keep only the even elements, and accumulate the results
into a list:

     (define (even-fibs n)
       (accumulate cons
                   nil
                   (filter even?
                           (map fib
                                (enumerate-interval 0 n)))))

   The value of expressing programs as sequence operations is that this
helps us make program designs that are modular, that is, designs that
are constructed by combining relatively independent pieces.  We can
encourage modular design by providing a library of standard components
together with a conventional interface for connecting the components in
flexible ways.

   Modular construction is a powerful strategy for controlling
complexity in engineering design.  In real signal-processing
applications, for example, designers regularly build systems by
cascading elements selected from standardized families of filters and
transducers.  Similarly, sequence operations provide a library of
standard program elements that we can mix and match.  For instance, we
can reuse pieces from the `sum-odd-squares' and `even-fibs' procedures
in a program that constructs a list of the squares of the first n + 1
Fibonacci numbers:

     (define (list-fib-squares n)
       (accumulate cons
                   nil
                   (map square
                        (map fib
                             (enumerate-interval 0 n)))))

     (list-fib-squares 10)
     (0 1 1 4 9 25 64 169 441 1156 3025)

   We can rearrange the pieces and use them in computing the product of
the odd integers in a sequence:

     (define (product-of-squares-of-odd-elements sequence)
       (accumulate *
                   1
                   (map square
                        (filter odd? sequence))))

     (product-of-squares-of-odd-elements (list 1 2 3 4 5))
     225

   We can also formulate conventional data-processing applications in
terms of sequence operations.  Suppose we have a sequence of personnel
records and we want to find the salary of the highest-paid programmer.
Assume that we have a selector `salary' that returns the salary of a
record, and a predicate `programmer?' that tests if a record is for a
programmer.  Then we can write

     (define (salary-of-highest-paid-programmer records)
       (accumulate max
                   0
                   (map salary
                        (filter programmer? records))))

   These examples give just a hint of the vast range of operations that
can be expressed as sequence operations.(2)

   Sequences, implemented here as lists, serve as a conventional
interface that permits us to combine processing modules.  Additionally,
when we uniformly represent structures as sequences, we have localized
the data-structure dependencies in our programs to a small number of
sequence operations.  By changing these, we can experiment with
alternative representations of sequences, while leaving the overall
design of our programs intact.  We will exploit this capability in
section *Note 3-5::, when we generalize the sequence-processing
paradigm to admit infinite sequences.

     *Exercise 2.33:* Fill in the missing expressions to complete the
     following definitions of some basic list-manipulation operations
     as accumulations:

          (define (map p sequence)
            (accumulate (lambda (x y) <??>) nil sequence))

          (define (append seq1 seq2)
            (accumulate cons <??> <??>))

          (define (length sequence)
            (accumulate <??> 0 sequence))

     *Exercise 2.34:* Evaluating a polynomial in x at a given value of
     x can be formulated as an accumulation.  We evaluate the polynomial

          a_n r^n | a_(n-1) r^(n-1) + ... + a_1 r + a_0

     using a well-known algorithm called "Horner's rule", which
     structures the computation as

          (... (a_n r + a_(n-1)) r + ... + a_1) r + a_0

     In other words, we start with a_n, multiply by x, add a_(n-1),
     multiply by x, and so on, until we reach a_0.(3)

     Fill in the following template to produce a procedure that
     evaluates a polynomial using Horner's rule.  Assume that the
     coefficients of the polynomial are arranged in a sequence, from
     a_0 through a_n.

          (define (horner-eval x coefficient-sequence)
            (accumulate (lambda (this-coeff higher-terms) <??>)
                        0
                        coefficient-sequence))

     For example, to compute 1 + 3x + 5x^3 + x^(5) at x = 2 you would
     evaluate

          (horner-eval 2 (list 1 3 0 5 0 1))

     *Exercise 2.35:* Redefine `count-leaves' from section *Note
     2-2-2:: as an accumulation:

          (define (count-leaves t)
            (accumulate <??> <??> (map <??> <??>)))

     *Exercise 2.36:* The procedure `accumulate-n' is similar to
     `accumulate' except that it takes as its third argument a sequence
     of sequences, which are all assumed to have the same number of
     elements.  It applies the designated accumulation procedure to
     combine all the first elements of the sequences, all the second
     elements of the sequences, and so on, and returns a sequence of
     the results.  For instance, if `s' is a sequence containing four
     sequences, `((1 2 3) (4 5 6) (7 8 9) (10 11 12)),' then the value
     of `(accumulate-n + 0 s)' should be the sequence `(22 26 30)'.
     Fill in the missing expressions in the following definition of
     `accumulate-n':

          (define (accumulate-n op init seqs)
            (if (null? (car seqs))
                nil
                (cons (accumulate op init <??>)
                      (accumulate-n op init <??>))))

Exercise 2.37
.............

Suppose we represent vectors v = (v_i) as sequences of numbers, and
matrices m = (m_(ij)) as sequences of vectors (the rows of the matrix).
For example, the matrix

     +-         -+
     |  1 2 3 4  |
     |  4 5 6 6  |
     |  6 7 8 9  |
     +-         -+

is represented as the sequence `((1 2 3 4) (4 5 6 6) (6 7 8 9))'.  With
this representation, we can use sequence operations to concisely
express the basic matrix and vector operations.  These operations
(which are described in any book on matrix algebra) are the following:

                                            __
     (dot-product v w)      returns the sum >_i v_i w_i

     (matrix-*-vector m v)  returns the vector t,
                                        __
                            where t_i = >_j m_(ij) v_j

     (matrix-*-matrix m n)  returns the matrix p,
                                           __
                            where p_(ij) = >_k m_(ik) n_(kj)

     (transpose m)          returns the matrix n,
                            where n_(ij) = m_(ji)

   We can define the dot product as(4)

     (define (dot-product v w)
       (accumulate + 0 (map * v w)))

   Fill in the missing expressions in the following procedures for
computing the other matrix operations.  (The procedure `accumulate-n'
is defined in *Note Exercise 2-36::.)

     (define (matrix-*-vector m v)
       (map <??> m))

     (define (transpose mat)
       (accumulate-n <??> <??> mat))

     (define (matrix-*-matrix m n)
       (let ((cols (transpose n)))
         (map <??> m)))

     *Exercise 2.38:* The `accumulate' procedure is also known as
     `fold-right', because it combines the first element of the
     sequence with the result of combining all the elements to the
     right.  There is also a `fold-left', which is similar to
     `fold-right', except that it combines elements working in the
     opposite direction:

          (define (fold-left op initial sequence)
            (define (iter result rest)
              (if (null? rest)
                  result
                  (iter (op result (car rest))
                        (cdr rest))))
            (iter initial sequence))

     What are the values of

          (fold-right / 1 (list 1 2 3))

          (fold-left / 1 (list 1 2 3))

          (fold-right list nil (list 1 2 3))

          (fold-left list nil (list 1 2 3))

     Give a property that `op' should satisfy to guarantee that
     `fold-right' and `fold-left' will produce the same values for any
     sequence.

     *Exercise 2.39:* Complete the following definitions of `reverse'
     (*Note Exercise 2-18::) in terms of `fold-right' and `fold-left'
     from *Note Exercise 2-38:::

          (define (reverse sequence)
            (fold-right (lambda (x y) <??>) nil sequence))

          (define (reverse sequence)
            (fold-left (lambda (x y) <??>) nil sequence))

Nested Mappings
...............

We can extend the sequence paradigm to include many computations that
are commonly expressed using nested loops.(5) Consider this problem:
Given a positive integer n, find all ordered pairs of distinct positive
integers i and j, where 1 <= j< i<= n, such that i + j is prime.  For
example, if n is 6, then the pairs are the following:

       i   | 2 3 4 4 5 6 6
       j   | 1 2 1 3 2 1 5
     ------+---------------
     i + j | 3 5 5 7 7 7 11

   A natural way to organize this computation is to generate the
sequence of all ordered pairs of positive integers less than or equal
to n, filter to select those pairs whose sum is prime, and then, for
each pair (i, j) that passes through the filter, produce the triple
(i,j,i + j).

   Here is a way to generate the sequence of pairs: For each integer i
<= n, enumerate the integers j<i, and for each such i and j generate
the pair (i,j).  In terms of sequence operations, we map along the
sequence `(enumerate-interval 1 n)'.  For each i in this sequence, we
map along the sequence `(enumerate-interval 1 (- i 1))'.  For each j in
this latter sequence, we generate the pair `(list i j)'.  This gives us
a sequence of pairs for each i.  Combining all the sequences for all
the i (by accumulating with `append') produces the required sequence of
pairs:(6)

     (accumulate append
                 nil
                 (map (lambda (i)
                        (map (lambda (j) (list i j))
                             (enumerate-interval 1 (- i 1))))
                      (enumerate-interval 1 n)))

   The combination of mapping and accumulating with `append' is so
common in this sort of program that we will isolate it as a separate
procedure:

     (define (flatmap proc seq)
       (accumulate append nil (map proc seq)))

   Now filter this sequence of pairs to find those whose sum is prime.
The filter predicate is called for each element of the sequence; its
argument is a pair and it must extract the integers from the pair.
Thus, the predicate to apply to each element in the sequence is

     (define (prime-sum? pair)
       (prime? (+ (car pair) (cadr pair))))

   Finally, generate the sequence of results by mapping over the
filtered pairs using the following procedure, which constructs a triple
consisting of the two elements of the pair along with their sum:

     (define (make-pair-sum pair)
       (list (car pair) (cadr pair) (+ (car pair) (cadr pair))))

   Combining all these steps yields the complete procedure:

     (define (prime-sum-pairs n)
       (map make-pair-sum
            (filter prime-sum?
                    (flatmap
                     (lambda (i)
                       (map (lambda (j) (list i j))
                            (enumerate-interval 1 (- i 1))))
                     (enumerate-interval 1 n)))))

   Nested mappings are also useful for sequences other than those that
enumerate intervals.  Suppose we wish to generate all the permutations
of a set S; that is, all the ways of ordering the items in the set.
For instance, the permutations of {1,2,3} are {1,2,3}, {1,3,2},
{2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}.  Here is a plan for generating
the permutations of S: For each item x in S, recursively generate the
sequence of permutations of S - x,(7) and adjoin x to the front of each
one.  This yields, for each x in S, the sequence of permutations of S
that begin with x.  Combining these sequences for all x gives all the
permutations of S:(8)

     (define (permutations s)
       (if (null? s)                    ; empty set?
           (list nil)                   ; sequence containing empty set
           (flatmap (lambda (x)
                      (map (lambda (p) (cons x p))
                           (permutations (remove x s))))
                    s)))

   Notice how this strategy reduces the problem of generating
permutations of S to the problem of generating the permutations of sets
with fewer elements than S.  In the terminal case, we work our way down
to the empty list, which represents a set of no elements.  For this, we
generate `(list nil)', which is a sequence with one item, namely the
set with no elements.  The `remove' procedure used in `permutations'
returns all the items in a given sequence except for a given item.
This can be expressed as a simple filter:

     (define (remove item sequence)
       (filter (lambda (x) (not (= x item)))
               sequence))

     *Exercise 2.40:* Define a procedure `unique-pairs' that, given an
     integer n, generates the sequence of pairs (i,j) with 1 <= j< i <=
     n.  Use `unique-pairs' to simplify the definition of
     `prime-sum-pairs' given above.

     *Exercise 2.41:* Write a procedure to find all ordered triples of
     distinct positive integers i, j, and k less than or equal to a
     given integer n that sum to a given integer s.

     *Figure 2.8:* A solution to the eight-queens puzzle.

          +---+---+---+---+---+---+---+---+
          |   |   |   |   |   | Q |   |   |
          +---+---+---+---+---+---+---+---+
          |   |   | Q |   |   |   |   |   |
          +---+---+---+---+---+---+---+---+
          | Q |   |   |   |   |   |   |   |
          +---+---+---+---+---+---+---+---+
          |   |   |   |   |   |   | Q |   |
          +---+---+---+---+---+---+---+---+
          |   |   |   |   | Q |   |   |   |
          +---+---+---+---+---+---+---+---+
          |   |   |   |   |   |   |   | Q |
          +---+---+---+---+---+---+---+---+
          |   | Q |   |   |   |   |   |   |
          +---+---+---+---+---+---+---+---+
          |   |   |   | Q |   |   |   |   |
          +---+---+---+---+---+---+---+---+

     *Exercise 2.42:* The "eight-queens puzzle" asks how to place eight
     queens on a chessboard so that no queen is in check from any other
     (i.e., no two queens are in the same row, column, or diagonal).
     One possible solution is shown in *Note Figure 2-8::.  One way to
     solve the puzzle is to work across the board, placing a queen in
     each column.  Once we have placed k - 1 queens, we must place the
     kth queen in a position where it does not check any of the queens
     already on the board.  We can formulate this approach recursively:
     Assume that we have already generated the sequence of all possible
     ways to place k - 1 queens in the first k - 1 columns of the
     board.  For each of these ways, generate an extended set of
     positions by placing a queen in each row of the kth column.  Now
     filter these, keeping only the positions for which the queen in
     the kth column is safe with respect to the other queens.  This
     produces the sequence of all ways to place k queens in the first k
     columns.  By continuing this process, we will produce not only one
     solution, but all solutions to the puzzle.

     We implement this solution as a procedure `queens', which returns a
     sequence of all solutions to the problem of placing n queens on an
     n*n chessboard.  `Queens' has an internal procedure `queen-cols'
     that returns the sequence of all ways to place queens in the first
     k columns of the board.

          (define (queens board-size)
            (define (queen-cols k)
              (if (= k 0)
                  (list empty-board)
                  (filter
                   (lambda (positions) (safe? k positions))
                   (flatmap
                    (lambda (rest-of-queens)
                      (map (lambda (new-row)
                             (adjoin-position new-row k rest-of-queens))
                           (enumerate-interval 1 board-size)))
                    (queen-cols (- k 1))))))
            (queen-cols board-size))

     In this procedure `rest-of-queens' is a way to place k - 1 queens
     in the first k - 1 columns, and `new-row' is a proposed row in
     which to place the queen for the kth column.  Complete the program
     by implementing the representation for sets of board positions,
     including the procedure `adjoin-position', which adjoins a new
     row-column position to a set of positions, and `empty-board',
     which represents an empty set of positions.  You must also write
     the procedure `safe?', which determines for a set of positions,
     whether the queen in the kth column is safe with respect to the
     others.  (Note that we need only check whether the new queen is
     safe--the other queens are already guaranteed safe with respect to
     each other.)

     *Exercise 2.43:* Louis Reasoner is having a terrible time doing
     *Note Exercise 2-42::.  His `queens' procedure seems to work, but
     it runs extremely slowly.  (Louis never does manage to wait long
     enough for it to solve even the 6*6 case.)  When Louis asks Eva Lu
     Ator for help, she points out that he has interchanged the order
     of the nested mappings in the `flatmap', writing it as

          (flatmap
           (lambda (new-row)
             (map (lambda (rest-of-queens)
                    (adjoin-position new-row k rest-of-queens))
                  (queen-cols (- k 1))))
           (enumerate-interval 1 board-size))

     Explain why this interchange makes the program run slowly.
     Estimate how long it will take Louis's program to solve the
     eight-queens puzzle, assuming that the program in *Note Exercise
     2-42:: solves the puzzle in time T.

   ---------- Footnotes ----------

   (1) This is, in fact, precisely the `fringe' procedure from *Note
Exercise 2-28::.  Here we've renamed it to emphasize that it is part of
a family of general sequence-manipulation procedures.

   (2) Richard Waters (1979) developed a program that automatically
analyzes traditional Fortran programs, viewing them in terms of maps,
filters, and accumulations.  He found that fully 90 percent of the code
in the Fortran Scientific Subroutine Package fits neatly into this
paradigm.  One of the reasons for the success of Lisp as a programming
language is that lists provide a standard medium for expressing ordered
collections so that they can be manipulated using higher-order
operations.  The programming language APL owes much of its power and
appeal to a similar choice. In APL all data are represented as arrays,
and there is a universal and convenient set of generic operators for
all sorts of array operations.

   (3) According to Knuth (1981), this rule was formulated by W. G.
Horner early in the nineteenth century, but the method was actually used
by Newton over a hundred years earlier.  Horner's rule evaluates the
polynomial using fewer additions and multiplications than does the
straightforward method of first computing a_n x^n, then adding
a_(n-1)x^(n-1), and so on.  In fact, it is possible to prove that any
algorithm for evaluating arbitrary polynomials must use at least as
many additions and multiplications as does Horner's rule, and thus
Horner's rule is an optimal algorithm for polynomial evaluation.  This
was proved (for the number of additions) by A. M. Ostrowski in a 1954
paper that essentially founded the modern study of optimal algorithms.
The analogous statement for multiplications was proved by V. Y. Pan in
1966.  The book by Borodin and Munro (1975) provides an overview of
these and other results about optimal algorithms.

   (4) This definition uses the extended version of `map' described in
*Note Footnote 12::.

   (5) This approach to nested mappings was shown to us by David
Turner, whose languages KRC and Miranda provide elegant formalisms for
dealing with these constructs.  The examples in this section (see also
*Note Exercise 2-42::) are adapted from Turner 1981.  In section *Note
3-5-3::, we'll see how this approach generalizes to infinite sequences.

   (6) We're representing a pair here as a list of two elements rather
than as a Lisp pair.  Thus, the "pair" (i,j) is represented as `(list i
j)', not `(cons i j)'.

   (7) The set S - x is the set of all elements of S, excluding x.

   (8) Semicolons in Scheme code are used to introduce "comments".
Everything from the semicolon to the end of the line is ignored by the
interpreter.  In this book we don't use many comments; we try to make
our programs self-documenting by using descriptive names.


File: sicp.info,  Node: 2-2-4,  Prev: 2-2-3,  Up: 2-2

2.2.4 Example: A Picture Language
---------------------------------

This section presents a simple language for drawing pictures that
illustrates the power of data abstraction and closure, and also
exploits higher-order procedures in an essential way.  The language is
designed to make it easy to experiment with patterns such as the ones
in *Note Figure 2-9::, which are composed of repeated elements that are
shifted and scaled.(1) In this language, the data objects being
combined are represented as procedures rather than as list structure.
Just as `cons', which satisfies the closure property, allowed us to
easily build arbitrarily complicated list structure, the operations in
this language, which also satisfy the closure property, allow us to
easily build arbitrarily complicated patterns.

     *Figure 2.9:* Designs generated with the picture language.

     [two graphic images not included]

The picture language
....................

When we began our study of programming in section *Note 1-1::, we
emphasized the importance of describing a language by focusing on the
language's primitives, its means of combination, and its means of
abstraction.  We'll follow that framework here.

   Part of the elegance of this picture language is that there is only
one kind of element, called a "painter".  A painter draws an image that
is shifted and scaled to fit within a designated parallelogram-shaped
frame.  For example, there's a primitive painter we'll call `wave' that
makes a crude line drawing, as shown in *Note Figure 2-10::.  The
actual shape of the drawing depends on the frame--all four images in
*Note Figure 2-10:: are produced by the same `wave' painter, but with
respect to four different frames.  Painters can be more elaborate than
this: The primitive painter called `rogers' paints a picture of MIT's
founder, William Barton Rogers, as shown in *Note Figure 2-11::.(2) The
four images in *Note Figure 2-11:: are drawn with respect to the same
four frames as the `wave' images in *Note Figure 2-10::.

   To combine images, we use various operations that construct new
painters from given painters.  For example, the `beside' operation
takes two painters and produces a new, compound painter that draws the
first painter's image in the left half of the frame and the second
painter's image in the right half of the frame.  Similarly, `below'
takes two painters and produces a compound painter that draws the first
painter's image below the second painter's image.  Some operations
transform a single painter to produce a new painter.  For example,
`flip-vert' takes a painter and produces a painter that draws its image
upside-down, and `flip-horiz' produces a painter that draws the
original painter's image left-to-right reversed.

     *Figure 2.10:* Images produced by the `wave' painter, with respect
     to four different frames.  The frames, shown with dotted lines,
     are not part of the images.

     [four graphic images not included]

     *Figure 2.11:* Images of William Barton Rogers, founder and first
     president of MIT, painted with respect to the same four frames as
     in *Note Figure 2-10:: (original image reprinted with the
     permission of the MIT Museum).

     [four graphic images not included]

   *Note Figure 2-12:: shows the drawing of a painter called `wave4'
that is built up in two stages starting from `wave':

     (define wave2 (beside wave (flip-vert wave)))
     (define wave4 (below wave2 wave2))

     *Figure 2.12:* Creating a complex figure, starting from the `wave'
     painter of *Note Figure 2-10::.

     [two graphic images not included]

          (define wave2                         (define wave4
            (beside wave (flip-vert wave)))       (below wave2 wave2))

   In building up a complex image in this manner we are exploiting the
fact that painters are closed under the language's means of
combination.  The `beside' or `below' of two painters is itself a
painter; therefore, we can use it as an element in making more complex
painters.  As with building up list structure using `cons', the closure
of our data under the means of combination is crucial to the ability to
create complex structures while using only a few operations.

   Once we can combine painters, we would like to be able to abstract
typical patterns of combining painters.  We will implement the painter
operations as Scheme procedures.  This means that we don't need a
special abstraction mechanism in the picture language: Since the means
of combination are ordinary Scheme procedures, we automatically have
the capability to do anything with painter operations that we can do
with procedures.  For example, we can abstract the pattern in `wave4' as

     (define (flipped-pairs painter)
       (let ((painter2 (beside painter (flip-vert painter))))
         (below painter2 painter2)))

and define `wave4' as an instance of this pattern:

     (define wave4 (flipped-pairs wave))

   We can also define recursive operations.  Here's one that makes
painters split and branch towards the right as shown in figures *Note
Figure 2-13:: and *Note Figure 2-14:::

     (define (right-split painter n)
       (if (= n 0)
           painter
           (let ((smaller (right-split painter (- n 1))))
             (beside painter (below smaller smaller)))))

     *Figure 2.13:* Recursive plans for `right-split' and
     `corner-split'.

          +-------------+-------------+    +------+------+-------------+
          |             |             |    | up-  | up-  |             |
          |             | right-split |    | split| split| corner-split|
          |             |             |    |      |      |             |
          |             |     n-1     |    |  n-1 |  n-1 |     n-1     |
          |             |             |    |      |      |             |
          |  identity   +-------------+    +------+------+-------------+
          |             |             |    |             | right-split |
          |             | right-split |    |             |     n-1     |
          |             |             |    |  identity   +-------------+
          |             |     n-1     |    |             | right-split |
          |             |             |    |             |     n-1     |
          +-------------+-------------+    +-------------+-------------+

                 right-split n                    corner-split n

   We can produce balanced patterns by branching upwards as well as
towards the right (see *Note Exercise 2-44:: and figures *Note Figure
2-13:: and *Note Figure 2-14::):

     (define (corner-split painter n)
       (if (= n 0)
           painter
           (let ((up (up-split painter (- n 1)))
                 (right (right-split painter (- n 1))))
             (let ((top-left (beside up up))
                   (bottom-right (below right right))
                   (corner (corner-split painter (- n 1))))
               (beside (below painter top-left)
                       (below bottom-right corner))))))

     *Figure 2.14:* The recursive operations `right-split' and
     `corner-split' applied to the painters `wave' and `rogers'.
     Combining four `corner-split' figures produces symmetric
     `square-limit' designs as shown in *Note Figure 2-9::.

     [two graphic images not included]

          (right-split wave 4)         (right-split rogers 4)

     [two graphic images not included]

          (corner-split wave 4)        (corner-split rogers 4)

   By placing four copies of a `corner-split' appropriately, we obtain a
pattern called `square-limit', whose application to `wave' and `rogers'
is shown in *Note Figure 2-9:::

     (define (square-limit painter n)
       (let ((quarter (corner-split painter n)))
         (let ((half (beside (flip-horiz quarter) quarter)))
           (below (flip-vert half) half))))

     *Exercise 2.44:* Define the procedure `up-split' used by
     `corner-split'.  It is similar to `right-split', except that it
     switches the roles of `below' and `beside'.

Higher-order operations
.......................

In addition to abstracting patterns of combining painters, we can work
at a higher level, abstracting patterns of combining painter
operations.  That is, we can view the painter operations as elements to
manipulate and can write means of combination for these
elements--procedures that take painter operations as arguments and
create new painter operations.

   For example, `flipped-pairs' and `square-limit' each arrange four
copies of a painter's image in a square pattern; they differ only in
how they orient the copies.  One way to abstract this pattern of
painter combination is with the following procedure, which takes four
one-argument painter operations and produces a painter operation that
transforms a given painter with those four operations and arranges the
results in a square.  `Tl', `tr', `bl', and `br' are the
transformations to apply to the top left copy, the top right copy, the
bottom left copy, and the bottom right copy, respectively.

     (define (square-of-four tl tr bl br)
       (lambda (painter)
         (let ((top (beside (tl painter) (tr painter)))
               (bottom (beside (bl painter) (br painter))))
           (below bottom top))))

   Then `flipped-pairs' can be defined in terms of `square-of-four' as
follows:(3)

     (define (flipped-pairs painter)
       (let ((combine4 (square-of-four identity flip-vert
                                       identity flip-vert)))
         (combine4 painter)))

and `square-limit' can be expressed as(4)

     (define (square-limit painter n)
       (let ((combine4 (square-of-four flip-horiz identity
                                       rotate180 flip-vert)))
         (combine4 (corner-split painter n))))

     *Exercise 2.45:* `Right-split' and `up-split' can be expressed as
     instances of a general splitting operation.  Define a procedure
     `split' with the property that evaluating

          (define right-split (split beside below))
          (define up-split (split below beside))

     produces procedures `right-split' and `up-split' with the same
     behaviors as the ones already defined.

Frames
......

Before we can show how to implement painters and their means of
combination, we must first consider frames.  A frame can be described
by three vectors--an origin vector and two edge vectors.  The origin
vector specifies the offset of the frame's origin from some absolute
origin in the plane, and the edge vectors specify the offsets of the
frame's corners from its origin.  If the edges are perpendicular, the
frame will be rectangular.  Otherwise the frame will be a more general
parallelogram.

   *Note Figure 2-15:: shows a frame and its associated vectors.  In
accordance with data abstraction, we need not be specific yet about how
frames are represented, other than to say that there is a constructor
`make-frame', which takes three vectors and produces a frame, and three
corresponding selectors `origin-frame', `edge1-frame', and
`edge2-frame' (see *Note Exercise 2-47::).

     *Figure 2.15:* A frame is described by three vectors - an origin
     and two edges.

                                   __
                               __--  \
                           __--       \
                __     __--            \   __
               |\  __--                 \__-|
                 \-                  __--
          frame   \              __--
          edge2    \         __--    frame
          vector    \    __--        edge1
                     \_--            vector
                      -   <--+
                    frame    |
                    origin   +-- (0,0) point
                    vector       on display screen

   We will use coordinates in the unit square (0<= x,y<= 1) to specify
images.  With each frame, we associate a "frame coordinate map", which
will be used to shift and scale images to fit the frame.  The map
transforms the unit square into the frame by mapping the vector v =
(x,y) to the vector sum

     Origin(Frame) + r * Edge_1(Frame) + y * Edge_2(Frame)

For example, (0,0) is mapped to the origin of the frame, (1,1) to the
vertex diagonally opposite the origin, and (0.5,0.5) to the center of
the frame.  We can create a frame's coordinate map with the following
procedure:(5)

     (define (frame-coord-map frame)
       (lambda (v)
         (add-vect
          (origin-frame frame)
          (add-vect (scale-vect (xcor-vect v)
                                (edge1-frame frame))
                    (scale-vect (ycor-vect v)
                                (edge2-frame frame))))))

   Observe that applying `frame-coord-map' to a frame returns a
procedure that, given a vector, returns a vector.  If the argument
vector is in the unit square, the result vector will be in the frame.
For example,

     ((frame-coord-map a-frame) (make-vect 0 0))

returns the same vector as

     (origin-frame a-frame)

     *Exercise 2.46:* A two-dimensional vector v running from the
     origin to a point can be represented as a pair consisting of an
     x-coordinate and a y-coordinate.  Implement a data abstraction for
     vectors by giving a constructor `make-vect' and corresponding
     selectors `xcor-vect' and `ycor-vect'.  In terms of your selectors
     and constructor, implement procedures `add-vect', `sub-vect', and
     `scale-vect' that perform the operations vector addition, vector
     subtraction, and multiplying a vector by a scalar:

          (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)
          (x_1, y_1) - (x_2, y_2) = (x_1 - x_2, y_1 - y_2)
                       s * (x, y) = (sx, sy)

     *Exercise 2.47:* Here are two possible constructors for frames:

          (define (make-frame origin edge1 edge2)
            (list origin edge1 edge2))

          (define (make-frame origin edge1 edge2)
            (cons origin (cons edge1 edge2)))

     For each constructor supply the appropriate selectors to produce an
     implementation for frames.

Painters
........

A painter is represented as a procedure that, given a frame as
argument, draws a particular image shifted and scaled to fit the frame.
That is to say, if `p' is a painter and `f' is a frame, then we
produce `p''s image in `f' by calling `p' with `f' as argument.

   The details of how primitive painters are implemented depend on the
particular characteristics of the graphics system and the type of image
to be drawn.  For instance, suppose we have a procedure `draw-line'
that draws a line on the screen between two specified points.  Then we
can create painters for line drawings, such as the `wave' painter in
*Note Figure 2-10::, from lists of line segments as follows:(6)

     (define (segments->painter segment-list)
       (lambda (frame)
         (for-each
          (lambda (segment)
            (draw-line
             ((frame-coord-map frame) (start-segment segment))
             ((frame-coord-map frame) (end-segment segment))))
          segment-list)))

   The segments are given using coordinates with respect to the unit
square.  For each segment in the list, the painter transforms the
segment endpoints with the frame coordinate map and draws a line
between the transformed points.

   Representing painters as procedures erects a powerful abstraction
barrier in the picture language.  We can create and intermix all sorts
of primitive painters, based on a variety of graphics capabilities. The
details of their implementation do not matter.  Any procedure can serve
as a painter, provided that it takes a frame as argument and draws
something scaled to fit the frame.(7)

     *Exercise 2.48:* A directed line segment in the plane can be
     represented as a pair of vectors--the vector running from the
     origin to the start-point of the segment, and the vector running
     from the origin to the end-point of the segment.  Use your vector
     representation from *Note Exercise 2-46:: to define a
     representation for segments with a constructor `make-segment' and
     selectors `start-segment' and `end-segment'.

     *Exercise 2.49:* Use `segments->painter' to define the following
     primitive painters:

       a. The painter that draws the outline of the designated frame.

       b. The painter that draws an "X" by connecting opposite corners
          of the frame.

       c. The painter that draws a diamond shape by connecting the
          midpoints of the sides of the frame.

       d. The `wave' painter.


Transforming and combining painters
...................................

An operation on painters (such as `flip-vert' or `beside') works by
creating a painter that invokes the original painters with respect to
frames derived from the argument frame.  Thus, for example, `flip-vert'
doesn't have to know how a painter works in order to flip it--it just
has to know how to turn a frame upside down: The flipped painter just
uses the original painter, but in the inverted frame.

   Painter operations are based on the procedure `transform-painter',
which takes as arguments a painter and information on how to transform
a frame and produces a new painter.  The transformed painter, when
called on a frame, transforms the frame and calls the original painter
on the transformed frame.  The arguments to `transform-painter' are
points (represented as vectors) that specify the corners of the new
frame: When mapped into the frame, the first point specifies the new
frame's origin and the other two specify the ends of its edge vectors.
Thus, arguments within the unit square specify a frame contained within
the original frame.

     (define (transform-painter painter origin corner1 corner2)
       (lambda (frame)
         (let ((m (frame-coord-map frame)))
           (let ((new-origin (m origin)))
             (painter
              (make-frame new-origin
                          (sub-vect (m corner1) new-origin)
                          (sub-vect (m corner2) new-origin)))))))

   Here's how to flip painter images vertically:

     (define (flip-vert painter)
       (transform-painter painter
                          (make-vect 0.0 1.0)   ; new `origin'
                          (make-vect 1.0 1.0)   ; new end of `edge1'
                          (make-vect 0.0 0.0))) ; new end of `edge2'

   Using `transform-painter', we can easily define new transformations.
For example, we can define a painter that shrinks its image to the
upper-right quarter of the frame it is given:

     (define (shrink-to-upper-right painter)
       (transform-painter painter
                          (make-vect 0.5 0.5)
                          (make-vect 1.0 0.5)
                          (make-vect 0.5 1.0)))

   Other transformations rotate images counterclockwise by 90 degrees(8)

     (define (rotate90 painter)
       (transform-painter painter
                          (make-vect 1.0 0.0)
                          (make-vect 1.0 1.0)
                          (make-vect 0.0 0.0)))

or squash images towards the center of the frame:(9)

     (define (squash-inwards painter)
       (transform-painter painter
                          (make-vect 0.0 0.0)
                          (make-vect 0.65 0.35)
                          (make-vect 0.35 0.65)))

   Frame transformation is also the key to defining means of combining
two or more painters.  The `beside' procedure, for example, takes two
painters, transforms them to paint in the left and right halves of an
argument frame respectively, and produces a new, compound painter.
When the compound painter is given a frame, it calls the first
transformed painter to paint in the left half of the frame and calls
the second transformed painter to paint in the right half of the frame:

     (define (beside painter1 painter2)
       (let ((split-point (make-vect 0.5 0.0)))
         (let ((paint-left
                (transform-painter painter1
                                   (make-vect 0.0 0.0)
                                   split-point
                                   (make-vect 0.0 1.0)))
               (paint-right
                (transform-painter painter2
                                   split-point
                                   (make-vect 1.0 0.0)
                                   (make-vect 0.5 1.0))))
           (lambda (frame)
             (paint-left frame)
             (paint-right frame)))))

   Observe how the painter data abstraction, and in particular the
representation of painters as procedures, makes `beside' easy to
implement.  The `beside' procedure need not know anything about the
details of the component painters other than that each painter will
draw something in its designated frame.

     *Exercise 2.50:* Define the transformation `flip-horiz', which
     flips painters horizontally, and transformations that rotate
     painters counterclockwise by 180 degrees and 270 degrees.

     *Exercise 2.51:* Define the `below' operation for painters.
     `Below' takes two painters as arguments.  The resulting painter,
     given a frame, draws with the first painter in the bottom of the
     frame and with the second painter in the top.  Define `below' in
     two different ways--first by writing a procedure that is analogous
     to the `beside' procedure given above, and again in terms of
     `beside' and suitable rotation operations (from *Note Exercise
     2-50::).

Levels of language for robust design
....................................

The picture language exercises some of the critical ideas we've
introduced about abstraction with procedures and data.  The fundamental
data abstractions, painters, are implemented using procedural
representations, which enables the language to handle different basic
drawing capabilities in a uniform way.  The means of combination
satisfy the closure property, which permits us to easily build up
complex designs.  Finally, all the tools for abstracting procedures are
available to us for abstracting means of combination for painters.

   We have also obtained a glimpse of another crucial idea about
languages and program design.  This is the approach of "stratified
design", the notion that a complex system should be structured as a
sequence of levels that are described using a sequence of languages.
Each level is constructed by combining parts that are regarded as
primitive at that level, and the parts constructed at each level are
used as primitives at the next level.  The language used at each level
of a stratified design has primitives, means of combination, and means
of abstraction appropriate to that level of detail.

   Stratified design pervades the engineering of complex systems.  For
example, in computer engineering, resistors and transistors are
combined (and described using a language of analog circuits) to produce
parts such as and-gates and or-gates, which form the primitives of a
language for digital-circuit design.(10) These parts are combined to
build processors, bus structures, and memory systems, which are in turn
combined to form computers, using languages appropriate to computer
architecture.  Computers are combined to form distributed systems, using
languages appropriate for describing network interconnections, and so
on.

   As a tiny example of stratification, our picture language uses
primitive elements (primitive painters) that are created using a
language that specifies points and lines to provide the lists of line
segments for `segments->painter', or the shading details for a painter
like `rogers'.  The bulk of our description of the picture language
focused on combining these primitives, using geometric combiners such
as `beside' and `below'.  We also worked at a higher level, regarding
`beside' and `below' as primitives to be manipulated in a language
whose operations, such as `square-of-four', capture common patterns of
combining geometric combiners.

   Stratified design helps make programs "robust", that is, it makes it
likely that small changes in a specification will require
correspondingly small changes in the program.  For instance, suppose we
wanted to change the image based on `wave' shown in *Note Figure 2-9::.
We could work at the lowest level to change the detailed appearance of
the `wave' element; we could work at the middle level to change the way
`corner-split' replicates the `wave'; we could work at the highest
level to change how `square-limit' arranges the four copies of the
corner.  In general, each level of a stratified design provides a
different vocabulary for expressing the characteristics of the system,
and a different kind of ability to change it.

     *Exercise 2.52:* Make changes to the square limit of `wave' shown
     in *Note Figure 2-9:: by working at each of the levels described
     above.  In particular:

       a. Add some segments to the primitive `wave' painter of *Note
          Exercise 2-49:: (to add a smile, for example).

       b. Change the pattern constructed by `corner-split' (for
          example, by using only one copy of the `up-split' and
          `right-split' images instead of two).

       c. Modify the version of `square-limit' that uses
          `square-of-four' so as to assemble the corners in a different
          pattern.  (For example, you might make the big Mr. Rogers
          look outward from each corner of the square.)


   ---------- Footnotes ----------

   (1) The picture language is based on the language Peter Henderson
created to construct images like M.C. Escher's "Square Limit" woodcut
(see Henderson 1982).  The woodcut incorporates a repeated scaled
pattern, similar to the arrangements drawn using the `square-limit'
procedure in this section.

   (2) William Barton Rogers (1804-1882) was the founder and first
president of MIT.  A geologist and talented teacher, he taught at
William and Mary College and at the University of Virginia.  In 1859 he
moved to Boston, where he had more time for research, worked on a plan
for establishing a "polytechnic institute," and served as
Massachusetts's first State Inspector of Gas Meters.

   When MIT was established in 1861, Rogers was elected its first
president.  Rogers espoused an ideal of "useful learning" that was
different from the university education of the time, with its
overemphasis on the classics, which, as he wrote, "stand in the way of
the broader, higher and more practical instruction and discipline of
the natural and social sciences."  This education was likewise to be
different from narrow trade-school education.  In Rogers's words:

     The world-enforced distinction between the practical and the
     scientific worker is utterly futile, and the whole experience of
     modern times has demonstrated its utter worthlessness.

   Rogers served as president of MIT until 1870, when he resigned due to
ill health.  In 1878 the second president of MIT, John Runkle, resigned
under the pressure of a financial crisis brought on by the Panic of
1873 and strain of fighting off attempts by Harvard to take over MIT.
Rogers returned to hold the office of president until 1881.

   Rogers collapsed and died while addressing MIT's graduating class at
the commencement exercises of 1882.  Runkle quoted Rogers's last words
in a memorial address delivered that same year:

     "As I stand here today and see what the Institute is, ... I call
     to mind the beginnings of science.  I remember one hundred and
     fifty years ago Stephen Hales published a pamphlet on the subject
     of illuminating gas, in which he stated that his researches had
     demonstrated that 128 grains of bituminous coal - " "Bituminous
     coal," these were his last words on earth.  Here he bent forward,
     as if consulting some notes on the table before him, then slowly
     regaining an erect position, threw up his hands, and was
     translated from the scene of his earthly labors and triumphs to
     "the tomorrow of death," where the mysteries of life are solved,
     and the disembodied spirit finds unending satisfaction in
     contemplating the new and still unfathomable mysteries of the
     infinite future.

   In the words of Francis A. Walker (MIT's third president):

     All his life he had borne himself most faithfully and heroically,
     and he died as so good a knight would surely have wished, in
     harness, at his post, and in the very part and act of public duty.

   (3) Equivalently, we could write

     (define flipped-pairs
       (square-of-four identity flip-vert identity flip-vert))

   (4) `Rotate180' rotates a painter by 180 degrees (see *Note Exercise
2-50::).  Instead of `rotate180' we could say `(compose flip-vert
flip-horiz)', using the `compose' procedure from *Note Exercise 1-42::.

   (5) `Frame-coord-map' uses the vector operations described in *Note
Exercise 2-46:: below, which we assume have been implemented using some
representation for vectors.  Because of data abstraction, it doesn't
matter what this vector representation is, so long as the vector
operations behave correctly.

   (6) `Segments->painter' uses the representation for line segments
described in *Note Exercise 2-48:: below.  It also uses the `for-each'
procedure described in *Note Exercise 2-23::.

   (7) For example, the `rogers' painter of *Note Figure 2-11:: was
constructed from a gray-level image.  For each point in a given frame,
the `rogers' painter determines the point in the image that is mapped
to it under the frame coordinate map, and shades it accordingly.  By
allowing different types of painters, we are capitalizing on the
abstract data idea discussed in section *Note 2-1-3::, where we argued
that a rational-number representation could be anything at all that
satisfies an appropriate condition.  Here we're using the fact that a
painter can be implemented in any way at all, so long as it draws
something in the designated frame.  Section *Note 2-1-3:: also showed
how pairs could be implemented as procedures.  Painters are our second
example of a procedural representation for data.

   (8) `Rotate90' is a pure rotation only for square frames, because it
also stretches and shrinks the image to fit into the rotated frame.

   (9) The diamond-shaped images in figures *Note Figure 2-10:: and
*Note Figure 2-11:: were created with `squash-inwards' applied to
`wave' and `rogers'.

   (10) Section *Note 3-3-4:: describes one such language.


File: sicp.info,  Node: 2-3,  Next: 2-4,  Prev: 2-2,  Up: Chapter 2

2.3 Symbolic Data
=================

All the compound data objects we have used so far were constructed
ultimately from numbers.  In this section we extend the
representational capability of our language by introducing the ability
to work with arbitrary symbols as data.

* Menu:

* 2-3-1::            Quotation
* 2-3-2::            Example: Symbolic Differentiation
* 2-3-3::            Example: Representing Sets
* 2-3-4::            Example: Huffman Encoding Trees


File: sicp.info,  Node: 2-3-1,  Next: 2-3-2,  Prev: 2-3,  Up: 2-3

2.3.1 Quotation
---------------

If we can form compound data using symbols, we can have lists such as

     (a b c d)
     (23 45 17)
     ((Norah 12) (Molly 9) (Anna 7) (Lauren 6) (Charlotte 4))

   Lists containing symbols can look just like the expressions of our
language:

     (* (+ 23 45) (+ x 9))

     (define (fact n) (if (= n 1) 1 (* n (fact (- n 1)))))

   In order to manipulate symbols we need a new element in our
language: the ability to "quote" a data object.  Suppose we want to
construct the list `(a b)'.  We can't accomplish this with `(list a
b)', because this expression constructs a list of the "values" of `a'
and `b' rather than the symbols themselves.  This issue is well known
in the context of natural languages, where words and sentences may be
regarded either as semantic entities or as character strings (syntactic
entities).  The common practice in natural languages is to use
quotation marks to indicate that a word or a sentence is to be treated
literally as a string of characters.  For instance, the first letter of
"John" is clearly "J."  If we tell somebody "say your name aloud," we
expect to hear that person's name.  However, if we tell somebody "say
`your name' aloud," we expect to hear the words "your name."  Note that
we are forced to nest quotation marks to describe what somebody else
might say.(1)

   We can follow this same practice to identify lists and symbols that
are to be treated as data objects rather than as expressions to be
evaluated.  However, our format for quoting differs from that of
natural languages in that we place a quotation mark (traditionally, the
single quote symbol `'') only at the beginning of the object to be
quoted.  We can get away with this in Scheme syntax because we rely on
blanks and parentheses to delimit objects.  Thus, the meaning of the
single quote character is to quote the next object.(2)

   Now we can distinguish between symbols and their values:

     (define a 1)

     (define b 2)

     (list a b)
     (1 2)

     (list 'a 'b)
     (a b)

     (list 'a b)
     (a 2)

   Quotation also allows us to type in compound objects, using the
conventional printed representation for lists:(3)

     (car '(a b c))
     a

     (cdr '(a b c))
     (b c)

   In keeping with this, we can obtain the empty list by evaluating
`'()', and thus dispense with the variable `nil'.

   One additional primitive used in manipulating symbols is `eq?', which
takes two symbols as arguments and tests whether they are the same.(4)
Using `eq?', we can implement a useful procedure called `memq'.  This
takes two arguments, a symbol and a list.  If the symbol is not
contained in the list (i.e., is not `eq?' to any item in the list),
then `memq' returns false.  Otherwise, it returns the sublist of the
list beginning with the first occurrence of the symbol:

     (define (memq item x)
       (cond ((null? x) false)
             ((eq? item (car x)) x)
             (else (memq item (cdr x)))))

   For example, the value of

     (memq 'apple '(pear banana prune))

is false, whereas the value of

     (memq 'apple '(x (apple sauce) y apple pear))

is `(apple pear)'.

     *Exercise 2.53:* What would the interpreter print in response to
     evaluating each of the following expressions?

          (list 'a 'b 'c)

          (list (list 'george))

          (cdr '((x1 x2) (y1 y2)))

          (cadr '((x1 x2) (y1 y2)))

          (pair? (car '(a short list)))

          (memq 'red '((red shoes) (blue socks)))

          (memq 'red '(red shoes blue socks))

     *Exercise 2.54:* Two lists are said to be `equal?' if they contain
     equal elements arranged in the same order.  For example,

          (equal? '(this is a list) '(this is a list))

     is true, but

          (equal? '(this is a list) '(this (is a) list))

     is false.  To be more precise, we can define `equal?'  recursively
     in terms of the basic `eq?' equality of symbols by saying that `a'
     and `b' are `equal?' if they are both symbols and the symbols are
     `eq?', or if they are both lists such that `(car a)' is `equal?'
     to `(car b)' and `(cdr a)' is `equal?' to `(cdr b)'.  Using this
     idea, implement `equal?' as a procedure.(5)

     *Exercise 2.55:* Eva Lu Ator types to the interpreter the
     expression

          (car ''abracadabra)

     To her surprise, the interpreter prints back `quote'.  Explain.

   ---------- Footnotes ----------

   (1) Allowing quotation in a language wreaks havoc with the ability
to reason about the language in simple terms, because it destroys the
notion that equals can be substituted for equals.  For example, three
is one plus two, but the word "three" is not the phrase "one plus two."
Quotation is powerful because it gives us a way to build expressions
that manipulate other expressions (as we will see when we write an
interpreter in *Note Chapter 4::). But allowing statements in a
language that talk about other statements in that language makes it
very difficult to maintain any coherent principle of what "equals can
be substituted for equals" should mean.  For example, if we know that
the evening star is the morning star, then from the statement "the
evening star is Venus" we can deduce "the morning star is Venus."
However, given that "John knows that the evening star is Venus" we
cannot infer that "John knows that the morning star is Venus."

   (2) The single quote is different from the double quote we have been
using to enclose character strings to be printed.  Whereas the single
quote can be used to denote lists or symbols, the double quote is used
only with character strings.  In this book, the only use for character
strings is as items to be printed.

   (3) Strictly, our use of the quotation mark violates the general
rule that all compound expressions in our language should be delimited
by parentheses and look like lists.  We can recover this consistency by
introducing a special form `quote', which serves the same purpose as
the quotation mark.  Thus, we would type `(quote a)' instead of `'a',
and we would type `(quote (a b c))' instead of `'(a b c)'.  This is
precisely how the interpreter works.  The quotation mark is just a
single-character abbreviation for wrapping the next complete expression
with `quote' to form `(quote <EXPRESSION>)'.  This is important because
it maintains the principle that any expression seen by the interpreter
can be manipulated as a data object.  For instance, we could construct
the expression `(car '(a b c))', which is the same as `(car (quote (a b
c)))', by evaluating `(list 'car (list 'quote '(a b c)))'.

   (4) We can consider two symbols to be "the same" if they consist of
the same characters in the same order.  Such a definition skirts a deep
issue that we are not yet ready to address: the meaning of "sameness"
in a programming language.  We will return to this in *Note Chapter 3::
(section *Note 3-1-3::).

   (5) In practice, programmers use `equal?' to compare lists that
contain numbers as well as symbols.  Numbers are not considered to be
symbols.  The question of whether two numerically equal numbers (as
tested by `=') are also `eq?' is highly implementation-dependent.  A
better definition of `equal?' (such as the one that comes as a
primitive in Scheme) would also stipulate that if `a' and `b' are both
numbers, then `a' and `b' are `equal?' if they are numerically equal.


File: sicp.info,  Node: 2-3-2,  Next: 2-3-3,  Prev: 2-3-1,  Up: 2-3

2.3.2 Example: Symbolic Differentiation
---------------------------------------

As an illustration of symbol manipulation and a further illustration of
data abstraction, consider the design of a procedure that performs
symbolic differentiation of algebraic expressions.  We would like the
procedure to take as arguments an algebraic expression and a variable
and to return the derivative of the expression with respect to the
variable.  For example, if the arguments to the procedure are ax^2 + bx
+ c and x, the procedure should return 2ax + b.  Symbolic
differentiation is of special historical significance in Lisp.  It was
one of the motivating examples behind the development of a computer
language for symbol manipulation.  Furthermore, it marked the beginning
of the line of research that led to the development of powerful systems
for symbolic mathematical work, which are currently being used by a
growing number of applied mathematicians and physicists.

   In developing the symbolic-differentiation program, we will follow
the same strategy of data abstraction that we followed in developing
the rational-number system of section *Note 2-1-1::.  That is, we will
first define a differentiation algorithm that operates on abstract
objects such as "sums," "products," and "variables" without worrying
about how these are to be represented.  Only afterward will we address
the representation problem.

The differentiation program with abstract data
..............................................

In order to keep things simple, we will consider a very simple
symbolic-differentiation program that handles expressions that are
built up using only the operations of addition and multiplication with
two arguments.  Differentiation of any such expression can be carried
out by applying the following reduction rules:

     dc
     -- = 0  for c a constant, or a variable different from x
     dx

     dx
     -- = 1
     dx

     d(u + v)   du   dv
     -------- = -- + --
        dx      dx   dx

     d(uv)     / dv \     / du \
     ----- = u | -- | + v | -- |
      dx       \ dx /     \ dx /

   Observe that the latter two rules are recursive in nature.  That is,
to obtain the derivative of a sum we first find the derivatives of the
terms and add them.  Each of the terms may in turn be an expression
that needs to be decomposed.  Decomposing into smaller and smaller
pieces will eventually produce pieces that are either constants or
variables, whose derivatives will be either 0 or 1.

   To embody these rules in a procedure we indulge in a little wishful
thinking, as we did in designing the rational-number implementation.
If we had a means for representing algebraic expressions, we should be
able to tell whether an expression is a sum, a product, a constant, or
a variable.  We should be able to extract the parts of an expression.
For a sum, for example we want to be able to extract the addend (first
term) and the augend (second term).  We should also be able to
construct expressions from parts.  Let us assume that we already have
procedures to implement the following selectors, constructors, and
predicates:

     (variable? e)          Is `e' a variable?
     (same-variable? v1 v2) Are `v1' and `v2' the same variable?
     (sum? e)               Is `e' a sum?
     (addend e)             Addend of the sum `e'.
     (augend e)             Augend of the sum `e'.
     (make-sum a1 a2)       Construct the sum of `a1' and `a2'.
     (product? e)           Is `e' a product?
     (multiplier e)         Multiplier of the product `e'.
     (multiplicand e)       Multiplicand of the product `e'.
     (make-product m1 m2)   Construct the product of `m1' and `m2'.

   Using these, and the primitive predicate `number?', which identifies
numbers, we can express the differentiation rules as the following
procedure:

     (define (deriv exp var)
       (cond ((number? exp) 0)
             ((variable? exp)
              (if (same-variable? exp var) 1 0))
             ((sum? exp)
              (make-sum (deriv (addend exp) var)
                        (deriv (augend exp) var)))
             ((product? exp)
              (make-sum
                (make-product (multiplier exp)
                              (deriv (multiplicand exp) var))
                (make-product (deriv (multiplier exp) var)
                              (multiplicand exp))))
             (else
              (error "unknown expression type -- DERIV" exp))))

   This `deriv' procedure incorporates the complete differentiation
algorithm.  Since it is expressed in terms of abstract data, it will
work no matter how we choose to represent algebraic expressions, as
long as we design a proper set of selectors and constructors.  This is
the issue we must address next.

Representing algebraic expressions
..................................

We can imagine many ways to use list structure to represent algebraic
expressions.  For example, we could use lists of symbols that mirror
the usual algebraic notation, representing ax + b as the list `(a * x +
b)' .  However, one especially straightforward choice is to use the same
parenthesized prefix notation that Lisp uses for combinations; that is,
to represent ax + b as `(+ (* a x) b)'.  Then our data representation
for the differentiation problem is as follows:

   * The variables are symbols.  They are identified by the primitive
     predicate `symbol?':

          (define (variable? x) (symbol? x))

   * Two variables are the same if the symbols representing them are
     `eq?':

          (define (same-variable? v1 v2)
            (and (variable? v1) (variable? v2) (eq? v1 v2)))

   * Sums and products are constructed as lists:

          (define (make-sum a1 a2) (list '+ a1 a2))

          (define (make-product m1 m2) (list '* m1 m2))

   * A sum is a list whose first element is the symbol `+':

          (define (sum? x)
            (and (pair? x) (eq? (car x) '+)))

   * The addend is the second item of the sum list:

          (define (addend s) (cadr s))

   * The augend is the third item of the sum list:

          (define (augend s) (caddr s))

   * A product is a list whose first element is the symbol `*':

          (define (product? x)
            (and (pair? x) (eq? (car x) '*)))

   * The multiplier is the second item of the product list:

          (define (multiplier p) (cadr p))

   * The multiplicand is the third item of the product list:

          (define (multiplicand p) (caddr p))


   Thus, we need only combine these with the algorithm as embodied by
`deriv' in order to have a working symbolic-differentiation program.
Let us look at some examples of its behavior:

     (deriv '(+ x 3) 'x)
     (+ 1 0)

     (deriv '(* x y) 'x)
     (+ (* x 0) (* 1 y))

     (deriv '(* (* x y) (+ x 3)) 'x)
        (+ (* (* x y) (+ 1 0))
        (* (+ (* x 0) (* 1 y))
           (+  x 3)))

   The program produces answers that are correct; however, they are
unsimplified.  It is true that

     d(xy)
     ----- = x * 0 + 1 * y
      dx

but we would like the program to know that x * 0 = 0, 1 * y = y, and 0
+ y = y.  The answer for the second example should have been simply
`y'.  As the third example shows, this becomes a serious issue when the
expressions are complex.

   Our difficulty is much like the one we encountered with the
rational-number implementation: we haven't reduced answers to simplest
form.  To accomplish the rational-number reduction, we needed to change
only the constructors and the selectors of the implementation.  We can
adopt a similar strategy here.  We won't change `deriv' at all.
Instead, we will change `make-sum' so that if both summands are
numbers, `make-sum' will add them and return their sum.  Also, if one
of the summands is 0, then `make-sum' will return the other summand.

     (define (make-sum a1 a2)
       (cond ((=number? a1 0) a2)
             ((=number? a2 0) a1)
             ((and (number? a1) (number? a2)) (+ a1 a2))
             (else (list '+ a1 a2))))

   This uses the procedure `=number?', which checks whether an
expression is equal to a given number:

     (define (=number? exp num)
       (and (number? exp) (= exp num)))

   Similarly, we will change `make-product' to build in the rules that 0
times anything is 0 and 1 times anything is the thing itself:

     (define (make-product m1 m2)
       (cond ((or (=number? m1 0) (=number? m2 0)) 0)
             ((=number? m1 1) m2)
             ((=number? m2 1) m1)
             ((and (number? m1) (number? m2)) (* m1 m2))
             (else (list '* m1 m2))))

   Here is how this version works on our three examples:

     (deriv '(+ x 3) 'x)
     1

     (deriv '(* x y) 'x)
     y

     (deriv '(* (* x y) (+ x 3)) 'x)
     (+ (* x y) (* y (+ x 3)))

   Although this is quite an improvement, the third example shows that
there is still a long way to go before we get a program that puts
expressions into a form that we might agree is "simplest."  The problem
of algebraic simplification is complex because, among other reasons, a
form that may be simplest for one purpose may not be for another.

     *Exercise 2.56:* Show how to extend the basic differentiator to
     handle more kinds of expressions.  For instance, implement the
     differentiation rule

          n_1   n_2
          --- = ---  if and only if n_1 d_2 = n_2 d_1
          d_1   d_2

     by adding a new clause to the `deriv' program and defining
     appropriate procedures `exponentiation?', `base', `exponent', and
     `make-exponentiation'.  (You may use the symbol `**' to denote
     exponentiation.)  Build in the rules that anything raised to the
     power 0 is 1 and anything raised to the power 1 is the thing
     itself.

     *Exercise 2.57:* Extend the differentiation program to handle sums
     and products of arbitrary numbers of (two or more) terms.  Then
     the last example above could be expressed as

          (deriv '(* x y (+ x 3)) 'x)

     Try to do this by changing only the representation for sums and
     products, without changing the `deriv' procedure at all.  For
     example, the `addend' of a sum would be the first term, and the
     `augend' would be the sum of the rest of the terms.

     *Exercise 2.58:* Suppose we want to modify the differentiation
     program so that it works with ordinary mathematical notation, in
     which `+' and `*' are infix rather than prefix operators.  Since
     the differentiation program is defined in terms of abstract data,
     we can modify it to work with different representations of
     expressions solely by changing the predicates, selectors, and
     constructors that define the representation of the algebraic
     expressions on which the differentiator is to operate.

       a. Show how to do this in order to differentiate algebraic
          expressions presented in infix form, such as `(x + (3 * (x +
          (y + 2))))'.  To simplify the task, assume that `+' and `*'
          always take two arguments and that expressions are fully
          parenthesized.

       b. The problem becomes substantially harder if we allow standard
          algebraic notation, such as `(x + 3 * (x + y + 2))', which
          drops unnecessary parentheses and assumes that multiplication
          is done before addition.  Can you design appropriate
          predicates, selectors, and constructors for this notation
          such that our derivative program still works?



File: sicp.info,  Node: 2-3-3,  Next: 2-3-4,  Prev: 2-3-2,  Up: 2-3

2.3.3 Example: Representing Sets
--------------------------------

In the previous examples we built representations for two kinds of
compound data objects: rational numbers and algebraic expressions.  In
one of these examples we had the choice of simplifying (reducing) the
expressions at either construction time or selection time, but other
than that the choice of a representation for these structures in terms
of lists was straightforward. When we turn to the representation of
sets, the choice of a representation is not so obvious.  Indeed, there
are a number of possible representations, and they differ significantly
from one another in several ways.

   Informally, a set is simply a collection of distinct objects.  To
give a more precise definition we can employ the method of data
abstraction.  That is, we define "set" by specifying the operations
that are to be used on sets.  These are `union-set',
`intersection-set', `element-of-set?', and `adjoin-set'.
`Element-of-set?' is a predicate that determines whether a given
element is a member of a set.  `Adjoin-set' takes an object and a set
as arguments and returns a set that contains the elements of the
original set and also the adjoined element.  `Union-set' computes the
union of two sets, which is the set containing each element that
appears in either argument.  `Intersection-set' computes the
intersection of two sets, which is the set containing only elements
that appear in both arguments.  From the viewpoint of data abstraction,
we are free to design any representation that implements these
operations in a way consistent with the interpretations given above.(1)

Sets as unordered lists
.......................

One way to represent a set is as a list of its elements in which no
element appears more than once.  The empty set is represented by the
empty list.  In this representation, `element-of-set?' is similar to
the procedure `memq' of section *Note 2-3-1::.  It uses `equal?'
instead of `eq?' so that the set elements need not be symbols:

     (define (element-of-set? x set)
       (cond ((null? set) false)
             ((equal? x (car set)) true)
             (else (element-of-set? x (cdr set)))))

   Using this, we can write `adjoin-set'.  If the object to be adjoined
is already in the set, we just return the set.  Otherwise, we use
`cons' to add the object to the list that represents the set:

     (define (adjoin-set x set)
       (if (element-of-set? x set)
           set
           (cons x set)))

   For `intersection-set' we can use a recursive strategy.  If we know
how to form the intersection of `set2' and the `cdr' of `set1', we only
need to decide whether to include the `car' of `set1' in this.  But
this depends on whether `(car set1)' is also in `set2'.  Here is the
resulting procedure:

     (define (intersection-set set1 set2)
       (cond ((or (null? set1) (null? set2)) '())
             ((element-of-set? (car set1) set2)
              (cons (car set1)
                    (intersection-set (cdr set1) set2)))
             (else (intersection-set (cdr set1) set2))))

   In designing a representation, one of the issues we should be
concerned with is efficiency.  Consider the number of steps required by
our set operations.  Since they all use `element-of-set?', the speed of
this operation has a major impact on the efficiency of the set
implementation as a whole.  Now, in order to check whether an object is
a member of a set, `element-of-set?' may have to scan the entire set.
(In the worst case, the object turns out not to be in the set.)  Hence,
if the set has n elements, `element-of-set?'  might take up to n steps.
Thus, the number of steps required grows as [theta](n).  The number of
steps required by `adjoin-set', which uses this operation, also grows
as [theta](n).  For `intersection-set', which does an `element-of-set?'
check for each element of `set1', the number of steps required grows as
the product of the sizes of the sets involved, or [theta](n^2) for two
sets of size n.  The same will be true of `union-set'.

     *Exercise 2.59:* Implement the `union-set' operation for the
     unordered-list representation of sets.

     *Exercise 2.60:* We specified that a set would be represented as a
     list with no duplicates.  Now suppose we allow duplicates.  For
     instance, the set {1,2,3} could be represented as the list `(2 3 2
     1 3 2 2)'.  Design procedures `element-of-set?', `adjoin-set',
     `union-set', and `intersection-set' that operate on this
     representation.  How does the efficiency of each compare with the
     corresponding procedure for the non-duplicate representation?  Are
     there applications for which you would use this representation in
     preference to the non-duplicate one?

Sets as ordered lists
.....................

One way to speed up our set operations is to change the representation
so that the set elements are listed in increasing order.  To do this,
we need some way to compare two objects so that we can say which is
bigger.  For example, we could compare symbols lexicographically, or we
could agree on some method for assigning a unique number to an object
and then compare the elements by comparing the corresponding numbers.
To keep our discussion simple, we will consider only the case where the
set elements are numbers, so that we can compare elements using `>' and
`<'.  We will represent a set of numbers by listing its elements in
increasing order.  Whereas our first representation above allowed us to
represent the set {1,3,6,10} by listing the elements in any order, our
new representation allows only the list `(1 3 6 10)'.

   One advantage of ordering shows up in `element-of-set?': In checking
for the presence of an item, we no longer have to scan the entire set.
If we reach a set element that is larger than the item we are looking
for, then we know that the item is not in the set:

     (define (element-of-set? x set)
       (cond ((null? set) false)
             ((= x (car set)) true)
             ((< x (car set)) false)
             (else (element-of-set? x (cdr set)))))

   How many steps does this save?  In the worst case, the item we are
looking for may be the largest one in the set, so the number of steps
is the same as for the unordered representation.  On the other hand, if
we search for items of many different sizes we can expect that
sometimes we will be able to stop searching at a point near the
beginning of the list and that other times we will still need to
examine most of the list.  On the average we should expect to have to
examine about half of the items in the set.  Thus, the average number
of steps required will be about n/2.  This is still [theta](n) growth,
but it does save us, on the average, a factor of 2 in number of steps
over the previous implementation.

   We obtain a more impressive speedup with `intersection-set'.  In the
unordered representation this operation required [theta](n^2) steps,
because we performed a complete scan of `set2' for each element of
`set1'.  But with the ordered representation, we can use a more clever
method.  Begin by comparing the initial elements, `x1' and `x2', of the
two sets.  If `x1' equals `x2', then that gives an element of the
intersection, and the rest of the intersection is the intersection of
the `cdr's of the two sets.  Suppose, however, that `x1' is less than
`x2'.  Since `x2' is the smallest element in `set2', we can immediately
conclude that `x1' cannot appear anywhere in `set2' and hence is not in
the intersection.  Hence, the intersection is equal to the intersection
of `set2' with the `cdr' of `set1'.  Similarly, if `x2' is less than
`x1', then the intersection is given by the intersection of `set1' with
the `cdr' of `set2'.  Here is the procedure:

     (define (intersection-set set1 set2)
       (if (or (null? set1) (null? set2))
           '()
           (let ((x1 (car set1)) (x2 (car set2)))
             (cond ((= x1 x2)
                    (cons x1
                          (intersection-set (cdr set1)
                                            (cdr set2))))
                   ((< x1 x2)
                    (intersection-set (cdr set1) set2))
                   ((< x2 x1)
                    (intersection-set set1 (cdr set2)))))))

   To estimate the number of steps required by this process, observe
that at each step we reduce the intersection problem to computing
intersections of smaller sets--removing the first element from `set1'
or `set2' or both.  Thus, the number of steps required is at most the
sum of the sizes of `set1' and `set2', rather than the product of the
sizes as with the unordered representation.  This is [theta](n) growth
rather than [theta](n^2)--a considerable speedup, even for sets of
moderate size.

     *Exercise 2.61:* Give an implementation of `adjoin-set' using the
     ordered representation.  By analogy with `element-of-set?' show
     how to take advantage of the ordering to produce a procedure that
     requires on the average about half as many steps as with the
     unordered representation.

     *Exercise 2.62:* Give a [theta](n) implementation of `union-set'
     for sets represented as ordered lists.

Sets as binary trees
....................

We can do better than the ordered-list representation by arranging the
set elements in the form of a tree.  Each node of the tree holds one
element of the set, called the "entry" at that node, and a link to each
of two other (possibly empty) nodes.  The "left" link points to
elements smaller than the one at the node, and the "right" link to
elements greater than the one at the node.  *Note Figure 2-16:: shows
some trees that represent the set {1,3,5,7,9,11}.  The same set may be
represented by a tree in a number of different ways.  The only thing we
require for a valid representation is that all elements in the left
subtree be smaller than the node entry and that all elements in the
right subtree be larger.

     *Figure 2.16:* Various binary trees that represent the set
     {1,3,5,7,9,11}.

             7          3             5
             /\         /\            /\
            3  9       1  7          3  9
           /\   \         /\        /   /\
          1  5  11       5  9      1   7  11
                             \
                             11

   The advantage of the tree representation is this: Suppose we want to
check whether a number x is contained in a set.  We begin by comparing
x with the entry in the top node.  If x is less than this, we know that
we need only search the left subtree; if x is greater, we need only
search the right subtree.  Now, if the tree is "balanced," each of
these subtrees will be about half the size of the original.  Thus, in
one step we have reduced the problem of searching a tree of size n to
searching a tree of size n/2.  Since the size of the tree is halved at
each step, we should expect that the number of steps needed to search a
tree of size n grows as [theta](`log' n).(2) For large sets, this will
be a significant speedup over the previous representations.

   We can represent trees by using lists.  Each node will be a list of
three items: the entry at the node, the left subtree, and the right
subtree.  A left or a right subtree of the empty list will indicate
that there is no subtree connected there.  We can describe this
representation by the following procedures:(3)

     (define (entry tree) (car tree))

     (define (left-branch tree) (cadr tree))

     (define (right-branch tree) (caddr tree))

     (define (make-tree entry left right)
       (list entry left right))

   Now we can write the `element-of-set?' procedure using the strategy
described above:

     (define (element-of-set? x set)
       (cond ((null? set) false)
             ((= x (entry set)) true)
             ((< x (entry set))
              (element-of-set? x (left-branch set)))
             ((> x (entry set))
              (element-of-set? x (right-branch set)))))

   Adjoining an item to a set is implemented similarly and also requires
[theta](`log' n) steps.  To adjoin an item `x', we compare `x' with the
node entry to determine whether `x' should be added to the right or to
the left branch, and having adjoined `x' to the appropriate branch we
piece this newly constructed branch together with the original entry
and the other branch.  If `x' is equal to the entry, we just return the
node.  If we are asked to adjoin `x' to an empty tree, we generate a
tree that has `x' as the entry and empty right and left branches.  Here
is the procedure:

     (define (adjoin-set x set)
       (cond ((null? set) (make-tree x '() '()))
             ((= x (entry set)) set)
             ((< x (entry set))
              (make-tree (entry set)
                         (adjoin-set x (left-branch set))
                         (right-branch set)))
             ((> x (entry set))
              (make-tree (entry set)
                         (left-branch set)
                         (adjoin-set x (right-branch set))))))

   The above claim that searching the tree can be performed in a
logarithmic number of steps rests on the assumption that the tree is
"balanced," i.e., that the left and the right subtree of every tree
have approximately the same number of elements, so that each subtree
contains about half the elements of its parent.  But how can we be
certain that the trees we construct will be balanced?  Even if we start
with a balanced tree, adding elements with `adjoin-set' may produce an
unbalanced result.  Since the position of a newly adjoined element
depends on how the element compares with the items already in the set,
we can expect that if we add elements "randomly" the tree will tend to
be balanced on the average.  But this is not a guarantee.  For example,
if we start with an empty set and adjoin the numbers 1 through 7 in
sequence we end up with the highly unbalanced tree shown in *Note
Figure 2-17::.  In this tree all the left subtrees are empty, so it has
no advantage over a simple ordered list.  One way to solve this problem
is to define an operation that transforms an arbitrary tree into a
balanced tree with the same elements.  Then we can perform this
transformation after every few `adjoin-set' operations to keep our set
in balance.  There are also other ways to solve this problem, most of
which involve designing new data structures for which searching and
insertion both can be done in [theta](`log' n) steps.(4)

     *Figure 2.17:* Unbalanced tree produced by adjoining 1 through 7
     in sequence.

          1
           \
            2
             \
              4
               \
                5
                 \
                  6
                   \
                    7

     *Exercise 2.63:* Each of the following two procedures converts a
     binary tree to a list.

          (define (tree->list-1 tree)
            (if (null? tree)
                '()
                (append (tree->list-1 (left-branch tree))
                        (cons (entry tree)
                              (tree->list-1 (right-branch tree))))))

          (define (tree->list-2 tree)
            (define (copy-to-list tree result-list)
              (if (null? tree)
                  result-list
                  (copy-to-list (left-branch tree)
                                (cons (entry tree)
                                      (copy-to-list (right-branch tree)
                                                    result-list)))))
            (copy-to-list tree '()))

       a. Do the two procedures produce the same result for every tree?
          If not, how do the results differ?  What lists do the two
          procedures produce for the trees in *Note Figure 2-16::?

       b. Do the two procedures have the same order of growth in the
          number of steps required to convert a balanced tree with n
          elements to a list?  If not, which one grows more slowly?


     *Exercise 2.64:* The following procedure `list->tree' converts an
     ordered list to a balanced binary tree.  The helper procedure
     `partial-tree' takes as arguments an integer n and list of at
     least n elements and constructs a balanced tree containing the
     first n elements of the list.  The result returned by
     `partial-tree' is a pair (formed with `cons') whose `car' is the
     constructed tree and whose `cdr' is the list of elements not
     included in the tree.

          (define (list->tree elements)
            (car (partial-tree elements (length elements))))

          (define (partial-tree elts n)
            (if (= n 0)
                (cons '() elts)
                (let ((left-size (quotient (- n 1) 2)))
                  (let ((left-result (partial-tree elts left-size)))
                    (let ((left-tree (car left-result))
                          (non-left-elts (cdr left-result))
                          (right-size (- n (+ left-size 1))))
                      (let ((this-entry (car non-left-elts))
                            (right-result (partial-tree (cdr non-left-elts)
                                                        right-size)))
                        (let ((right-tree (car right-result))
                              (remaining-elts (cdr right-result)))
                          (cons (make-tree this-entry left-tree right-tree)
                                remaining-elts))))))))

       a. Write a short paragraph explaining as clearly as you can how
          `partial-tree' works.  Draw the tree produced by `list->tree'
          for the list `(1 3 5 7 9 11)'.

       b. What is the order of growth in the number of steps required by
          `list->tree' to convert a list of n elements?


     *Exercise 2.65:* Use the results of *Note Exercise 2-63:: and
     *Note Exercise 2-64:: to give [theta](n) implementations of
     `union-set' and `intersection-set' for sets implemented as
     (balanced) binary trees.(5)

Sets and information retrieval
..............................

We have examined options for using lists to represent sets and have
seen how the choice of representation for a data object can have a
large impact on the performance of the programs that use the data.
Another reason for concentrating on sets is that the techniques
discussed here appear again and again in applications involving
information retrieval.

   Consider a data base containing a large number of individual
records, such as the personnel files for a company or the transactions
in an accounting system.  A typical data-management system spends a
large amount of time accessing or modifying the data in the records and
therefore requires an efficient method for accessing records.  This is
done by identifying a part of each record to serve as an identifying "key".
A key can be anything that uniquely identifies the record.  For a
personnel file, it might be an employee's ID number.  For an accounting
system, it might be a transaction number.  Whatever the key is, when we
define the record as a data structure we should include a `key'
selector procedure that retrieves the key associated with a given
record.

   Now we represent the data base as a set of records. To locate the
record with a given key we use a procedure `lookup', which takes as
arguments a key and a data base and which returns the record that has
that key, or false if there is no such record.  `Lookup' is implemented
in almost the same way as `element-of-set?'.  For example, if the set
of records is implemented as an unordered list, we could use

     (define (lookup given-key set-of-records)
       (cond ((null? set-of-records) false)
             ((equal? given-key (key (car set-of-records)))
              (car set-of-records))
             (else (lookup given-key (cdr set-of-records)))))

   Of course, there are better ways to represent large sets than as
unordered lists.  Information-retrieval systems in which records have
to be "randomly accessed" are typically implemented by a tree-based
method, such as the binary-tree representation discussed previously.
In designing such a system the methodology of data abstraction can be a
great help.  The designer can create an initial implementation using a
simple, straightforward representation such as unordered lists.  This
will be unsuitable for the eventual system, but it can be useful in
providing a "quick and dirty" data base with which to test the rest of
the system.  Later on, the data representation can be modified to be
more sophisticated.  If the data base is accessed in terms of abstract
selectors and constructors, this change in representation will not
require any changes to the rest of the system.

     *Exercise 2.66:* Implement the `lookup' procedure for the case
     where the set of records is structured as a binary tree, ordered
     by the numerical values of the keys.

   ---------- Footnotes ----------

   (1) If we want to be more formal, we can specify "consistent with
the interpretations given above" to mean that the operations satisfy a
collection of rules such as these:

   * For any set `S' and any object `x', `(element-of-set? x
     (adjoin-set x S))' is true (informally: "Adjoining an object to a
     set produces a set that contains the object").

   * For any sets `S' and `T' and any object `x', `(element-of-set? x
     (union-set S T))' is equal to `(or (element-of-set? x S)
     (element-of-set? x T))' (informally: "The elements of `(union S
     T)' are the elements that are in `S' or in `T'").

   * For any object `x', `(element-of-set? x '())' is false
     (informally: "No object is an element of the empty set").


   (2) Halving the size of the problem at each step is the
distinguishing characteristic of logarithmic growth, as we saw with the
fast-exponentiation algorithm of section *Note 1-2-4:: and the
half-interval search method of section *Note 1-3-3::.

   (3) We are representing sets in terms of trees, and trees in terms
of lists--in effect, a data abstraction built upon a data abstraction.
We can regard the procedures `entry', `left-branch', `right-branch',
and `make-tree' as a way of isolating the abstraction of a "binary
tree" from the particular way we might wish to represent such a tree in
terms of list structure.

   (4) Examples of such structures include "B-trees" and "red-black
trees".  There is a large literature on data structures devoted to this
problem.  See Cormen, Leiserson, and Rivest 1990.

   (5) *Note Exercise 2-63:: through *Note Exercise 2-65:: are due to
Paul Hilfinger.


File: sicp.info,  Node: 2-3-4,  Prev: 2-3-3,  Up: 2-3

2.3.4 Example: Huffman Encoding Trees
-------------------------------------

This section provides practice in the use of list structure and data
abstraction to manipulate sets and trees.  The application is to
methods for representing data as sequences of ones and zeros (bits).
For example, the ASCII standard code used to represent text in
computers encodes each character as a sequence of seven bits.  Using
seven bits allows us to distinguish 2^(7), or 128, possible different
characters.  In general, if we want to distinguish n different symbols,
we will need to use `log'_2 n bits per symbol.  If all our messages are
made up of the eight symbols A, B, C, D, E, F, G, and H, we can choose
a code with three bits per character, for example

     A 000 C 010 E 100 G 110
     B 001 D 011 F 101 H 111

With this code, the message

   BACADAEAFABBAAAGAH

is encoded as the string of 54 bits

   001000010000011000100000101000001001000000000110000111

   Codes such as ASCII and the A-through-H code above are known as "fixed-length"
codes, because they represent each symbol in the message with the same
number of bits.  It is sometimes advantageous to use "variable-length"
codes, in which different symbols may be represented by different
numbers of bits.  For example, Morse code does not use the same number
of dots and dashes for each letter of the alphabet.  In particular, E,
the most frequent letter, is represented by a single dot.  In general,
if our messages are such that some symbols appear very frequently and
some very rarely, we can encode data more efficiently (i.e., using
fewer bits per message) if we assign shorter codes to the frequent
symbols.  Consider the following alternative code for the letters A
through H:

     A 0   C 1010  E 1100  G 1110
     B 100 D 1011  F 1101  H 1111

With this code, the same message as above is encoded as the string

   100010100101101100011010100100000111001111

   This string contains 42 bits, so it saves more than 20% in space in
comparison with the fixed-length code shown above.

   One of the difficulties of using a variable-length code is knowing
when you have reached the end of a symbol in reading a sequence of
zeros and ones.  Morse code solves this problem by using a special "separator
code" (in this case, a pause) after the sequence of dots and dashes for
each letter.  Another solution is to design the code in such a way that
no complete code for any symbol is the beginning (or "prefix") of the
code for another symbol.  Such a code is called a "prefix code".  In
the example above, A is encoded by 0 and B is encoded by 100, so no
other symbol can have a code that begins with 0 or with 100.

   In general, we can attain significant savings if we use
variable-length prefix codes that take advantage of the relative
frequencies of the symbols in the messages to be encoded.  One
particular scheme for doing this is called the Huffman encoding method,
after its discoverer, David Huffman.  A Huffman code can be represented
as a binary tree whose leaves are the symbols that are encoded.  At
each non-leaf node of the tree there is a set containing all the
symbols in the leaves that lie below the node.  In addition, each
symbol at a leaf is assigned a weight (which is its relative
frequency), and each non-leaf node contains a weight that is the sum of
all the weights of the leaves lying below it.  The weights are not used
in the encoding or the decoding process.  We will see below how they
are used to help construct the tree.

     *Figure 2.18:* A Huffman encoding tree.

                     {A B C D E F G H} 17
                              *
                             / \
                            /   \
                          A 8    * {B C D E F G H} 9
                      __________/ \_____________
                     /                          \
          {B C D} 5 *                            * {E F G H} 4
                   / \                       ___/ \___
                  /   \                     /         \
                B 3    * {C D} 2   {E F} 2 *           * {G H} 2
                      / \                 / \         / \
                     /   \               /   \       /   \
                   C 1   D 1           E 1   F 1   G 1   H 1

   *Note Figure 2-18:: shows the Huffman tree for the A-through-H code
given above.  The weights at the leaves indicate that the tree was
designed for messages in which A appears with relative frequency 8, B
with relative frequency 3, and the other letters each with relative
frequency 1.

   Given a Huffman tree, we can find the encoding of any symbol by
starting at the root and moving down until we reach the leaf that holds
the symbol.  Each time we move down a left branch we add a 0 to the
code, and each time we move down a right branch we add a 1.  (We decide
which branch to follow by testing to see which branch either is the
leaf node for the symbol or contains the symbol in its set.)  For
example, starting from the root of the tree in *Note Figure 2-18::, we
arrive at the leaf for D by following a right branch, then a left
branch, then a right branch, then a right branch; hence, the code for D
is 1011.

   To decode a bit sequence using a Huffman tree, we begin at the root
and use the successive zeros and ones of the bit sequence to determine
whether to move down the left or the right branch.  Each time we come
to a leaf, we have generated a new symbol in the message, at which
point we start over from the root of the tree to find the next symbol.
For example, suppose we are given the tree above and the sequence
10001010.  Starting at the root, we move down the right branch, (since
the first bit of the string is 1), then down the left branch (since the
second bit is 0), then down the left branch (since the third bit is
also 0).  This brings us to the leaf for B, so the first symbol of the
decoded message is B.  Now we start again at the root, and we make a
left move because the next bit in the string is 0.  This brings us to
the leaf for A.  Then we start again at the root with the rest of the
string 1010, so we move right, left, right, left and reach C.  Thus,
the entire message is BAC.

Generating Huffman trees
........................

Given an "alphabet" of symbols and their relative frequencies, how do we
construct the "best" code?  (In other words, which tree will encode
messages with the fewest bits?)  Huffman gave an algorithm for doing
this and showed that the resulting code is indeed the best
variable-length code for messages where the relative frequency of the
symbols matches the frequencies with which the code was constructed.
We will not prove this optimality of Huffman codes here, but we will
show how Huffman trees are constructed.(1)

   The algorithm for generating a Huffman tree is very simple. The idea
is to arrange the tree so that the symbols with the lowest frequency
appear farthest away from the root. Begin with the set of leaf nodes,
containing symbols and their frequencies, as determined by the initial
data from which the code is to be constructed. Now find two leaves with
the lowest weights and merge them to produce a node that has these two
nodes as its left and right branches. The weight of the new node is the
sum of the two weights. Remove the two leaves from the original set and
replace them by this new node. Now continue this process. At each step,
merge two nodes with the smallest weights, removing them from the set
and replacing them with a node that has these two as its left and right
branches. The process stops when there is only one node left, which is
the root of the entire tree.  Here is how the Huffman tree of *Note
Figure 2-18:: was generated:

     Initial leaves {(A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (G 1) (H 1)}
     Merge          {(A 8) (B 3) ({C D} 2) (E 1) (F 1) (G 1) (H 1)}
     Merge          {(A 8) (B 3) ({C D} 2) ({E F} 2) (G 1) (H 1)}
     Merge          {(A 8) (B 3) ({C D} 2) ({E F} 2) ({G H} 2)}
     Merge          {(A 8) (B 3) ({C D} 2) ({E F G H} 4)}
     Merge          {(A 8) ({B C D} 5) ({E F G H} 4)}
     Merge          {(A 8) ({B C D E F G H} 9)}
     Final merge    {({A B C D E F G H} 17)}

   The algorithm does not always specify a unique tree, because there
may not be unique smallest-weight nodes at each step.  Also, the choice
of the order in which the two nodes are merged (i.e., which will be the
right branch and which will be the left branch) is arbitrary.

Representing Huffman trees
..........................

In the exercises below we will work with a system that uses Huffman
trees to encode and decode messages and generates Huffman trees
according to the algorithm outlined above.  We will begin by discussing
how trees are represented.

   Leaves of the tree are represented by a list consisting of the symbol
`leaf', the symbol at the leaf, and the weight:

     (define (make-leaf symbol weight)
       (list 'leaf symbol weight))

     (define (leaf? object)
       (eq? (car object) 'leaf))

     (define (symbol-leaf x) (cadr x))

     (define (weight-leaf x) (caddr x))

   A general tree will be a list of a left branch, a right branch, a
set of symbols, and a weight.  The set of symbols will be simply a list
of the symbols, rather than some more sophisticated set representation.
When we make a tree by merging two nodes, we obtain the weight of the
tree as the sum of the weights of the nodes, and the set of symbols as
the union of the sets of symbols for the nodes.  Since our symbol sets
are represented as lists, we can form the union by using the `append'
procedure we defined in section *Note 2-2-1:::

     (define (make-code-tree left right)
       (list left
             right
             (append (symbols left) (symbols right))
             (+ (weight left) (weight right))))

   If we make a tree in this way, we have the following selectors:

     (define (left-branch tree) (car tree))

     (define (right-branch tree) (cadr tree))

     (define (symbols tree)
       (if (leaf? tree)
           (list (symbol-leaf tree))
           (caddr tree)))

     (define (weight tree)
       (if (leaf? tree)
           (weight-leaf tree)
           (cadddr tree)))

   The procedures `symbols' and `weight' must do something slightly
different depending on whether they are called with a leaf or a general
tree.  These are simple examples of "generic procedures" (procedures
that can handle more than one kind of data), which we will have much
more to say about in sections *Note 2-4:: and *Note 2-5::.

The decoding procedure
......................

The following procedure implements the decoding algorithm.  It takes as
arguments a list of zeros and ones, together with a Huffman tree.

     (define (decode bits tree)
       (define (decode-1 bits current-branch)
         (if (null? bits)
             '()
             (let ((next-branch
                    (choose-branch (car bits) current-branch)))
               (if (leaf? next-branch)
                   (cons (symbol-leaf next-branch)
                         (decode-1 (cdr bits) tree))
                   (decode-1 (cdr bits) next-branch)))))
       (decode-1 bits tree))

     (define (choose-branch bit branch)
       (cond ((= bit 0) (left-branch branch))
             ((= bit 1) (right-branch branch))
             (else (error "bad bit -- CHOOSE-BRANCH" bit))))

   The procedure `decode-1' takes two arguments: the list of remaining
bits and the current position in the tree.  It keeps moving "down" the
tree, choosing a left or a right branch according to whether the next
bit in the list is a zero or a one.  (This is done with the procedure
`choose-branch'.)  When it reaches a leaf, it returns the symbol at
that leaf as the next symbol in the message by `cons'ing it onto the
result of decoding the rest of the message, starting at the root of the
tree.  Note the error check in the final clause of `choose-branch',
which complains if the procedure finds something other than a zero or a
one in the input data.

Sets of weighted elements
.........................

In our representation of trees, each non-leaf node contains a set of
symbols, which we have represented as a simple list.  However, the
tree-generating algorithm discussed above requires that we also work
with sets of leaves and trees, successively merging the two smallest
items.  Since we will be required to repeatedly find the smallest item
in a set, it is convenient to use an ordered representation for this
kind of set.

   We will represent a set of leaves and trees as a list of elements,
arranged in increasing order of weight.  The following `adjoin-set'
procedure for constructing sets is similar to the one described in
*Note Exercise 2-61::; however, items are compared by their weights,
and the element being added to the set is never already in it.

     (define (adjoin-set x set)
       (cond ((null? set) (list x))
             ((< (weight x) (weight (car set))) (cons x set))
             (else (cons (car set)
                         (adjoin-set x (cdr set))))))

   The following procedure takes a list of symbol-frequency pairs such
as `((A 4) (B 2) (C 1) (D 1))' and constructs an initial ordered set of
leaves, ready to be merged according to the Huffman algorithm:

     (define (make-leaf-set pairs)
       (if (null? pairs)
           '()
           (let ((pair (car pairs)))
             (adjoin-set (make-leaf (car pair)    ; symbol
                                    (cadr pair))  ; frequency
                         (make-leaf-set (cdr pairs))))))

     *Exercise 2.67:* Define an encoding tree and a sample message:

          (define sample-tree
            (make-code-tree (make-leaf 'A 4)
                            (make-code-tree
                             (make-leaf 'B 2)
                             (make-code-tree (make-leaf 'D 1)
                                             (make-leaf 'C 1)))))

          (define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))

     Use the `decode' procedure to decode the message, and give the
     result.

     *Exercise 2.68:* The `encode' procedure takes as arguments a
     message and a tree and produces the list of bits that gives the
     encoded message.

          (define (encode message tree)
            (if (null? message)
                '()
                (append (encode-symbol (car message) tree)
                        (encode (cdr message) tree))))

     `Encode-symbol' is a procedure, which you must write, that returns
     the list of bits that encodes a given symbol according to a given
     tree.  You should design `encode-symbol' so that it signals an
     error if the symbol is not in the tree at all.  Test your
     procedure by encoding the result you obtained in *Note Exercise
     2-67:: with the sample tree and seeing whether it is the same as
     the original sample message.

     *Exercise 2.69:* The following procedure takes as its argument a
     list of symbol-frequency pairs (where no symbol appears in more
     than one pair) and generates a Huffman encoding tree according to
     the Huffman algorithm.

          (define (generate-huffman-tree pairs)
            (successive-merge (make-leaf-set pairs)))

     `Make-leaf-set' is the procedure given above that transforms the
     list of pairs into an ordered set of leaves.  `Successive-merge'
     is the procedure you must write, using `make-code-tree' to
     successively merge the smallest-weight elements of the set until
     there is only one element left, which is the desired Huffman tree.
     (This procedure is slightly tricky, but not really complicated.
     If you find yourself designing a complex procedure, then you are
     almost certainly doing something wrong.  You can take significant
     advantage of the fact that we are using an ordered set
     representation.)

     *Exercise 2.70:* The following eight-symbol alphabet with
     associated relative frequencies was designed to efficiently encode
     the lyrics of 1950s rock songs.  (Note that the "symbols" of an
     "alphabet" need not be individual letters.)

          A     2 NA   16
          BOOM  1 SHA  3
          GET   2 YIP  9
          JOB   2 WAH  1

     Use `generate-huffman-tree' (*Note Exercise 2-69::) to generate a
     corresponding Huffman tree, and use `encode' (*Note Exercise
     2-68::) to encode the following message:

          Get a job

          Sha na na na na na na na na

          Get a job

          Sha na na na na na na na na

          Wah yip yip yip yip yip yip yip yip yip

          Sha boom

     How many bits are required for the encoding?  What is the smallest
     number of bits that would be needed to encode this song if we used
     a fixed-length code for the eight-symbol alphabet?

     *Exercise 2.71:* Suppose we have a Huffman tree for an alphabet of
     n symbols, and that the relative frequencies of the symbols are 1,
     2, 4, ..., 2^(n-1).  Sketch the tree for n=5; for n=10.  In such a
     tree (for general n) how may bits are required to encode the most
     frequent symbol?  the least frequent symbol?

     *Exercise 2.72:* Consider the encoding procedure that you designed
     in *Note Exercise 2-68::.  What is the order of growth in the
     number of steps needed to encode a symbol?  Be sure to include the
     number of steps needed to search the symbol list at each node
     encountered.  To answer this question in general is difficult.
     Consider the special case where the relative frequencies of the n
     symbols are as described in *Note Exercise 2-71::, and give the
     order of growth (as a function of n) of the number of steps needed
     to encode the most frequent and least frequent symbols in the
     alphabet.

   ---------- Footnotes ----------

   (1) See Hamming 1980 for a discussion of the mathematical properties
of Huffman codes.


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2.4 Multiple Representations for Abstract Data
==============================================

We have introduced data abstraction, a methodology for structuring
systems in such a way that much of a program can be specified
independent of the choices involved in implementing the data objects
that the program manipulates.  For example, we saw in section *Note
2-1-1:: how to separate the task of designing a program that uses
rational numbers from the task of implementing rational numbers in
terms of the computer language's primitive mechanisms for constructing
compound data.  The key idea was to erect an abstraction barrier - in
this case, the selectors and constructors for rational numbers
(`make-rat', `numer', `denom')--that isolates the way rational numbers
are used from their underlying representation in terms of list
structure.  A similar abstraction barrier isolates the details of the
procedures that perform rational arithmetic (`add-rat', `sub-rat',
`mul-rat', and `div-rat') from the "higher-level" procedures that use
rational numbers.  The resulting program has the structure shown in
*Note Figure 2-1::.

   These data-abstraction barriers are powerful tools for controlling
complexity.  By isolating the underlying representations of data
objects, we can divide the task of designing a large program into
smaller tasks that can be performed separately.  But this kind of data
abstraction is not yet powerful enough, because it may not always make
sense to speak of "the underlying representation" for a data object.

   For one thing, there might be more than one useful representation
for a data object, and we might like to design systems that can deal
with multiple representations.  To take a simple example, complex
numbers may be represented in two almost equivalent ways: in
rectangular form (real and imaginary parts) and in polar form
(magnitude and angle).  Sometimes rectangular form is more appropriate
and sometimes polar form is more appropriate.  Indeed, it is perfectly
plausible to imagine a system in which complex numbers are represented
in both ways, and in which the procedures for manipulating complex
numbers work with either representation.

   More importantly, programming systems are often designed by many
people working over extended periods of time, subject to requirements
that change over time.  In such an environment, it is simply not
possible for everyone to agree in advance on choices of data
representation.  So in addition to the data-abstraction barriers that
isolate representation from use, we need abstraction barriers that
isolate different design choices from each other and permit different
choices to coexist in a single program.  Furthermore, since large
programs are often created by combining pre-existing modules that were
designed in isolation, we need conventions that permit programmers to
incorporate modules into larger systems "additively", that is, without
having to redesign or reimplement these modules.

   In this section, we will learn how to cope with data that may be
represented in different ways by different parts of a program.  This
requires constructing "generic procedures"--procedures that can operate
on data that may be represented in more than one way.  Our main
technique for building generic procedures will be to work in terms of
data objects that have tags "type tags", that is, data objects that
include explicit information about how they are to be processed.  We
will also discuss "data-directed" programming, a powerful and
convenient implementation strategy for additively assembling systems
with generic operations.

   We begin with the simple complex-number example. We will see how
type tags and data-directed style enable us to design separate
rectangular and polar representations for complex numbers while
maintaining the notion of an abstract "complex-number" data object.  We
will accomplish this by defining arithmetic procedures for complex
numbers (`add-complex', `sub-complex', `mul-complex', and
`div-complex') in terms of generic selectors that access parts of a
complex number independent of how the number is represented.  The
resulting complex-number system, as shown in *Note Figure 2-19::,
contains two different kinds of abstraction barriers.  The "horizontal"
abstraction barriers play the same role as the ones in *Note Figure
2-1::.  They isolate "higher-level" operations from "lower-level"
representations.  In addition, there is a "vertical" barrier that gives
us the ability to separately design and install alternative
representations.

     *Figure 2.19:* Data-abstraction barriers in the complex-number
     system.

                     Programs that use complex numbers
            +-------------------------------------------------+
          --| add-complex sub-complex mul-complex div-complex |--
            +-------------------------------------------------+
                        Complex arithmetic package
          ---------------------------+---------------------------
                    Rectangular      |         Polar
                  representation     |     representation
          ---------------------------+---------------------------
              List structure and primitive machine arithmetic

   In section *Note 2-5:: we will show how to use type tags and
data-directed style to develop a generic arithmetic package.  This
provides procedures (`add', `mul', and so on) that can be used to
manipulate all sorts of "numbers" and can be easily extended when a new
kind of number is needed.  In section *Note 2-5-3::, we'll show how to
use generic arithmetic in a system that performs symbolic algebra.

* Menu:

* 2-4-1::            Representations for Complex Numbers
* 2-4-2::            Tagged data
* 2-4-3::            Data-Directed Programming and Additivity


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2.4.1 Representations for Complex Numbers
-----------------------------------------

We will develop a system that performs arithmetic operations on complex
numbers as a simple but unrealistic example of a program that uses
generic operations.  We begin by discussing two plausible
representations for complex numbers as ordered pairs: rectangular form
(real part and imaginary part) and polar form (magnitude and angle).(1)
Section *Note 2-4-2:: will show how both representations can be made
to coexist in a single system through the use of type tags and generic
operations.

   Like rational numbers, complex numbers are naturally represented as
ordered pairs.  The set of complex numbers can be thought of as a
two-dimensional space with two orthogonal axes, the "real" axis and the
"imaginary" axis. (See *Note Figure 2-20::.)  From this point of view,
the complex number z = x + iy (where i^2 = - 1) can be thought of as
the point in the plane whose real coordinate is x and whose imaginary
coordinate is y.  Addition of complex numbers reduces in this
representation to addition of coordinates:

     Real-part(z_1 + z_2) = Real-part(z_1) + Real-part(z_2)

     Imaginary-part(z_1 + z_2) = Imaginary-part(z_1) + Imaginary-part(z_2)

   When multiplying complex numbers, it is more natural to think in
terms of representing a complex number in polar form, as a magnitude
and an angle (r and A in *Note Figure 2-20::).  The product of two
complex numbers is the vector obtained by stretching one complex number
by the length of the other and then rotating it through the angle of
the other:

     Magnitude(z_1 * z_2) = Magnitude(z_1) * Magnitude(z_2)

     Angle(z_1 * z_2) = Angle(z_1) + Angle(z_2)

     *Figure 2.20:* Complex numbers as points in the plane.

           Imaginary
              ^
              |
            y |.........................* z = x + ?y = r e^(?A)
              |                    __-- .
              |                __--     .
              |          r __--         .
              |        __--             .
              |    __-- \               .
              |__--    A |              .
          ----+----------+-------------------> Real
                                        x

   Thus, there are two different representations for complex numbers,
which are appropriate for different operations.  Yet, from the
viewpoint of someone writing a program that uses complex numbers, the
principle of data abstraction suggests that all the operations for
manipulating complex numbers should be available regardless of which
representation is used by the computer.  For example, it is often
useful to be able to find the magnitude of a complex number that is
specified by rectangular coordinates.  Similarly, it is often useful to
be able to determine the real part of a complex number that is
specified by polar coordinates.

   To design such a system, we can follow the same data-abstraction
strategy we followed in designing the rational-number package in
section *Note 2-1-1::.  Assume that the operations on complex numbers
are implemented in terms of four selectors: `real-part', `imag-part',
`magnitude', and `angle'.  Also assume that we have two procedures for
constructing complex numbers: `make-from-real-imag' returns a complex
number with specified real and imaginary parts, and `make-from-mag-ang'
returns a complex number with specified magnitude and angle.  These
procedures have the property that, for any complex number `z', both

     (make-from-real-imag (real-part z) (imag-part z))

and

     (make-from-mag-ang (magnitude z) (angle z))

produce complex numbers that are equal to `z'.

   Using these constructors and selectors, we can implement arithmetic
on complex numbers using the "abstract data" specified by the
constructors and selectors, just as we did for rational numbers in
section *Note 2-1-1::.  As shown in the formulas above, we can add and
subtract complex numbers in terms of real and imaginary parts while
multiplying and dividing complex numbers in terms of magnitudes and
angles:

     (define (add-complex z1 z2)
       (make-from-real-imag (+ (real-part z1) (real-part z2))
                            (+ (imag-part z1) (imag-part z2))))

     (define (sub-complex z1 z2)
       (make-from-real-imag (- (real-part z1) (real-part z2))
                            (- (imag-part z1) (imag-part z2))))

     (define (mul-complex z1 z2)
       (make-from-mag-ang (* (magnitude z1) (magnitude z2))
                          (+ (angle z1) (angle z2))))

     (define (div-complex z1 z2)
       (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
                          (- (angle z1) (angle z2))))

   To complete the complex-number package, we must choose a
representation and we must implement the constructors and selectors in
terms of primitive numbers and primitive list structure.  There are two
obvious ways to do this: We can represent a complex number in
"rectangular form" as a pair (real part, imaginary part) or in "polar
form" as a pair (magnitude, angle).  Which shall we choose?

   In order to make the different choices concrete, imagine that there
are two programmers, Ben Bitdiddle and Alyssa P. Hacker, who are
independently designing representations for the complex-number system.
Ben chooses to represent complex numbers in rectangular form.  With
this choice, selecting the real and imaginary parts of a complex number
is straightforward, as is constructing a complex number with given real
and imaginary parts.  To find the magnitude and the angle, or to
construct a complex number with a given magnitude and angle, he uses
the trigonometric relations

                           __________
     x = r cos A     r = ./ x^2 + y^2

     y = r sin A     A = arctan(y,x)

which relate the real and imaginary parts (x, y) to the magnitude and
the angle (r, A).(2)  Ben's representation is therefore given by the
following selectors and constructors:

     (define (real-part z) (car z))

     (define (imag-part z) (cdr z))

     (define (magnitude z)
       (sqrt (+ (square (real-part z)) (square (imag-part z)))))

     (define (angle z)
       (atan (imag-part z) (real-part z)))

     (define (make-from-real-imag x y) (cons x y))

     (define (make-from-mag-ang r a)
       (cons (* r (cos a)) (* r (sin a))))

   Alyssa, in contrast, chooses to represent complex numbers in polar
form.  For her, selecting the magnitude and angle is straightforward,
but she has to use the trigonometric relations to obtain the real and
imaginary parts.  Alyssa's representation is:

     (define (real-part z)
       (* (magnitude z) (cos (angle z))))

     (define (imag-part z)
       (* (magnitude z) (sin (angle z))))

     (define (magnitude z) (car z))

     (define (angle z) (cdr z))

     (define (make-from-real-imag x y)
       (cons (sqrt (+ (square x) (square y)))
             (atan y x)))

     (define (make-from-mag-ang r a) (cons r a))

   The discipline of data abstraction ensures that the same
implementation of `add-complex', `sub-complex', `mul-complex', and
`div-complex' will work with either Ben's representation or Alyssa's
representation.

   ---------- Footnotes ----------

   (1) In actual computational systems, rectangular form is preferable
to polar form most of the time because of roundoff errors in conversion
between rectangular and polar form.  This is why the complex-number
example is unrealistic.  Nevertheless, it provides a clear illustration
of the design of a system using generic operations and a good
introduction to the more substantial systems to be developed later in
this chapter.

   (2) The arctangent function referred to here, computed by Scheme's
`atan' procedure, is defined so as to take two arguments y and x and to
return the angle whose tangent is y/x.  The signs of the arguments
determine the quadrant of the angle.


File: sicp.info,  Node: 2-4-2,  Next: 2-4-3,  Prev: 2-4-1,  Up: 2-4

2.4.2 Tagged data
-----------------

One way to view data abstraction is as an application of the "principle
of least commitment."  In implementing the complex-number system in
section *Note 2-4-1::, we can use either Ben's rectangular
representation or Alyssa's polar representation.  The abstraction
barrier formed by the selectors and constructors permits us to defer to
the last possible moment the choice of a concrete representation for
our data objects and thus retain maximum flexibility in our system
design.

   The principle of least commitment can be carried to even further
extremes.  If we desire, we can maintain the ambiguity of
representation even _after_ we have designed the selectors and
constructors, and elect to use both Ben's representation _and_ Alyssa's
representation.  If both representations are included in a single
system, however, we will need some way to distinguish data in polar
form from data in rectangular form.  Otherwise, if we were asked, for
instance, to find the `magnitude' of the pair (3,4), we wouldn't know
whether to answer 5 (interpreting the number in rectangular form) or 3
(interpreting the number in polar form).  A straightforward way to
accomplish this distinction is to include a "type tag"--the symbol
`rectangular' or `polar'--as part of each complex number.  Then when we
need to manipulate a complex number we can use the tag to decide which
selector to apply.

   In order to manipulate tagged data, we will assume that we have
procedures `type-tag' and `contents' that extract from a data object
the tag and the actual contents (the polar or rectangular coordinates,
in the case of a complex number).  We will also postulate a procedure
`attach-tag' that takes a tag and contents and produces a tagged data
object.  A straightforward way to implement this is to use ordinary
list structure:

     (define (attach-tag type-tag contents)
       (cons type-tag contents))

     (define (type-tag datum)
       (if (pair? datum)
           (car datum)
           (error "Bad tagged datum -- TYPE-TAG" datum)))

     (define (contents datum)
       (if (pair? datum)
           (cdr datum)
           (error "Bad tagged datum -- CONTENTS" datum)))

   Using these procedures, we can define predicates `rectangular?'  and
`polar?', which recognize polar and rectangular numbers, respectively:

     (define (rectangular? z)
       (eq? (type-tag z) 'rectangular))

     (define (polar? z)
       (eq? (type-tag z) 'polar))

   With type tags, Ben and Alyssa can now modify their code so that
their two different representations can coexist in the same system.
Whenever Ben constructs a complex number, he tags it as rectangular.
Whenever Alyssa constructs a complex number, she tags it as polar.  In
addition, Ben and Alyssa must make sure that the names of their
procedures do not conflict.  One way to do this is for Ben to append
the suffix `rectangular' to the name of each of his representation
procedures and for Alyssa to append `polar' to the names of hers.  Here
is Ben's revised rectangular representation from section *Note 2-4-1:::

     (define (real-part-rectangular z) (car z))

     (define (imag-part-rectangular z) (cdr z))

     (define (magnitude-rectangular z)
       (sqrt (+ (square (real-part-rectangular z))
                (square (imag-part-rectangular z)))))

     (define (angle-rectangular z)
       (atan (imag-part-rectangular z)
             (real-part-rectangular z)))

     (define (make-from-real-imag-rectangular x y)
       (attach-tag 'rectangular (cons x y)))

     (define (make-from-mag-ang-rectangular r a)
       (attach-tag 'rectangular
                   (cons (* r (cos a)) (* r (sin a)))))

and here is Alyssa's revised polar representation:

     (define (real-part-polar z)
       (* (magnitude-polar z) (cos (angle-polar z))))

     (define (imag-part-polar z)
       (* (magnitude-polar z) (sin (angle-polar z))))

     (define (magnitude-polar z) (car z))

     (define (angle-polar z) (cdr z))

     (define (make-from-real-imag-polar x y)
       (attach-tag 'polar
                    (cons (sqrt (+ (square x) (square y)))
                          (atan y x))))

     (define (make-from-mag-ang-polar r a)
       (attach-tag 'polar (cons r a)))

   Each generic selector is implemented as a procedure that checks the
tag of its argument and calls the appropriate procedure for handling
data of that type.  For example, to obtain the real part of a complex
number, `real-part' examines the tag to determine whether to use Ben's
`real-part-rectangular' or Alyssa's `real-part-polar'.  In either case,
we use `contents' to extract the bare, untagged datum and send this to
the rectangular or polar procedure as required:

     (define (real-part z)
       (cond ((rectangular? z)
              (real-part-rectangular (contents z)))
             ((polar? z)
              (real-part-polar (contents z)))
             (else (error "Unknown type -- REAL-PART" z))))

     (define (imag-part z)
       (cond ((rectangular? z)
              (imag-part-rectangular (contents z)))
             ((polar? z)
              (imag-part-polar (contents z)))
             (else (error "Unknown type -- IMAG-PART" z))))

     (define (magnitude z)
       (cond ((rectangular? z)
              (magnitude-rectangular (contents z)))
             ((polar? z)
              (magnitude-polar (contents z)))
             (else (error "Unknown type -- MAGNITUDE" z))))

     (define (angle z)
       (cond ((rectangular? z)
              (angle-rectangular (contents z)))
             ((polar? z)
              (angle-polar (contents z)))
             (else (error "Unknown type -- ANGLE" z))))

   To implement the complex-number arithmetic operations, we can use
the same procedures `add-complex', `sub-complex', `mul-complex', and
`div-complex' from section *Note 2-4-1::, because the selectors they
call are generic, and so will work with either representation.  For
example, the procedure `add-complex' is still

     (define (add-complex z1 z2)
       (make-from-real-imag (+ (real-part z1) (real-part z2))
                            (+ (imag-part z1) (imag-part z2))))

   Finally, we must choose whether to construct complex numbers using
Ben's representation or Alyssa's representation.  One reasonable choice
is to construct rectangular numbers whenever we have real and imaginary
parts and to construct polar numbers whenever we have magnitudes and
angles:

     (define (make-from-real-imag x y)
       (make-from-real-imag-rectangular x y))

     (define (make-from-mag-ang r a)
       (make-from-mag-ang-polar r a))

     *Figure 2.21:* Structure of the generic complex-arithmetic system.

              +--------------------------------------------------+
          ----| add-complex sub-complex mul-complex- div-complex |----
              +--------------------------------------------------+
                          Complex arithmetic package
                           +-----------------------+
                           | real-part   imag-part |
          -----------------|                       |------------------
                           | magnitude   angle     |
                           +-----------+-----------+
                     Rectangular       |          Polar
                    representation     |     representation
          -----------------------------+------------------------------
                 List structure and primitive machine arithmetic

   The resulting complex-number system has the structure shown in *Note
Figure 2-21::.  The system has been decomposed into three relatively
independent parts: the complex-number-arithmetic operations, Alyssa's
polar implementation, and Ben's rectangular implementation.  The polar
and rectangular implementations could have been written by Ben and
Alyssa working separately, and both of these can be used as underlying
representations by a third programmer implementing the
complex-arithmetic procedures in terms of the abstract
constructor/selector interface.

   Since each data object is tagged with its type, the selectors
operate on the data in a generic manner.  That is, each selector is
defined to have a behavior that depends upon the particular type of
data it is applied to.  Notice the general mechanism for interfacing
the separate representations: Within a given representation
implementation (say, Alyssa's polar package) a complex number is an
untyped pair (magnitude, angle).  When a generic selector operates on a
number of `polar' type, it strips off the tag and passes the contents on
to Alyssa's code.  Conversely, when Alyssa constructs a number for
general use, she tags it with a type so that it can be appropriately
recognized by the higher-level procedures.  This discipline of
stripping off and attaching tags as data objects are passed from level
to level can be an important organizational strategy, as we shall see
in section *Note 2-5::.


File: sicp.info,  Node: 2-4-3,  Prev: 2-4-2,  Up: 2-4

2.4.3 Data-Directed Programming and Additivity
----------------------------------------------

The general strategy of checking the type of a datum and calling an
appropriate procedure is called "dispatching on type".  This is a
powerful strategy for obtaining modularity in system design.  Oh the
other hand, implementing the dispatch as in section *Note 2-4-2:: has
two significant weaknesses.  One weakness is that the generic interface
procedures (`real-part', `imag-part', `magnitude', and `angle') must
know about all the different representations.  For instance, suppose we
wanted to incorporate a new representation for complex numbers into our
complex-number system.  We would need to identify this new
representation with a type, and then add a clause to each of the
generic interface procedures to check for the new type and apply the
appropriate selector for that representation.

   Another weakness of the technique is that even though the individual
representations can be designed separately, we must guarantee that no
two procedures in the entire system have the same name.  This is why
Ben and Alyssa had to change the names of their original procedures
from section *Note 2-4-1::.

   The issue underlying both of these weaknesses is that the technique
for implementing generic interfaces is not "additive".  The person
implementing the generic selector procedures must modify those
procedures each time a new representation is installed, and the people
interfacing the individual representations must modify their code to
avoid name conflicts.  In each of these cases, the changes that must be
made to the code are straightforward, but they must be made
nonetheless, and this is a source of inconvenience and error.  This is
not much of a problem for the complex-number system as it stands, but
suppose there were not two but hundreds of different representations
for complex numbers.  And suppose that there were many generic
selectors to be maintained in the abstract-data interface.  Suppose, in
fact, that no one programmer knew all the interface procedures or all
the representations.  The problem is real and must be addressed in such
programs as large-scale data-base-management systems.

   What we need is a means for modularizing the system design even
further.  This is provided by the programming technique known as programming
"data-directed programming".  To understand how data-directed
programming works, begin with the observation that whenever we deal
with a set of generic operations that are common to a set of different
types we are, in effect, dealing with a two-dimensional table that
contains the possible operations on one axis and the possible types on
the other axis.  The entries in the table are the procedures that
implement each operation for each type of argument presented.  In the
complex-number system developed in the previous section, the
correspondence between operation name, data type, and actual procedure
was spread out among the various conditional clauses in the generic
interface procedures.  But the same information could have been
organized in a table, as shown in *Note Figure 2-22::.

   Data-directed programming is the technique of designing programs to
work with such a table directly.  Previously, we implemented the
mechanism that interfaces the complex-arithmetic code with the two
representation packages as a set of procedures that each perform an
explicit dispatch on type.  Here we will implement the interface as a
single procedure that looks up the combination of the operation name
and argument type in the table to find the correct procedure to apply,
and then applies it to the contents of the argument.  If we do this,
then to add a new representation package to the system we need not
change any existing procedures; we need only add new entries to the
table.

     *Figure 2.22:* Table of operations for the complex-number system.

                     |               Types
          Operations | Polar           | Rectangular
          ===========+=================+======================
          real-part  | real-part-polar | real-part-rectangular
          imag-part  | imag-part-polar | imag-part-rectangular
          magnitude  | magnitude-polar | magnitude-rectangular
          angle      | angle-polar     | angle-rectangular

   To implement this plan, assume that we have two procedures, `put' and
`get', for manipulating the operation-and-type table:

   * `(put <OP> <TYPE> <ITEM>)' installs the `<ITEM>' in the table,
     indexed by the `<OP>' and the `<TYPE>'.

   * `(get <OP> <TYPE>)' looks up the `<OP>', `<TYPE>' entry in the
     table and returns the item found there.  If no item is found,
     `get' returns false.


   For now, we can assume that `put' and `get' are included in our
language.  In *Note Chapter 3:: (section *Note 3-3-3::, *Note Exercise
3-24::) we will see how to implement these and other operations for
manipulating tables.

   Here is how data-directed programming can be used in the
complex-number system.  Ben, who developed the rectangular
representation, implements his code just as he did originally.  He
defines a collection of procedures, or a "package", and interfaces
these to the rest of the system by adding entries to the table that
tell the system how to operate on rectangular numbers.  This is
accomplished by calling the following procedure:

     (define (install-rectangular-package)
       ;; internal procedures
       (define (real-part z) (car z))
       (define (imag-part z) (cdr z))
       (define (make-from-real-imag x y) (cons x y))
       (define (magnitude z)
         (sqrt (+ (square (real-part z))
                  (square (imag-part z)))))
       (define (angle z)
         (atan (imag-part z) (real-part z)))
       (define (make-from-mag-ang r a)
         (cons (* r (cos a)) (* r (sin a))))

       ;; interface to the rest of the system
       (define (tag x) (attach-tag 'rectangular x))
       (put 'real-part '(rectangular) real-part)
       (put 'imag-part '(rectangular) imag-part)
       (put 'magnitude '(rectangular) magnitude)
       (put 'angle '(rectangular) angle)
       (put 'make-from-real-imag 'rectangular
            (lambda (x y) (tag (make-from-real-imag x y))))
       (put 'make-from-mag-ang 'rectangular
            (lambda (r a) (tag (make-from-mag-ang r a))))
       'done)

   Notice that the internal procedures here are the same procedures
from section *Note 2-4-1:: that Ben wrote when he was working in
isolation.  No changes are necessary in order to interface them to the
rest of the system.  Moreover, since these procedure definitions are
internal to the installation procedure, Ben needn't worry about name
conflicts with other procedures outside the rectangular package.  To
interface these to the rest of the system, Ben installs his `real-part'
procedure under the operation name `real-part' and the type
`(rectangular)', and similarly for the other selectors.(1)  The
interface also defines the constructors to be used by the external
system.(2)  These are identical to Ben's internally defined
constructors, except that they attach the tag.

   Alyssa's polar package is analogous:

     (define (install-polar-package)
       ;; internal procedures
       (define (magnitude z) (car z))
       (define (angle z) (cdr z))
       (define (make-from-mag-ang r a) (cons r a))
       (define (real-part z)
         (* (magnitude z) (cos (angle z))))
       (define (imag-part z)
         (* (magnitude z) (sin (angle z))))
       (define (make-from-real-imag x y)
         (cons (sqrt (+ (square x) (square y)))
               (atan y x)))

       ;; interface to the rest of the system
       (define (tag x) (attach-tag 'polar x))
       (put 'real-part '(polar) real-part)
       (put 'imag-part '(polar) imag-part)
       (put 'magnitude '(polar) magnitude)
       (put 'angle '(polar) angle)
       (put 'make-from-real-imag 'polar
            (lambda (x y) (tag (make-from-real-imag x y))))
       (put 'make-from-mag-ang 'polar
            (lambda (r a) (tag (make-from-mag-ang r a))))
       'done)

   Even though Ben and Alyssa both still use their original procedures
defined with the same names as each other's (e.g., `real-part'), these
definitions are now internal to different procedures (see section *Note
1-1-8::), so there is no name conflict.

   The complex-arithmetic selectors access the table by means of a
general "operation" procedure called `apply-generic', which applies a
generic operation to some arguments.  `Apply-generic' looks in the
table under the name of the operation and the types of the arguments
and applies the resulting procedure if one is present:(3)

     (define (apply-generic op . args)
       (let ((type-tags (map type-tag args)))
         (let ((proc (get op type-tags)))
           (if proc
               (apply proc (map contents args))
               (error
                 "No method for these types -- APPLY-GENERIC"
                 (list op type-tags))))))

   Using `apply-generic', we can define our generic selectors as
follows:

     (define (real-part z) (apply-generic 'real-part z))
     (define (imag-part z) (apply-generic 'imag-part z))
     (define (magnitude z) (apply-generic 'magnitude z))
     (define (angle z) (apply-generic 'angle z))

   Observe that these do not change at all if a new representation is
added to the system.

   We can also extract from the table the constructors to be used by
the programs external to the packages in making complex numbers from
real and imaginary parts and from magnitudes and angles.  As in section
*Note 2-4-2::, we construct rectangular numbers whenever we have real
and imaginary parts, and polar numbers whenever we have magnitudes and
angles:

     (define (make-from-real-imag x y)
       ((get 'make-from-real-imag 'rectangular) x y))

     (define (make-from-mag-ang r a)
       ((get 'make-from-mag-ang 'polar) r a))

     *Exercise 2.73:* Section *Note 2-3-2:: described a program that
     performs symbolic differentiation:

          (define (deriv exp var)
            (cond ((number? exp) 0)
                  ((variable? exp) (if (same-variable? exp var) 1 0))
                  ((sum? exp)
                   (make-sum (deriv (addend exp) var)
                             (deriv (augend exp) var)))
                  ((product? exp)
                   (make-sum
                     (make-product (multiplier exp)
                                   (deriv (multiplicand exp) var))
                     (make-product (deriv (multiplier exp) var)
                                   (multiplicand exp))))
                  <MORE RULES CAN BE ADDED HERE>
                  (else (error "unknown expression type -- DERIV" exp))))

     We can regard this program as performing a dispatch on the type of
     the expression to be differentiated.  In this situation the "type
     tag" of the datum is the algebraic operator symbol (such as `+')
     and the operation being performed is `deriv'.  We can transform
     this program into data-directed style by rewriting the basic
     derivative procedure as

          (define (deriv exp var)
             (cond ((number? exp) 0)
                   ((variable? exp) (if (same-variable? exp var) 1 0))
                   (else ((get 'deriv (operator exp)) (operands exp)
                                                      var))))

          (define (operator exp) (car exp))

          (define (operands exp) (cdr exp))

       a. Explain what was done above.  Why can't we assimilate the
          predicates `number?' and `same-variable?' into the
          data-directed dispatch?

       b. Write the procedures for derivatives of sums and products,
          and the auxiliary code required to install them in the table
          used by the program above.

       c. Choose any additional differentiation rule that you like,
          such as the one for exponents (*Note Exercise 2-56::), and
          install it in this data-directed system.

       d. In this simple algebraic manipulator the type of an
          expression is the algebraic operator that binds it together.
          Suppose, however, we indexed the procedures in the opposite
          way, so that the dispatch line in `deriv' looked like

               ((get (operator exp) 'deriv) (operands exp) var)

          What corresponding changes to the derivative system are
          required?


     *Exercise 2.74:* Insatiable Enterprises, Inc., is a highly
     decentralized conglomerate company consisting of a large number of
     independent divisions located all over the world.  The company's
     computer facilities have just been interconnected by means of a
     clever network-interfacing scheme that makes the entire network
     appear to any user to be a single computer.  Insatiable's
     president, in her first attempt to exploit the ability of the
     network to extract administrative information from division files,
     is dismayed to discover that, although all the division files have
     been implemented as data structures in Scheme, the particular data
     structure used varies from division to division.  A meeting of
     division managers is hastily called to search for a strategy to
     integrate the files that will satisfy headquarters' needs while
     preserving the existing autonomy of the divisions.

     Show how such a strategy can be implemented with data-directed
     programming.  As an example, suppose that each division's
     personnel records consist of a single file, which contains a set
     of records keyed on employees' names.  The structure of the set
     varies from division to division.  Furthermore, each employee's
     record is itself a set (structured differently from division to
     division) that contains information keyed under identifiers such
     as `address' and `salary'.  In particular:

       a. Implement for headquarters a `get-record' procedure that
          retrieves a specified employee's record from a specified
          personnel file.  The procedure should be applicable to any
          division's file.  Explain how the individual divisions' files
          should be structured.  In particular, what type information
          must be supplied?

       b. Implement for headquarters a `get-salary' procedure that
          returns the salary information from a given employee's record
          from any division's personnel file.  How should the record be
          structured in order to make this operation work?

       c. Implement for headquarters a `find-employee-record'
          procedure.  This should search all the divisions' files for
          the record of a given employee and return the record.  Assume
          that this procedure takes as arguments an employee's name and
          a list of all the divisions' files.

       d. When Insatiable takes over a new company, what changes must
          be made in order to incorporate the new personnel information
          into the central system?


Message passing
...............

The key idea of data-directed programming is to handle generic
operations in programs by dealing explicitly with operation-and-type
tables, such as the table in *Note Figure 2-22::.  The style of
programming we used in section *Note 2-4-2:: organized the required
dispatching on type by having each operation take care of its own
dispatching.  In effect, this decomposes the operation-and-type table
into rows, with each generic operation procedure representing a row of
the table.

   An alternative implementation strategy is to decompose the table
into columns and, instead of using "intelligent operations" that
dispatch on data types, to work with "intelligent data objects" that
dispatch on operation names.  We can do this by arranging things so
that a data object, such as a rectangular number, is represented as a
procedure that takes as input the required operation name and performs
the operation indicated.  In such a discipline, `make-from-real-imag'
could be written as

     (define (make-from-real-imag x y)
       (define (dispatch op)
         (cond ((eq? op 'real-part) x)
               ((eq? op 'imag-part) y)
               ((eq? op 'magnitude)
                (sqrt (+ (square x) (square y))))
               ((eq? op 'angle) (atan y x))
               (else
                (error "Unknown op -- MAKE-FROM-REAL-IMAG" op))))
       dispatch)

   The corresponding `apply-generic' procedure, which applies a generic
operation to an argument, now simply feeds the operation's name to the
data object and lets the object do the work:(4)

     (define (apply-generic op arg) (arg op))

   Note that the value returned by `make-from-real-imag' is a
procedure--the internal `dispatch' procedure.  This is the procedure
that is invoked when `apply-generic' requests an operation to be
performed.

   This style of programming is called "message passing".  The name
comes from the image that a data object is an entity that receives the
requested operation name as a "message."  We have already seen an
example of message passing in section *Note 2-1-3::, where we saw how
`cons', `car', and `cdr' could be defined with no data objects but only
procedures.  Here we see that message passing is not a mathematical
trick but a useful technique for organizing systems with generic
operations.  In the remainder of this chapter we will continue to use
data-directed programming, rather than message passing, to discuss
generic arithmetic operations.  In *Note Chapter 3:: we will return to
message passing, and we will see that it can be a powerful tool for
structuring simulation programs.

     *Exercise 2.75:* Implement the constructor `make-from-mag-ang' in
     message-passing style.  This procedure should be analogous to the
     `make-from-real-imag' procedure given above.

     *Exercise 2.76:* As a large system with generic operations
     evolves, new types of data objects or new operations may be needed.
     For each of the three strategies--generic operations with explicit
     dispatch, data-directed style, and message-passing-style--describe
     the changes that must be made to a system in order to add new
     types or new operations.  Which organization would be most
     appropriate for a system in which new types must often be added?
     Which would be most appropriate for a system in which new
     operations must often be added?

   ---------- Footnotes ----------

   (1) We use the list `(rectangular)' rather than the symbol
`rectangular' to allow for the possibility of operations with multiple
arguments, not all of the same type.

   (2) The type the constructors are installed under needn't be a list
because a constructor is always used to make an object of one
particular type.

   (3) `Apply-generic' uses the dotted-tail notation described in *Note
Exercise 2-20::, because different generic operations may take
different numbers of arguments.  In `apply-generic', `op' has as its
value the first argument to `apply-generic' and `args' has as its value
a list of the remaining arguments.

   `Apply-generic' also uses the primitive procedure `apply', which
takes two arguments, a procedure and a list.  `Apply' applies the
procedure, using the elements in the list as arguments.  For example,

     (apply + (list 1 2 3 4))

returns 10.

   (4) One limitation of this organization is it permits only generic
procedures of one argument.


File: sicp.info,  Node: 2-5,  Prev: 2-4,  Up: Chapter 2

2.5 Systems with Generic Operations
===================================

In the previous section, we saw how to design systems in which data
objects can be represented in more than one way.  The key idea is to
link the code that specifies the data operations to the several
representations by means of generic interface procedures.  Now we will
see how to use this same idea not only to define operations that are
generic over different representations but also to define operations
that are generic over different kinds of arguments.  We have already
seen several different packages of arithmetic operations: the primitive
arithmetic (`+', `-', `*', `/') built into our language, the
rational-number arithmetic (`add-rat', `sub-rat', `mul-rat', `div-rat')
of section *Note 2-1-1::, and the complex-number arithmetic that we
implemented in section *Note 2-4-3::.  We will now use data-directed
techniques to construct a package of arithmetic operations that
incorporates all the arithmetic packages we have already constructed.

   *Note Figure 2-23:: shows the structure of the system we shall
build.  Notice the abstraction barriers.  From the perspective of
someone using "numbers," there is a single procedure `add' that
operates on whatever numbers are supplied.  `Add' is part of a generic
interface that allows the separate ordinary-arithmetic,
rational-arithmetic, and complex-arithmetic packages to be accessed
uniformly by programs that use numbers.  Any individual arithmetic
package (such as the complex package) may itself be accessed through
generic procedures (such as `add-complex') that combine packages
designed for different representations (such as rectangular and polar).
Moreover, the structure of the system is additive, so that one can
design the individual arithmetic packages separately and combine them
to produce a generic arithmetic system.

     *Figure 2.23:* Generic arithmetic system.

                                  Programs that use numbers
                                     +-----------------+
          ---------------------------| add sub mul div |-------------------
                                     +-----------------+
                                  Generic arithmetic package
           +-----------------+   +-------------------------+
           | add-rat sub-rat |   | add-complex sub-complex |   +---------+
          -|                 |-+-|                         |-+-| + - * / |-
           | mul-rat div-rat | | | mul-complex div-complex | | +---------+
           +-----------------+ | +-------------------------+ |
                Rational       |     Complex artithmetic     |   Ordinary
               arithmetic      +--------------+--------------+  arithmetic
                               | Rectangular  |     Polar    |
          ---------------------+--------------+--------------+-------------

* Menu:

* 2-5-1::            Generic Arithmetic Operations
* 2-5-2::            Combining Data of Different Types
* 2-5-3::            Example: Symbolic Algebra


File: sicp.info,  Node: 2-5-1,  Next: 2-5-2,  Prev: 2-5,  Up: 2-5

2.5.1 Generic Arithmetic Operations
-----------------------------------

The task of designing generic arithmetic operations is analogous to
that of designing the generic complex-number operations.  We would
like, for instance, to have a generic addition procedure `add' that
acts like ordinary primitive addition `+' on ordinary numbers, like
`add-rat' on rational numbers, and like `add-complex' on complex
numbers.  We can implement `add', and the other generic arithmetic
operations, by following the same strategy we used in section *Note
2-4-3:: to implement the generic selectors for complex numbers.  We
will attach a type tag to each kind of number and cause the generic
procedure to dispatch to an appropriate package according to the data
type of its arguments.

   The generic arithmetic procedures are defined as follows:

     (define (add x y) (apply-generic 'add x y))
     (define (sub x y) (apply-generic 'sub x y))
     (define (mul x y) (apply-generic 'mul x y))
     (define (div x y) (apply-generic 'div x y))

   We begin by installing a package for handling "ordinary" numbers,
that is, the primitive numbers of our language.  We will tag these with
the symbol `scheme-number'.  The arithmetic operations in this package
are the primitive arithmetic procedures (so there is no need to define
extra procedures to handle the untagged numbers).  Since these
operations each take two arguments, they are installed in the table
keyed by the list `(scheme-number scheme-number)':

     (define (install-scheme-number-package)
       (define (tag x)
         (attach-tag 'scheme-number x))
       (put 'add '(scheme-number scheme-number)
            (lambda (x y) (tag (+ x y))))
       (put 'sub '(scheme-number scheme-number)
            (lambda (x y) (tag (- x y))))
       (put 'mul '(scheme-number scheme-number)
            (lambda (x y) (tag (* x y))))
       (put 'div '(scheme-number scheme-number)
            (lambda (x y) (tag (/ x y))))
       (put 'make 'scheme-number
            (lambda (x) (tag x)))
       'done)

   Users of the Scheme-number package will create (tagged) ordinary
numbers by means of the procedure:

     (define (make-scheme-number n)
       ((get 'make 'scheme-number) n))

   Now that the framework of the generic arithmetic system is in place,
we can readily include new kinds of numbers.  Here is a package that
performs rational arithmetic.  Notice that, as a benefit of additivity,
we can use without modification the rational-number code from section
*Note 2-1-1:: as the internal procedures in the package:

     (define (install-rational-package)
       ;; internal procedures
       (define (numer x) (car x))
       (define (denom x) (cdr x))
       (define (make-rat n d)
         (let ((g (gcd n d)))
           (cons (/ n g) (/ d g))))
       (define (add-rat x y)
         (make-rat (+ (* (numer x) (denom y))
                      (* (numer y) (denom x)))
                   (* (denom x) (denom y))))
       (define (sub-rat x y)
         (make-rat (- (* (numer x) (denom y))
                      (* (numer y) (denom x)))
                   (* (denom x) (denom y))))
       (define (mul-rat x y)
         (make-rat (* (numer x) (numer y))
                   (* (denom x) (denom y))))
       (define (div-rat x y)
         (make-rat (* (numer x) (denom y))
                   (* (denom x) (numer y))))

       ;; interface to rest of the system
       (define (tag x) (attach-tag 'rational x))
       (put 'add '(rational rational)
            (lambda (x y) (tag (add-rat x y))))
       (put 'sub '(rational rational)
            (lambda (x y) (tag (sub-rat x y))))
       (put 'mul '(rational rational)
            (lambda (x y) (tag (mul-rat x y))))
       (put 'div '(rational rational)
            (lambda (x y) (tag (div-rat x y))))

       (put 'make 'rational
            (lambda (n d) (tag (make-rat n d))))
       'done)

     (define (make-rational n d)
       ((get 'make 'rational) n d))

   We can install a similar package to handle complex numbers, using
the tag `complex'.  In creating the package, we extract from the table
the operations `make-from-real-imag' and `make-from-mag-ang' that were
defined by the rectangular and polar packages.  Additivity permits us
to use, as the internal operations, the same `add-complex',
`sub-complex', `mul-complex', and `div-complex' procedures from section
*Note 2-4-1::.

     (define (install-complex-package)
       ;; imported procedures from rectangular and polar packages
       (define (make-from-real-imag x y)
         ((get 'make-from-real-imag 'rectangular) x y))
       (define (make-from-mag-ang r a)
         ((get 'make-from-mag-ang 'polar) r a))

       ;; internal procedures
       (define (add-complex z1 z2)
         (make-from-real-imag (+ (real-part z1) (real-part z2))
                              (+ (imag-part z1) (imag-part z2))))
       (define (sub-complex z1 z2)
         (make-from-real-imag (- (real-part z1) (real-part z2))
                              (- (imag-part z1) (imag-part z2))))
       (define (mul-complex z1 z2)
         (make-from-mag-ang (* (magnitude z1) (magnitude z2))
                            (+ (angle z1) (angle z2))))
       (define (div-complex z1 z2)
         (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
                            (- (angle z1) (angle z2))))

       ;; interface to rest of the system
       (define (tag z) (attach-tag 'complex z))
       (put 'add '(complex complex)
            (lambda (z1 z2) (tag (add-complex z1 z2))))
       (put 'sub '(complex complex)
            (lambda (z1 z2) (tag (sub-complex z1 z2))))
       (put 'mul '(complex complex)
            (lambda (z1 z2) (tag (mul-complex z1 z2))))
       (put 'div '(complex complex)
            (lambda (z1 z2) (tag (div-complex z1 z2))))
       (put 'make-from-real-imag 'complex
            (lambda (x y) (tag (make-from-real-imag x y))))
       (put 'make-from-mag-ang 'complex
            (lambda (r a) (tag (make-from-mag-ang r a))))
       'done)

   Programs outside the complex-number package can construct complex
numbers either from real and imaginary parts or from magnitudes and
angles.  Notice how the underlying procedures, originally defined in
the rectangular and polar packages, are exported to the complex
package, and exported from there to the outside world.

     (define (make-complex-from-real-imag x y)
       ((get 'make-from-real-imag 'complex) x y))

     (define (make-complex-from-mag-ang r a)
       ((get 'make-from-mag-ang 'complex) r a))

   What we have here is a two-level tag system.  A typical complex
number, such as 3 + 4i in rectangular form, would be represented as
shown in *Note Figure 2-24::.  The outer tag (`complex') is used to
direct the number to the complex package.  Once within the complex
package, the next tag (`rectangular') is used to direct the number to
the rectangular package.  In a large and complicated system there might
be many levels, each interfaced with the next by means of generic
operations.  As a data object is passed "downward," the outer tag that
is used to direct it to the appropriate package is stripped off (by
applying `contents') and the next level of tag (if any) becomes visible
to be used for further dispatching.

     *Figure 2.24:* Representation of 3 + 4i in rectangular form.

               +---+---+     +---+---+     +---+---+
          ---->| * | *-+---->| * | *-+---->| * | * |
               +-|-+---+     +-|-+---+     +-|-+-|-+
                 |             |             |   |
                 V             V             V   V
           +---------+   +-------------+  +---+ +---+
           | complex |   | rectangular |  | 3 | | 4 |
           +---------+   +-------------+  +---+ +---+

   In the above packages, we used `add-rat', `add-complex', and the
other arithmetic procedures exactly as originally written.  Once these
definitions are internal to different installation procedures, however,
they no longer need names that are distinct from each other: we could
simply name them `add', `sub', `mul', and `div' in both packages.

     *Exercise 2.77:* Louis Reasoner tries to evaluate the expression
     `(magnitude z)' where `z' is the object shown in *Note Figure
     2-24::.  To his surprise, instead of the answer 5 he gets an error
     message from `apply-generic', saying there is no method for the
     operation `magnitude' on the types `(complex)'.  He shows this
     interaction to Alyssa P. Hacker, who says "The problem is that the
     complex-number selectors were never defined for `complex' numbers,
     just for `polar' and `rectangular' numbers.  All you have to do to
     make this work is add the following to the `complex' package:"

          (put 'real-part '(complex) real-part)
          (put 'imag-part '(complex) imag-part)
          (put 'magnitude '(complex) magnitude)
          (put 'angle '(complex) angle)

     Describe in detail why this works.  As an example, trace through
     all the procedures called in evaluating the expression `(magnitude
     z)' where `z' is the object shown in *Note Figure 2-24::.  In
     particular, how many times is `apply-generic' invoked?  What
     procedure is dispatched to in each case?

     *Exercise 2.78:* The internal procedures in the `scheme-number'
     package are essentially nothing more than calls to the primitive
     procedures `+', `-', etc.  It was not possible to use the
     primitives of the language directly because our type-tag system
     requires that each data object have a type attached to it.  In
     fact, however, all Lisp implementations do have a type system,
     which they use internally.  Primitive predicates such as `symbol?'
     and `number?'  determine whether data objects have particular
     types.  Modify the definitions of `type-tag', `contents', and
     `attach-tag' from section *Note 2-4-2:: so that our generic system
     takes advantage of Scheme's internal type system.  That is to say,
     the system should work as before except that ordinary numbers
     should be represented simply as Scheme numbers rather than as
     pairs whose `car' is the symbol `scheme-number'.

     *Exercise 2.79:* Define a generic equality predicate `equ?' that
     tests the equality of two numbers, and install it in the generic
     arithmetic package.  This operation should work for ordinary
     numbers, rational numbers, and complex numbers.

     *Exercise 2.80:* Define a generic predicate `=zero?' that tests if
     its argument is zero, and install it in the generic arithmetic
     package.  This operation should work for ordinary numbers, rational
     numbers, and complex numbers.


File: sicp.info,  Node: 2-5-2,  Next: 2-5-3,  Prev: 2-5-1,  Up: 2-5

2.5.2 Combining Data of Different Types
---------------------------------------

We have seen how to define a unified arithmetic system that encompasses
ordinary numbers, complex numbers, rational numbers, and any other type
of number we might decide to invent, but we have ignored an important
issue.  The operations we have defined so far treat the different data
types as being completely independent.  Thus, there are separate
packages for adding, say, two ordinary numbers, or two complex numbers.
What we have not yet considered is the fact that it is meaningful to
define operations that cross the type boundaries, such as the addition
of a complex number to an ordinary number.  We have gone to great pains
to introduce barriers between parts of our programs so that they can be
developed and understood separately.  We would like to introduce the
cross-type operations in some carefully controlled way, so that we can
support them without seriously violating our module boundaries.

   One way to handle cross-type operations is to design a different
procedure for each possible combination of types for which the
operation is valid.  For example, we could extend the complex-number
package so that it provides a procedure for adding complex numbers to
ordinary numbers and installs this in the table using the tag `(complex
scheme-number)':(1)

     ;; to be included in the complex package
     (define (add-complex-to-schemenum z x)
       (make-from-real-imag (+ (real-part z) x)
                            (imag-part z)))

     (put 'add '(complex scheme-number)
          (lambda (z x) (tag (add-complex-to-schemenum z x))))

   This technique works, but it is cumbersome.  With such a system, the
cost of introducing a new type is not just the construction of the
package of procedures for that type but also the construction and
installation of the procedures that implement the cross-type
operations.  This can easily be much more code than is needed to define
the operations on the type itself.  The method also undermines our
ability to combine separate packages additively, or least to limit the
extent to which the implementors of the individual packages need to
take account of other packages.  For instance, in the example above, it
seems reasonable that handling mixed operations on complex numbers and
ordinary numbers should be the responsibility of the complex-number
package.  Combining rational numbers and complex numbers, however,
might be done by the complex package, by the rational package, or by
some third package that uses operations extracted from these two
packages.  Formulating coherent policies on the division of
responsibility among packages can be an overwhelming task in designing
systems with many packages and many cross-type operations.

Coercion
........

In the general situation of completely unrelated operations acting on
completely unrelated types, implementing explicit cross-type operations,
cumbersome though it may be, is the best that one can hope for.
Fortunately, we can usually do better by taking advantage of additional
structure that may be latent in our type system.  Often the different
data types are not completely independent, and there may be ways by
which objects of one type may be viewed as being of another type.  This
process is called "coercion".  For example, if we are asked to
arithmetically combine an ordinary number with a complex number, we can
view the ordinary number as a complex number whose imaginary part is
zero.  This transforms the problem to that of combining two complex
numbers, which can be handled in the ordinary way by the
complex-arithmetic package.

   In general, we can implement this idea by designing coercion
procedures that transform an object of one type into an equivalent
object of another type.  Here is a typical coercion procedure, which
transforms a given ordinary number to a complex number with that real
part and zero imaginary part:

     (define (scheme-number->complex n)
       (make-complex-from-real-imag (contents n) 0))

   We install these coercion procedures in a special coercion table,
indexed under the names of the two types:

     (put-coercion 'scheme-number 'complex scheme-number->complex)

   (We assume that there are `put-coercion' and `get-coercion'
procedures available for manipulating this table.)  Generally some of
the slots in the table will be empty, because it is not generally
possible to coerce an arbitrary data object of each type into all other
types.  For example, there is no way to coerce an arbitrary complex
number to an ordinary number, so there will be no general
`complex->scheme-number' procedure included in the table.

   Once the coercion table has been set up, we can handle coercion in a
uniform manner by modifying the `apply-generic' procedure of section
*Note 2-4-3::.  When asked to apply an operation, we first check
whether the operation is defined for the arguments' types, just as
before.  If so, we dispatch to the procedure found in the
operation-and-type table.  Otherwise, we try coercion.  For simplicity,
we consider only the case where there are two arguments.(2)  We check
the coercion table to see if objects of the first type can be coerced
to the second type.  If so, we coerce the first argument and try the
operation again.  If objects of the first type cannot in general be
coerced to the second type, we try the coercion the other way around to
see if there is a way to coerce the second argument to the type of the
first argument.  Finally, if there is no known way to coerce either
type to the other type, we give up.  Here is the procedure:

     (define (apply-generic op . args)
       (let ((type-tags (map type-tag args)))
         (let ((proc (get op type-tags)))
           (if proc
               (apply proc (map contents args))
               (if (= (length args) 2)
                   (let ((type1 (car type-tags))
                         (type2 (cadr type-tags))
                         (a1 (car args))
                         (a2 (cadr args)))
                     (let ((t1->t2 (get-coercion type1 type2))
                           (t2->t1 (get-coercion type2 type1)))
                       (cond (t1->t2
                              (apply-generic op (t1->t2 a1) a2))
                             (t2->t1
                              (apply-generic op a1 (t2->t1 a2)))
                             (else
                              (error "No method for these types"
                                     (list op type-tags))))))
                   (error "No method for these types"
                          (list op type-tags)))))))

   This coercion scheme has many advantages over the method of defining
explicit cross-type operations, as outlined above.  Although we still
need to write coercion procedures to relate the types (possibly n^2
procedures for a system with n types), we need to write only one
procedure for each pair of types rather than a different procedure for
each collection of types and each generic operation.(3)  What we are
counting on here is the fact that the appropriate transformation
between types depends only on the types themselves, not on the
operation to be applied.

   On the other hand, there may be applications for which our coercion
scheme is not general enough.  Even when neither of the objects to be
combined can be converted to the type of the other it may still be
possible to perform the operation by converting both objects to a third
type.  In order to deal with such complexity and still preserve
modularity in our programs, it is usually necessary to build systems
that take advantage of still further structure in the relations among
types, as we discuss next.

Hierarchies of types
....................

The coercion scheme presented above relied on the existence of natural
relations between pairs of types.  Often there is more "global"
structure in how the different types relate to each other.  For
instance, suppose we are building a generic arithmetic system to handle
integers, rational numbers, real numbers, and complex numbers.  In such
a system, it is quite natural to regard an integer as a special kind of
rational number, which is in turn a special kind of real number, which
is in turn a special kind of complex number.  What we actually have is
a so-called "hierarchy of types", in which, for example, integers are a "subtype"
of rational numbers (i.e., any operation that can be applied to a
rational number can automatically be applied to an integer).
Conversely, we say that rational numbers form a "supertype" of
integers.  The particular hierarchy we have here is of a very simple
kind, in which each type has at most one supertype and at most one
subtype.  Such a structure, called a "tower", is illustrated in *Note
Figure 2-25::.

     *Figure 2.25:* A tower of types.

           complex
             ^
             |
            real
             ^
             |
          rational
             ^
             |
          integer

   If we have a tower structure, then we can greatly simplify the
problem of adding a new type to the hierarchy, for we need only specify
how the new type is embedded in the next supertype above it and how it
is the supertype of the type below it.  For example, if we want to add
an integer to a complex number, we need not explicitly define a special
coercion procedure `integer->complex'.  Instead, we define how an
integer can be transformed into a rational number, how a rational
number is transformed into a real number, and how a real number is
transformed into a complex number.  We then allow the system to
transform the integer into a complex number through these steps and
then add the two complex numbers.

   We can redesign our `apply-generic' procedure in the following way:
For each type, we need to supply a `raise' procedure, which "raises"
objects of that type one level in the tower.  Then when the system is
required to operate on objects of different types it can successively
raise the lower types until all the objects are at the same level in
the tower.  (*Note Exercise 2-83:: and *Note Exercise 2-84:: concern
the details of implementing such a strategy.)

   Another advantage of a tower is that we can easily implement the
notion that every type "inherits" all operations defined on a
supertype.  For instance, if we do not supply a special procedure for
finding the real part of an integer, we should nevertheless expect that
`real-part' will be defined for integers by virtue of the fact that
integers are a subtype of complex numbers.  In a tower, we can arrange
for this to happen in a uniform way by modifying `apply-generic'.  If
the required operation is not directly defined for the type of the
object given, we raise the object to its supertype and try again.  We
thus crawl up the tower, transforming our argument as we go, until we
either find a level at which the desired operation can be performed or
hit the top (in which case we give up).

   Yet another advantage of a tower over a more general hierarchy is
that it gives us a simple way to "lower" a data object to the simplest
representation.  For example, if we add 2 + 3i to 4 - 3i, it would be
nice to obtain the answer as the integer 6 rather than as the complex
number 6 + 0i.  *Note Exercise 2-85:: discusses a way to implement such
a lowering operation.  (The trick is that we need a general way to
distinguish those objects that can be lowered, such as 6 + 0i, from
those that cannot, such as 6 + 2i.)

     *Figure 2.26:* Relations among types of geometric figures.

                               polygon
                              /       \
                             /         \
                      triangle         quadrilateral
                      /     \              /     \
                     /       \            /       \
               isosceles   right      trapezoid   kite
               triangle    triangle       |         |
                |     \      |            |         |
                |      \     |            |         |
          equilateral   isosceles   parallelogram   |
          triangle      right          |       \    |
                        triangle       |        \   |
                                    rectangle  rhombus
                                          \    /
                                           \  /
                                          square

Inadequacies of hierarchies
...........................

If the data types in our system can be naturally arranged in a tower,
this greatly simplifies the problems of dealing with generic operations
on different types, as we have seen.  Unfortunately, this is usually
not the case.  *Note Figure 2-26:: illustrates a more complex
arrangement of mixed types, this one showing relations among different
types of geometric figures.  We see that, in general, a type may have
more than one subtype.  Triangles and quadrilaterals, for instance, are
both subtypes of polygons.  In addition, a type may have more than one
supertype.  For example, an isosceles right triangle may be regarded
either as an isosceles triangle or as a right triangle.  This
multiple-supertypes issue is particularly thorny, since it means that
there is no unique way to "raise" a type in the hierarchy.  Finding the
"correct" supertype in which to apply an operation to an object may
involve considerable searching through the entire type network on the
part of a procedure such as `apply-generic'.  Since there generally are
multiple subtypes for a type, there is a similar problem in coercing a
value "down" the type hierarchy.  Dealing with large numbers of
interrelated types while still preserving modularity in the design of
large systems is very difficult, and is an area of much current
research.(4)

     *Exercise 2.81:* Louis Reasoner has noticed that `apply-generic'
     may try to coerce the arguments to each other's type even if they
     already have the same type.  Therefore, he reasons, we need to put
     procedures in the coercion table to "coerce" arguments of each
     type to their own type.  For example, in addition to the
     `scheme-number->complex' coercion shown above, he would do:

          (define (scheme-number->scheme-number n) n)
          (define (complex->complex z) z)
          (put-coercion 'scheme-number 'scheme-number
                        scheme-number->scheme-number)
          (put-coercion 'complex 'complex complex->complex)

       a. With Louis's coercion procedures installed, what happens if
          `apply-generic' is called with two arguments of type
          `scheme-number' or two arguments of type `complex' for an
          operation that is not found in the table for those types?
          For example, assume that we've defined a generic
          exponentiation operation:

               (define (exp x y) (apply-generic 'exp x y))

          and have put a procedure for exponentiation in the
          Scheme-number package but not in any other package:

               ;; following added to Scheme-number package
               (put 'exp '(scheme-number scheme-number)
                    (lambda (x y) (tag (expt x y)))) ; using primitive `expt'

          What happens if we call `exp' with two complex numbers as
          arguments?

       b. Is Louis correct that something had to be done about coercion
          with arguments of the same type, or does `apply-generic' work
          correctly as is?

       c. Modify `apply-generic' so that it doesn't try coercion if the
          two arguments have the same type.


     *Exercise 2.82:* Show how to generalize `apply-generic' to handle
     coercion in the general case of multiple arguments.  One strategy
     is to attempt to coerce all the arguments to the type of the first
     argument, then to the type of the second argument, and so on.
     Give an example of a situation where this strategy (and likewise
     the two-argument version given above) is not sufficiently general.
     (Hint: Consider the case where there are some suitable mixed-type
     operations present in the table that will not be tried.)

     *Exercise 2.83:* Suppose you are designing a generic arithmetic
     system for dealing with the tower of types shown in *Note Figure
     2-25::: integer, rational, real, complex.  For each type (except
     complex), design a procedure that raises objects of that type one
     level in the tower.  Show how to install a generic `raise'
     operation that will work for each type (except complex).

     *Exercise 2.84:* Using the `raise' operation of *Note Exercise
     2-83::, modify the `apply-generic' procedure so that it coerces
     its arguments to have the same type by the method of successive
     raising, as discussed in this section.  You will need to devise a
     way to test which of two types is higher in the tower.  Do this in
     a manner that is "compatible" with the rest of the system and will
     not lead to problems in adding new levels to the tower.

     *Exercise 2.85:* This section mentioned a method for "simplifying"
     a data object by lowering it in the tower of types as far as
     possible.  Design a procedure `drop' that accomplishes this for the
     tower described in *Note Exercise 2-83::.  The key is to decide,
     in some general way, whether an object can be lowered.  For
     example, the complex number 1.5 + 0i can be lowered as far as
     `real', the complex number 1 + 0i can be lowered as far as
     `integer', and the complex number 2 + 3i cannot be lowered at all.
     Here is a plan for determining whether an object can be lowered:
     Begin by defining a generic operation `project' that "pushes" an
     object down in the tower.  For example, projecting a complex
     number would involve throwing away the imaginary part.  Then a
     number can be dropped if, when we `project' it and `raise' the
     result back to the type we started with, we end up with something
     equal to what we started with.  Show how to implement this idea in
     detail, by writing a `drop' procedure that drops an object as far
     as possible.  You will need to design the various projection
     operations(5) and install `project' as a generic operation in the
     system.  You will also need to make use of a generic equality
     predicate, such as described in *Note Exercise 2-79::.  Finally,
     use `drop' to rewrite `apply-generic' from *Note Exercise 2-84::
     so that it "simplifies" its answers.

     *Exercise 2.86:* Suppose we want to handle complex numbers whose
     real parts, imaginary parts, magnitudes, and angles can be either
     ordinary numbers, rational numbers, or other numbers we might wish
     to add to the system.  Describe and implement the changes to the
     system needed to accommodate this.  You will have to define
     operations such as `sine' and `cosine' that are generic over
     ordinary numbers and rational numbers.

   ---------- Footnotes ----------

   (1) We also have to supply an almost identical procedure to handle
the types `(scheme-number complex)'.

   (2) See *Note Exercise 2-82:: for generalizations.

   (3) If we are clever, we can usually get by with fewer than n^2
coercion procedures.  For instance, if we know how to convert from type
1 to type 2 and from type 2 to type 3, then we can use this knowledge to
convert from type 1 to type 3.  This can greatly decrease the number of
coercion procedures we need to supply explicitly when we add a new type
to the system.  If we are willing to build the required amount of
sophistication into our system, we can have it search the "graph" of
relations among types and automatically generate those coercion
procedures that can be inferred from the ones that are supplied
explicitly.

   (4) This statement, which also appears in the first edition of this
book, is just as true now as it was when we wrote it twelve years ago.
Developing a useful, general framework for expressing the relations
among different types of entities (what philosophers call "ontology")
seems intractably difficult.  The main difference between the confusion
that existed ten years ago and the confusion that exists now is that
now a variety of inadequate ontological theories have been embodied in
a plethora of correspondingly inadequate programming languages.  For
example, much of the complexity of object-oriented programming
languages--and the subtle and confusing differences among contemporary
object-oriented languages--centers on the treatment of generic
operations on interrelated types.  Our own discussion of computational
objects in *Note Chapter 3:: avoids these issues entirely.  Readers
familiar with object-oriented programming will notice that we have much
to say in *Note Chapter 3:: about local state, but we do not even
mention "classes" or "inheritance."  In fact, we suspect that these
problems cannot be adequately addressed in terms of computer-language
design alone, without also drawing on work in knowledge representation
and automated reasoning.

   (5) A real number can be projected to an integer using the `round'
primitive, which returns the closest integer to its argument.


File: sicp.info,  Node: 2-5-3,  Prev: 2-5-2,  Up: 2-5

2.5.3 Example: Symbolic Algebra
-------------------------------

The manipulation of symbolic algebraic expressions is a complex process
that illustrates many of the hardest problems that occur in the design
of large-scale systems.  An algebraic expression, in general, can be
viewed as a hierarchical structure, a tree of operators applied to
operands.  We can construct algebraic expressions by starting with a
set of primitive objects, such as constants and variables, and
combining these by means of algebraic operators, such as addition and
multiplication.  As in other languages, we form abstractions that
enable us to refer to compound objects in simple terms.  Typical
abstractions in symbolic algebra are ideas such as linear combination,
polynomial, rational function, or trigonometric function.  We can
regard these as compound "types," which are often useful for directing
the processing of expressions.  For example, we could describe the
expression

     x^2 sin (y^2 + 1) + r cos 2y + cos(y^3 - 2y^2)

as a polynomial in x with coefficients that are trigonometric functions
of polynomials in y whose coefficients are integers.

   We will not attempt to develop a complete algebraic-manipulation
system here.  Such systems are exceedingly complex programs, embodying
deep algebraic knowledge and elegant algorithms.  What we will do is
look at a simple but important part of algebraic manipulation: the
arithmetic of polynomials.  We will illustrate the kinds of decisions
the designer of such a system faces, and how to apply the ideas of
abstract data and generic operations to help organize this effort.

Arithmetic on polynomials
.........................

Our first task in designing a system for performing arithmetic on
polynomials is to decide just what a polynomial is.  Polynomials are
normally defined relative to certain variables (the "indeterminates" of
the polynomial).  For simplicity, we will restrict ourselves to
polynomials having just one indeterminate ("univariate
polynomials").(1) We will define a polynomial to be a sum of terms,
each of which is either a coefficient, a power of the indeterminate, or
a product of a coefficient and a power of the indeterminate.  A
coefficient is defined as an algebraic expression that is not dependent
upon the indeterminate of the polynomial.  For example,

     5x^2 + 3r + 7

is a simple polynomial in x, and

     (y^2 + 1)r^3 + (2y)x + 1

is a polynomial in x whose coefficients are polynomials in y.

   Already we are skirting some thorny issues.  Is the first of these
polynomials the same as the polynomial 5y^2 + 3y + 7, or not?  A
reasonable answer might be "yes, if we are considering a polynomial
purely as a mathematical function, but no, if we are considering a
polynomial to be a syntactic form."  The second polynomial is
algebraically equivalent to a polynomial in y whose coefficients are
polynomials in x.  Should our system recognize this, or not?
Furthermore, there are other ways to represent a polynomial--for
example, as a product of factors, or (for a univariate polynomial) as
the set of roots, or as a listing of the values of the polynomial at a
specified set of points.(2)  We can finesse these questions by deciding
that in our algebraic-manipulation system a "polynomial" will be a
particular syntactic form, not its underlying mathematical meaning.

   Now we must consider how to go about doing arithmetic on
polynomials.  In this simple system, we will consider only addition and
multiplication.  Moreover, we will insist that two polynomials to be
combined must have the same indeterminate.

   We will approach the design of our system by following the familiar
discipline of data abstraction.  We will represent polynomials using a
data structure called a "poly", which consists of a variable and a
collection of terms.  We assume that we have selectors `variable' and
`term-list' that extract those parts from a poly and a constructor
`make-poly' that assembles a poly from a given variable and a term
list.  A variable will be just a symbol, so we can use the
`same-variable?'  procedure of section *Note 2-3-2:: to compare
variables.  The following procedures define addition and multiplication
of polys:

     (define (add-poly p1 p2)
       (if (same-variable? (variable p1) (variable p2))
           (make-poly (variable p1)
                      (add-terms (term-list p1)
                                 (term-list p2)))
           (error "Polys not in same var -- ADD-POLY"
                  (list p1 p2))))

     (define (mul-poly p1 p2)
       (if (same-variable? (variable p1) (variable p2))
           (make-poly (variable p1)
                      (mul-terms (term-list p1)
                                 (term-list p2)))
           (error "Polys not in same var -- MUL-POLY"
                  (list p1 p2))))

   To incorporate polynomials into our generic arithmetic system, we
need to supply them with type tags.  We'll use the tag `polynomial',
and install appropriate operations on tagged polynomials in the
operation table.  We'll embed all our code in an installation procedure
for the polynomial package, similar to the ones in section *Note
2-5-1:::

     (define (install-polynomial-package)
       ;; internal procedures
       ;; representation of poly
       (define (make-poly variable term-list)
         (cons variable term-list))
       (define (variable p) (car p))
       (define (term-list p) (cdr p))
       <_procedures `same-variable?' and `variable?' from section 2.3.2_>

       ;; representation of terms and term lists
       <_procedures `adjoin-term' ... `coeff' from text below_>

       ;; continued on next page

       (define (add-poly p1 p2) ...)
       <_procedures used by `add-poly'_>
       (define (mul-poly p1 p2) ...)
       <_procedures used by `mul-poly'_>

       ;; interface to rest of the system
       (define (tag p) (attach-tag 'polynomial p))
       (put 'add '(polynomial polynomial)
            (lambda (p1 p2) (tag (add-poly p1 p2))))
       (put 'mul '(polynomial polynomial)
            (lambda (p1 p2) (tag (mul-poly p1 p2))))
       (put 'make 'polynomial
            (lambda (var terms) (tag (make-poly var terms))))
       'done)

   Polynomial addition is performed termwise.  Terms of the same order
(i.e., with the same power of the indeterminate) must be combined.
This is done by forming a new term of the same order whose coefficient
is the sum of the coefficients of the addends.  Terms in one addend for
which there are no terms of the same order in the other addend are
simply accumulated into the sum polynomial being constructed.

   In order to manipulate term lists, we will assume that we have a
constructor `the-empty-termlist' that returns an empty term list and a
constructor `adjoin-term' that adjoins a new term to a term list.  We
will also assume that we have a predicate `empty-termlist?' that tells
if a given term list is empty, a selector `first-term' that extracts
the highest-order term from a term list, and a selector `rest-terms'
that returns all but the highest-order term.  To manipulate terms, we
will suppose that we have a constructor `make-term' that constructs a
term with given order and coefficient, and selectors `order' and
`coeff' that return, respectively, the order and the coefficient of the
term.  These operations allow us to consider both terms and term lists
as data abstractions, whose concrete representations we can worry about
separately.

   Here is the procedure that constructs the term list for the sum of
two polynomials:(3)

     (define (add-terms L1 L2)
       (cond ((empty-termlist? L1) L2)
             ((empty-termlist? L2) L1)
             (else
              (let ((t1 (first-term L1)) (t2 (first-term L2)))
                (cond ((> (order t1) (order t2))
                       (adjoin-term
                        t1 (add-terms (rest-terms L1) L2)))
                      ((< (order t1) (order t2))
                       (adjoin-term
                        t2 (add-terms L1 (rest-terms L2))))
                      (else
                       (adjoin-term
                        (make-term (order t1)
                                   (add (coeff t1) (coeff t2)))
                        (add-terms (rest-terms L1)
                                   (rest-terms L2)))))))))

   The most important point to note here is that we used the generic
addition procedure `add' to add together the coefficients of the terms
being combined.  This has powerful consequences, as we will see below.

   In order to multiply two term lists, we multiply each term of the
first list by all the terms of the other list, repeatedly using
`mul-term-by-all-terms', which multiplies a given term by all terms in
a given term list.  The resulting term lists (one for each term of the
first list) are accumulated into a sum.  Multiplying two terms forms a
term whose order is the sum of the orders of the factors and whose
coefficient is the product of the coefficients of the factors:

     (define (mul-terms L1 L2)
       (if (empty-termlist? L1)
           (the-empty-termlist)
           (add-terms (mul-term-by-all-terms (first-term L1) L2)
                      (mul-terms (rest-terms L1) L2))))

     (define (mul-term-by-all-terms t1 L)
       (if (empty-termlist? L)
           (the-empty-termlist)
           (let ((t2 (first-term L)))
             (adjoin-term
              (make-term (+ (order t1) (order t2))
                         (mul (coeff t1) (coeff t2)))
              (mul-term-by-all-terms t1 (rest-terms L))))))

   This is really all there is to polynomial addition and
multiplication.  Notice that, since we operate on terms using the
generic procedures `add' and `mul', our polynomial package is
automatically able to handle any type of coefficient that is known
about by the generic arithmetic package.  If we include a coercion
mechanism such as one of those discussed in section *Note 2-5-2::, then
we also are automatically able to handle operations on polynomials of
different coefficient types, such as

                              /        2                 \
     [3x^2 + (2 + 3i)x + 7] * | x^4 + --- x^2 + (5 + 3i) |
                              \        3                 /

   Because we installed the polynomial addition and multiplication
procedures `add-poly' and `mul-poly' in the generic arithmetic system
as the `add' and `mul' operations for type `polynomial', our system is
also automatically able to handle polynomial operations such as

     [(y + 1)x^2 + (y^2 + 1)x + (y - 1)] * [(y - 2)x + (y^3 + 7)]

   The reason is that when the system tries to combine coefficients, it
will dispatch through `add' and `mul'.  Since the coefficients are
themselves polynomials (in y), these will be combined using `add-poly'
and `mul-poly'.  The result is a kind of "data-directed recursion" in
which, for example, a call to `mul-poly' will result in recursive calls
to `mul-poly' in order to multiply the coefficients.  If the
coefficients of the coefficients were themselves polynomials (as might
be used to represent polynomials in three variables), the data
direction would ensure that the system would follow through another
level of recursive calls, and so on through as many levels as the
structure of the data dictates.(4)

Representing term lists
.......................

Finally, we must confront the job of implementing a good representation
for term lists.  A term list is, in effect, a set of coefficients keyed
by the order of the term.  Hence, any of the methods for representing
sets, as discussed in section *Note 2-3-3::, can be applied to this
task.  On the other hand, our procedures `add-terms' and `mul-terms'
always access term lists sequentially from highest to lowest order.
Thus, we will use some kind of ordered list representation.

   How should we structure the list that represents a term list?  One
consideration is the "density" of the polynomials we intend to
manipulate.  A polynomial is said to be "dense" if it has nonzero
coefficients in terms of most orders.  If it has many zero terms it is
said to be "sparse".  For example,

     A : x^5 + 2x^4 + 3x^2 - 2x - 5

is a dense polynomial, whereas

     B : x^100 + 2x^2 + 1

is sparse.

   The term lists of dense polynomials are most efficiently represented
as lists of the coefficients.  For example, A above would be nicely
represented as `(1 2 0 3 -2 -5)'.  The order of a term in this
representation is the length of the sublist beginning with that term's
coefficient, decremented by 1.(5)  This would be a terrible
representation for a sparse polynomial such as B: There would be a
giant list of zeros punctuated by a few lonely nonzero terms.  A more
reasonable representation of the term list of a sparse polynomial is as
a list of the nonzero terms, where each term is a list containing the
order of the term and the coefficient for that order.  In such a
scheme, polynomial B is efficiently represented as `((100 1) (2 2) (0
1))'.  As most polynomial manipulations are performed on sparse
polynomials, we will use this method.  We will assume that term lists
are represented as lists of terms, arranged from highest-order to
lowest-order term.  Once we have made this decision, implementing the
selectors and constructors for terms and term lists is
straightforward:(6)

     (define (adjoin-term term term-list)
       (if (=zero? (coeff term))
           term-list
           (cons term term-list)))

     (define (the-empty-termlist) '())
     (define (first-term term-list) (car term-list))
     (define (rest-terms term-list) (cdr term-list))
     (define (empty-termlist? term-list) (null? term-list))

     (define (make-term order coeff) (list order coeff))
     (define (order term) (car term))
     (define (coeff term) (cadr term))

where `=zero?' is as defined in *Note Exercise 2-80::.  (See also *Note
Exercise 2-87:: below.)

   Users of the polynomial package will create (tagged) polynomials by
means of the procedure:

     (define (make-polynomial var terms)
       ((get 'make 'polynomial) var terms))

     *Exercise 2.87:* Install `=zero?' for polynomials in the generic
     arithmetic package.  This will allow `adjoin-term' to work for
     polynomials with coefficients that are themselves polynomials.

     *Exercise 2.88:* Extend the polynomial system to include
     subtraction of polynomials.  (Hint: You may find it helpful to
     define a generic negation operation.)

     *Exercise 2.89:* Define procedures that implement the term-list
     representation described above as appropriate for dense
     polynomials.

     *Exercise 2.90:* Suppose we want to have a polynomial system that
     is efficient for both sparse and dense polynomials.  One way to do
     this is to allow both kinds of term-list representations in our
     system.  The situation is analogous to the complex-number example
     of section *Note 2-4::, where we allowed both rectangular and
     polar representations.  To do this we must distinguish different
     types of term lists and make the operations on term lists generic.
     Redesign the polynomial system to implement this generalization.
     This is a major effort, not a local change.

     *Exercise 2.91:* A univariate polynomial can be divided by another
     one to produce a polynomial quotient and a polynomial remainder.
     For example,

          x^5 - 1
          ------- = x^3 + x, remainder x - 1
          x^2 - 1

     Division can be performed via long division.  That is, divide the
     highest-order term of the dividend by the highest-order term of
     the divisor.  The result is the first term of the quotient.  Next,
     multiply the result by the divisor, subtract that from the
     dividend, and produce the rest of the answer by recursively
     dividing the difference by the divisor.  Stop when the order of the
     divisor exceeds the order of the dividend and declare the dividend
     to be the remainder.  Also, if the dividend ever becomes zero,
     return zero as both quotient and remainder.

     We can design a `div-poly' procedure on the model of `add-poly' and
     `mul-poly'. The procedure checks to see if the two polys have the
     same variable.  If so, `div-poly' strips off the variable and
     passes the problem to `div-terms', which performs the division
     operation on term lists. `Div-poly' finally reattaches the
     variable to the result supplied by `div-terms'.  It is convenient
     to design `div-terms' to compute both the quotient and the
     remainder of a division.  `Div-terms' can take two term lists as
     arguments and return a list of the quotient term list and the
     remainder term list.

     Complete the following definition of `div-terms' by filling in the
     missing expressions.  Use this to implement `div-poly', which
     takes two polys as arguments and returns a list of the quotient
     and remainder polys.

          (define (div-terms L1 L2)
            (if (empty-termlist? L1)
                (list (the-empty-termlist) (the-empty-termlist))
                (let ((t1 (first-term L1))
                      (t2 (first-term L2)))
                  (if (> (order t2) (order t1))
                      (list (the-empty-termlist) L1)
                      (let ((new-c (div (coeff t1) (coeff t2)))
                            (new-o (- (order t1) (order t2))))
                        (let ((rest-of-result
                               <COMPUTE REST OF RESULT RECURSIVELY>
                               ))
                          <FORM COMPLETE RESULT>
                          ))))))

Hierarchies of types in symbolic algebra
........................................

Our polynomial system illustrates how objects of one type (polynomials)
may in fact be complex objects that have objects of many different
types as parts.  This poses no real difficulty in defining generic
operations.  We need only install appropriate generic operations for
performing the necessary manipulations of the parts of the compound
types.  In fact, we saw that polynomials form a kind of "recursive data
abstraction," in that parts of a polynomial may themselves be
polynomials.  Our generic operations and our data-directed programming
style can handle this complication without much trouble.

   On the other hand, polynomial algebra is a system for which the data
types cannot be naturally arranged in a tower.  For instance, it is
possible to have polynomials in x whose coefficients are polynomials in
y.  It is also possible to have polynomials in y whose coefficients are
polynomials in x.  Neither of these types is "above" the other in any
natural way, yet it is often necessary to add together elements from
each set.  There are several ways to do this.  One possibility is to
convert one polynomial to the type of the other by expanding and
rearranging terms so that both polynomials have the same principal
variable.  One can impose a towerlike structure on this by ordering the
variables and thus always converting any polynomial to a "canonical
form" with the highest-priority variable dominant and the
lower-priority variables buried in the coefficients.  This strategy
works fairly well, except that the conversion may expand a polynomial
unnecessarily, making it hard to read and perhaps less efficient to
work with.  The tower strategy is certainly not natural for this domain
or for any domain where the user can invent new types dynamically using
old types in various combining forms, such as trigonometric functions,
power series, and integrals.

   It should not be surprising that controlling coercion is a serious
problem in the design of large-scale algebraic-manipulation systems.
Much of the complexity of such systems is concerned with relationships
among diverse types.  Indeed, it is fair to say that we do not yet
completely understand coercion.  In fact, we do not yet completely
understand the concept of a data type.  Nevertheless, what we know
provides us with powerful structuring and modularity principles to
support the design of large systems.

     *Exercise 2.92:* By imposing an ordering on variables, extend the
     polynomial package so that addition and multiplication of
     polynomials works for polynomials in different variables.  (This
     is not easy!)

Extended exercise: Rational functions
.....................................

We can extend our generic arithmetic system to include functions
"rational functions".  These are "fractions" whose numerator and
denominator are polynomials, such as

      x + 1
     -------
     x^3 - 1

   The system should be able to add, subtract, multiply, and divide
rational functions, and to perform such computations as

      x + 1       x      x^3 + 2x^2 + 3x + 1
     ------- + ------- = -------------------
     x^3 - 1   x^2 - 1    x^4 + x^3 - x - 1

(Here the sum has been simplified by removing common factors.  Ordinary
"cross multiplication" would have produced a fourth-degree polynomial
over a fifth-degree polynomial.)

   If we modify our rational-arithmetic package so that it uses generic
operations, then it will do what we want, except for the problem of
reducing fractions to lowest terms.

     *Exercise 2.93:* Modify the rational-arithmetic package to use
     generic operations, but change `make-rat' so that it does not
     attempt to reduce fractions to lowest terms.  Test your system by
     calling `make-rational' on two polynomials to produce a rational
     function

          (define p1 (make-polynomial 'x '((2 1)(0 1))))
          (define p2 (make-polynomial 'x '((3 1)(0 1))))
          (define rf (make-rational p2 p1))

     Now add `rf' to itself, using `add'. You will observe that this
     addition procedure does not reduce fractions to lowest terms.

     We can reduce polynomial fractions to lowest terms using the same
     idea we used with integers: modifying `make-rat' to divide both
     the numerator and the denominator by their greatest common
     divisor.  The notion of "greatest common divisor" makes sense for
     polynomials.  In fact, we can compute the GCD of two polynomials
     using essentially the same Euclid's Algorithm that works for
     integers.(7)  The integer version is

          (define (gcd a b)
            (if (= b 0)
                a
                (gcd b (remainder a b))))

     Using this, we could make the obvious modification to define a GCD
     operation that works on term lists:

          (define (gcd-terms a b)
            (if (empty-termlist? b)
                a
                (gcd-terms b (remainder-terms a b))))

     where `remainder-terms' picks out the remainder component of the
     list returned by the term-list division operation `div-terms' that
     was implemented in *Note Exercise 2-91::.

     *Exercise 2.94:* Using `div-terms', implement the procedure
     `remainder-terms' and use this to define `gcd-terms' as above.
     Now write a procedure `gcd-poly' that computes the polynomial GCD
     of two polys.  (The procedure should signal an error if the two
     polys are not in the same variable.)  Install in the system a
     generic operation `greatest-common-divisor' that reduces to
     `gcd-poly' for polynomials and to ordinary `gcd' for ordinary
     numbers.  As a test, try

          (define p1 (make-polynomial 'x '((4 1) (3 -1) (2 -2) (1 2))))
          (define p2 (make-polynomial 'x '((3 1) (1 -1))))
          (greatest-common-divisor p1 p2)

     and check your result by hand.

     *Exercise 2.95:* Define P_1, P_2, and P_3 to be the polynomials

          P_1 : x^2 - 2x + 1

          P_2 : 11x^2 + 7

          P_3 : 13x + 5

     Now define Q_1 to be the product of P_1 and P_2 and Q_2 to be the
     product of P_1 and P_3, and use `greatest-common-divisor' (*Note
     Exercise 2-94::) to compute the GCD of Q_1 and Q_2.  Note that the
     answer is not the same as P_1.  This example introduces noninteger
     operations into the computation, causing difficulties with the GCD
     algorithm.(8)  To understand what is happening, try tracing
     `gcd-terms' while computing the GCD or try performing the division
     by hand.

     We can solve the problem exhibited in *Note Exercise 2-95:: if we
     use the following modification of the GCD algorithm (which really
     works only in the case of polynomials with integer coefficients).
     Before performing any polynomial division in the GCD computation,
     we multiply the dividend by an integer constant factor, chosen to
     guarantee that no fractions will arise during the division
     process.  Our answer will thus differ from the actual GCD by an
     integer constant factor, but this does not matter in the case of
     reducing rational functions to lowest terms; the GCD will be used
     to divide both the numerator and denominator, so the integer
     constant factor will cancel out.

     More precisely, if P and Q are polynomials, let O_1 be the order of
     P (i.e., the order of the largest term of P) and let O_2 be the
     order of Q.  Let c be the leading coefficient of Q.  Then it can be
     shown that, if we multiply P by the "integerizing factor" c^(1+O_1
     -O_2), the resulting polynomial can be divided by Q by using the
     `div-terms' algorithm without introducing any fractions.  The
     operation of multiplying the dividend by this constant and then
     dividing is sometimes called the "pseudodivision" of P by Q.  The
     remainder of the division is called the "pseudoremainder".

     *Exercise 2.96:*
       a. Implement the procedure `pseudoremainder-terms', which is
          just like `remainder-terms' except that it multiplies the
          dividend by the integerizing factor described above before
          calling `div-terms'.  Modify `gcd-terms' to use
          `pseudoremainder-terms', and verify that
          `greatest-common-divisor' now produces an answer with integer
          coefficients on the example in *Note Exercise 2-95::.

       b. The GCD now has integer coefficients, but they are larger
          than those of P_1.  Modify `gcd-terms' so that it removes
          common factors from the coefficients of the answer by
          dividing all the coefficients by their (integer) greatest
          common divisor.


     Thus, here is how to reduce a rational function to lowest terms:

        * Compute the GCD of the numerator and denominator, using the
          version of `gcd-terms' from *Note Exercise 2-96::.

        * When you obtain the GCD, multiply both numerator and
          denominator by the same integerizing factor before dividing
          through by the GCD, so that division by the GCD will not
          introduce any noninteger coefficients.  As the factor you can
          use the leading coefficient of the GCD raised to the power 1
          + O_1 - O_2, where O_2 is the order of the GCD and O_1 is the
          maximum of the orders of the numerator and denominator.  This
          will ensure that dividing the numerator and denominator by
          the GCD will not introduce any fractions.

        * The result of this operation will be a numerator and
          denominator with integer coefficients.  The coefficients will
          normally be very large because of all of the integerizing
          factors, so the last step is to remove the redundant factors
          by computing the (integer) greatest common divisor of all the
          coefficients of the numerator and the denominator and
          dividing through by this factor.


     *Exercise 2.97:*
       a. Implement this algorithm as a procedure `reduce-terms' that
          takes two term lists `n' and `d' as arguments and returns a
          list `nn', `dd', which are `n' and `d' reduced to lowest
          terms via the algorithm given above.  Also write a procedure
          `reduce-poly', analogous to `add-poly', that checks to see if
          the two polys have the same variable.  If so, `reduce-poly'
          strips off the variable and passes the problem to
          `reduce-terms', then reattaches the variable to the two term
          lists supplied by `reduce-terms'.

       b. Define a procedure analogous to `reduce-terms' that does what
          the original `make-rat' did for integers:

               (define (reduce-integers n d)
                 (let ((g (gcd n d)))
                   (list (/ n g) (/ d g))))

          and define `reduce' as a generic operation that calls
          `apply-generic' to dispatch to either `reduce-poly' (for
          `polynomial' arguments) or `reduce-integers' (for
          `scheme-number' arguments).  You can now easily make the
          rational-arithmetic package reduce fractions to lowest terms
          by having `make-rat' call `reduce' before combining the given
          numerator and denominator to form a rational number.  The
          system now handles rational expressions in either integers or
          polynomials.  To test your program, try the example at the
          beginning of this extended exercise:

               (define p1 (make-polynomial 'x '((1 1)(0 1))))
               (define p2 (make-polynomial 'x '((3 1)(0 -1))))
               (define p3 (make-polynomial 'x '((1 1))))
               (define p4 (make-polynomial 'x '((2 1)(0 -1))))

               (define rf1 (make-rational p1 p2))
               (define rf2 (make-rational p3 p4))

               (add rf1 rf2)

          See if you get the correct answer, correctly reduced to
          lowest terms.

          The GCD computation is at the heart of any system that does
          operations on rational functions.  The algorithm used above,
          although mathematically straightforward, is extremely slow.
          The slowness is due partly to the large number of division
          operations and partly to the enormous size of the
          intermediate coefficients generated by the pseudodivisions.
          One of the active areas in the development of
          algebraic-manipulation systems is the design of better
          algorithms for computing polynomial GCDs.(9)


   ---------- Footnotes ----------

   (1) On the other hand, we will allow polynomials whose coefficients
are themselves polynomials in other variables.  This will give us
essentially the same representational power as a full multivariate
system, although it does lead to coercion problems, as discussed below.

   (2) For univariate polynomials, giving the value of a polynomial at
a given set of points can be a particularly good representation.  This
makes polynomial arithmetic extremely simple.  To obtain, for example,
the sum of two polynomials represented in this way, we need only add
the values of the polynomials at corresponding points.  To transform
back to a more familiar representation, we can use the Lagrange
interpolation formula, which shows how to recover the coefficients of a
polynomial of degree n given the values of the polynomial at n + 1
points.

   (3) This operation is very much like the ordered `union-set'
operation we developed in exercise *Note Exercise 2-62::.  In fact, if
we think of the terms of the polynomial as a set ordered according to
the power of the indeterminate, then the program that produces the term
list for a sum is almost identical to `union-set'.

   (4) To make this work completely smoothly, we should also add to our
generic arithmetic system the ability to coerce a "number" to a
polynomial by regarding it as a polynomial of degree zero whose
coefficient is the number.  This is necessary if we are going to
perform operations such as

     [x^2 + (y + 1)x + 5] + [x^2 + 2x + 1]

which requires adding the coefficient y + 1 to the coefficient 2.

   (5) In these polynomial examples, we assume that we have implemented
the generic arithmetic system using the type mechanism suggested in
*Note Exercise 2-78::.  Thus, coefficients that are ordinary numbers
will be represented as the numbers themselves rather than as pairs
whose `car' is the symbol `scheme-number'.

   (6) Although we are assuming that term lists are ordered, we have
implemented `adjoin-term' to simply `cons' the new term onto the
existing term list.  We can get away with this so long as we guarantee
that the procedures (such as `add-terms') that use `adjoin-term' always
call it with a higher-order term than appears in the list.  If we did
not want to make such a guarantee, we could have implemented
`adjoin-term' to be similar to the `adjoin-set' constructor for the
ordered-list representation of sets (*Note Exercise 2-61::).

   (7) The fact that Euclid's Algorithm works for polynomials is
formalized in algebra by saying that polynomials form a kind of
algebraic domain called a "Euclidean ring".  A Euclidean ring is a
domain that admits addition, subtraction, and commutative
multiplication, together with a way of assigning to each element x of
the ring a positive integer "measure" m(x) with the properties that
m(xy)>= m(x) for any nonzero x and y and that, given any x and y, there
exists a q such that y = qx + r and either r = 0 or m(r)< m(x).  From
an abstract point of view, this is what is needed to prove that
Euclid's Algorithm works.  For the domain of integers, the measure m of
an integer is the absolute value of the integer itself.  For the domain
of polynomials, the measure of a polynomial is its degree.

   (8) In an implementation like MIT Scheme, this produces a polynomial
that is indeed a divisor of Q_1 and Q_2, but with rational
coefficients.  In many other Scheme systems, in which division of
integers can produce limited-precision decimal numbers, we may fail to
get a valid divisor.

   (9) One extremely efficient and elegant method for computing
polynomial GCDs was discovered by Richard Zippel (1979).  The method is
a probabilistic algorithm, as is the fast test for primality that we
discussed in *Note Chapter 1::.  Zippel's book (1993) describes this
method, together with other ways to compute polynomial GCDs.


File: sicp.info,  Node: Chapter 3,  Next: Chapter 4,  Prev: Chapter 2,  Up: Top

3 Modularity, Objects, and State
********************************

     [greek not included here]

     (Even while it changes, it stands still.)

     --Heraclitus

     Plus c,a change, plus c'est la me*me chose.

     --Alphonse Karr

   The preceding chapters introduced the basic elements from which
programs are made.  We saw how primitive procedures and primitive data
are combined to construct compound entities, and we learned that
abstraction is vital in helping us to cope with the complexity of large
systems.  But these tools are not sufficient for designing programs.
Effective program synthesis also requires organizational principles
that can guide us in formulating the overall design of a program.  In
particular, we need strategies to help us structure large systems so
that they will be "modular", that is, so that they can be divided
"naturally" into coherent parts that can be separately developed and
maintained.

   One powerful design strategy, which is particularly appropriate to
the construction of programs for modeling physical systems, is to base
the structure of our programs on the structure of the system being
modeled.  For each object in the system, we construct a corresponding
computational object.  For each system action, we define a symbolic
operation in our computational model.  Our hope in using this strategy
is that extending the model to accommodate new objects or new actions
will require no strategic changes to the program, only the addition of
the new symbolic analogs of those objects or actions.  If we have been
successful in our system organization, then to add a new feature or
debug an old one we will have to work on only a localized part of the
system.

   To a large extent, then, the way we organize a large program is
dictated by our perception of the system to be modeled.  In this
chapter we will investigate two prominent organizational strategies
arising from two rather different "world views" of the structure of
systems.  The first organizational strategy concentrates on "objects",
viewing a large system as a collection of distinct objects whose
behaviors may change over time.  An alternative organizational strategy
concentrates on the "streams" of information that flow in the system,
much as an electrical engineer views a signal-processing system.

   Both the object-based approach and the stream-processing approach
raise significant linguistic issues in programming.  With objects, we
must be concerned with how a computational object can change and yet
maintain its identity.  This will force us to abandon our old
substitution model of computation (section *Note 1-1-5::) in favor of a
more mechanistic but less theoretically tractable "environment model"
of computation.  The difficulties of dealing with objects, change, and
identity are a fundamental consequence of the need to grapple with time
in our computational models.  These difficulties become even greater
when we allow the possibility of concurrent execution of programs.  The
stream approach can be most fully exploited when we decouple simulated
time in our model from the order of the events that take place in the
computer during evaluation.  We will accomplish this using a technique
known as "delayed evaluation".

* Menu:

* 3-1::              Assignment and Local State
* 3-2::              The Environment Model of Evaluation
* 3-3::              Modeling with Mutable Data
* 3-4::              Concurrency: Time Is of the Essence
* 3-5::              Streams


File: sicp.info,  Node: 3-1,  Next: 3-2,  Prev: Chapter 3,  Up: Chapter 3

3.1 Assignment and Local State
==============================

We ordinarily view the world as populated by independent objects, each
of which has a state that changes over time.  An object is said to
"have state" if its behavior is influenced by its history.  A bank
account, for example, has state in that the answer to the question "Can
I withdraw $100?"  depends upon the history of deposit and withdrawal
transactions.  We can characterize an object's state by one or more "state
variables", which among them maintain enough information about history
to determine the object's current behavior.  In a simple banking
system, we could characterize the state of an account by a current
balance rather than by remembering the entire history of account
transactions.

   In a system composed of many objects, the objects are rarely
completely independent.  Each may influence the states of others
through interactions, which serve to couple the state variables of one
object to those of other objects.  Indeed, the view that a system is
composed of separate objects is most useful when the state variables of
the system can be grouped into closely coupled subsystems that are only
loosely coupled to other subsystems.

   This view of a system can be a powerful framework for organizing
computational models of the system.  For such a model to be modular, it
should be decomposed into computational objects that model the actual
objects in the system.  Each computational object must have its own "local
state variables" describing the actual object's state.  Since the
states of objects in the system being modeled change over time, the
state variables of the corresponding computational objects must also
change.  If we choose to model the flow of time in the system by the
elapsed time in the computer, then we must have a way to construct
computational objects whose behaviors change as our programs run.  In
particular, if we wish to model state variables by ordinary symbolic
names in the programming language, then the language must provide an operator
"assignment operator" to enable us to change the value associated with
a name.

* Menu:

* 3-1-1::            Local State Variables
* 3-1-2::            The Benefits of Introducing Assignment
* 3-1-3::            The Costs of Introducing Assignment


File: sicp.info,  Node: 3-1-1,  Next: 3-1-2,  Prev: 3-1,  Up: 3-1

3.1.1 Local State Variables
---------------------------

To illustrate what we mean by having a computational object with
time-varying state, let us model the situation of withdrawing money
from a bank account.  We will do this using a procedure `withdraw',
which takes as argument an `amount' to be withdrawn.  If there is
enough money in the account to accommodate the withdrawal, then
`withdraw' should return the balance remaining after the withdrawal.
Otherwise, `withdraw' should return the message _Insufficient funds_.
For example, if we begin with $100 in the account, we should obtain the
following sequence of responses using `withdraw':

     (withdraw 25)
     75

     (withdraw 25)
     50

     (withdraw 60)
     "Insufficient funds"

     (withdraw 15)
     35

   Observe that the expression `(withdraw 25)', evaluated twice, yields
different values.  This is a new kind of behavior for a procedure.
Until now, all our procedures could be viewed as specifications for
computing mathematical functions.  A call to a procedure computed the
value of the function applied to the given arguments, and two calls to
the same procedure with the same arguments always produced the same
result.(1)

   To implement `withdraw', we can use a variable `balance' to indicate
the balance of money in the account and define `withdraw' as a procedure
that accesses `balance'.  The `withdraw' procedure checks to see if
`balance' is at least as large as the requested `amount'.  If so,
`withdraw' decrements `balance' by `amount' and returns the new value
of `balance'.  Otherwise, `withdraw' returns the _Insufficient funds_
message.  Here are the definitions of `balance' and `withdraw':

     (define balance 100)

     (define (withdraw amount)
       (if (>= balance amount)
           (begin (set! balance (- balance amount))
                  balance)
           "Insufficient funds"))

   Decrementing `balance' is accomplished by the expression

     (set! balance (- balance amount))

   This uses the `set!' special form, whose syntax is

     (set! <NAME> <NEW-VALUE>)

   Here <NAME> is a symbol and <NEW-VALUE> is any expression.  `Set!'
changes <NAME> so that its value is the result obtained by evaluating
<NEW-VALUE>.  In the case at hand, we are changing `balance' so that
its new value will be the result of subtracting `amount' from the
previous value of `balance'.(2)

   `Withdraw' also uses the `begin' special form to cause two
expressions to be evaluated in the case where the `if' test is true:
first decrementing `balance' and then returning the value of `balance'.
In general, evaluating the expression

     (begin <EXP_1> <EXP_2> ... <EXP_K>)

causes the expressions <EXP_1> through <EXP_K> to be evaluated in
sequence and the value of the final expression <EXP_K> to be returned
as the value of the entire `begin' form.(3)

   Although `withdraw' works as desired, the variable `balance' presents
a problem.  As specified above, `balance' is a name defined in the
global environment and is freely accessible to be examined or modified
by any procedure.  It would be much better if we could somehow make
`balance' internal to `withdraw', so that `withdraw' would be the only
procedure that could access `balance' directly and any other procedure
could access `balance' only indirectly (through calls to `withdraw').
This would more accurately model the notion that `balance' is a local
state variable used by `withdraw' to keep track of the state of the
account.

   We can make `balance' internal to `withdraw' by rewriting the
definition as follows:

     (define new-withdraw
       (let ((balance 100))
         (lambda (amount)
           (if (>= balance amount)
               (begin (set! balance (- balance amount))
                      balance)
               "Insufficient funds"))))

   What we have done here is use `let' to establish an environment with
a local variable `balance', bound to the initial value 100.  Within this
local environment, we use `lambda' to create a procedure that takes
`amount' as an argument and behaves like our previous `withdraw'
procedure.  This procedure--returned as the result of evaluating the
`let' expression--is `new-withdraw', which behaves in precisely the
same way as `withdraw' but whose variable `balance' is not accessible
by any other procedure.(4)

   Combining `set!' with local variables is the general programming
technique we will use for constructing computational objects with local
state.  Unfortunately, using this technique raises a serious problem:
When we first introduced procedures, we also introduced the
substitution model of evaluation (section *Note 1-1-5::) to provide an
interpretation of what procedure application means.  We said that
applying a procedure should be interpreted as evaluating the body of
the procedure with the formal parameters replaced by their values.  The
trouble is that, as soon as we introduce assignment into our language,
substitution is no longer an adequate model of procedure application.
(We will see why this is so in section *Note 3-1-3::.)  As a
consequence, we technically have at this point no way to understand why
the `new-withdraw' procedure behaves as claimed above.  In order to
really understand a procedure such as `new-withdraw', we will need to
develop a new model of procedure application.  In section *Note 3-2::
we will introduce such a model, together with an explanation of `set!'
and local variables.  First, however, we examine some variations on the
theme established by `new-withdraw'.

   The following procedure, `make-withdraw', creates "withdrawal
processors."  The formal parameter `balance' in `make-withdraw'
specifies the initial amount of money in the account.(5)

     (define (make-withdraw balance)
       (lambda (amount)
         (if (>= balance amount)
             (begin (set! balance (- balance amount))
                    balance)
             "Insufficient funds")))

   `Make-withdraw' can be used as follows to create two objects `W1' and
`W2':

     (define W1 (make-withdraw 100))
     (define W2 (make-withdraw 100))

     (W1 50)
     50

     (W2 70)
     30

     (W2 40)
     "Insufficient funds"

     (W1 40)
     10

   Observe that `W1' and `W2' are completely independent objects, each
with its own local state variable `balance'.  Withdrawals from one do
not affect the other.

   We can also create objects that handle deposits as well as
withdrawals, and thus we can represent simple bank accounts.  Here is a
procedure that returns a "bank-account object" with a specified initial
balance:

     (define (make-account balance)
       (define (withdraw amount)
         (if (>= balance amount)
             (begin (set! balance (- balance amount))
                    balance)
             "Insufficient funds"))
       (define (deposit amount)
         (set! balance (+ balance amount))
         balance)
       (define (dispatch m)
         (cond ((eq? m 'withdraw) withdraw)
               ((eq? m 'deposit) deposit)
               (else (error "Unknown request -- MAKE-ACCOUNT"
                            m))))
       dispatch)

   Each call to `make-account' sets up an environment with a local state
variable `balance'.  Within this environment, `make-account' defines
procedures `deposit' and `withdraw' that access `balance' and an
additional procedure `dispatch' that takes a "message" as input and
returns one of the two local procedures.  The `dispatch' procedure
itself is returned as the value that represents the bank-account
object.  This is precisely the "message-passing" style of programming
that we saw in section *Note 2-4-3::, although here we are using it in
conjunction with the ability to modify local variables.

   `Make-account' can be used as follows:

     (define acc (make-account 100))

     ((acc 'withdraw) 50)
     50

     ((acc 'withdraw) 60)
     "Insufficient funds"

     ((acc 'deposit) 40)
     90

     ((acc 'withdraw) 60)
     30

   Each call to `acc' returns the locally defined `deposit' or
`withdraw' procedure, which is then applied to the specified `amount'.
As was the case with `make-withdraw', another call to `make-account'

     (define acc2 (make-account 100))

will produce a completely separate account object, which maintains its
own local `balance'.

     *Exercise 3.1:* An "accumulator" is a procedure that is called
     repeatedly with a single numeric argument and accumulates its
     arguments into a sum.  Each time it is called, it returns the
     currently accumulated sum.  Write a procedure `make-accumulator'
     that generates accumulators, each maintaining an independent sum.
     The input to `make-accumulator' should specify the initial value
     of the sum; for example

          (define A (make-accumulator 5))

          (A 10)
          15

          (A 10)
          25

     *Exercise 3.2:* In software-testing applications, it is useful to
     be able to count the number of times a given procedure is called
     during the course of a computation.  Write a procedure
     `make-monitored' that takes as input a procedure, `f', that itself
     takes one input.  The result returned by `make-monitored' is a
     third procedure, say `mf', that keeps track of the number of times
     it has been called by maintaining an internal counter.  If the
     input to `mf' is the special symbol `how-many-calls?', then `mf'
     returns the value of the counter.  If the input is the special
     symbol `reset-count', then `mf' resets the counter to zero.  For
     any other input, `mf' returns the result of calling `f' on that
     input and increments the counter.  For instance, we could make a
     monitored version of the `sqrt' procedure:

          (define s (make-monitored sqrt))

          (s 100)
          10

          (s 'how-many-calls?)
          1

     *Exercise 3.3:* Modify the `make-account' procedure so that it
     creates password-protected accounts.  That is, `make-account'
     should take a symbol as an additional argument, as in

          (define acc (make-account 100 'secret-password))

     The resulting account object should process a request only if it
     is accompanied by the password with which the account was created,
     and should otherwise return a complaint:

          ((acc 'secret-password 'withdraw) 40)
          60

          ((acc 'some-other-password 'deposit) 50)
          "Incorrect password"

     *Exercise 3.4:* Modify the `make-account' procedure of *Note
     Exercise 3-3:: by adding another local state variable so that, if
     an account is accessed more than seven consecutive times with an
     incorrect password, it invokes the procedure `call-the-cops'.

   ---------- Footnotes ----------

   (1) Actually, this is not quite true.  One exception was the
random-number generator in section *Note 1-2-6::.  Another exception
involved the operation/type tables we introduced in section *Note
2-4-3::, where the values of two calls to `get' with the same arguments
depended on intervening calls to `put'.  On the other hand, until we
introduce assignment, we have no way to create such procedures
ourselves.

   (2) The value of a `set!' expression is implementation-dependent.
`Set!' should be used only for its effect, not for its value.

   The name `set!' reflects a naming convention used in Scheme:
Operations that change the values of variables (or that change data
structures, as we will see in section *Note 3-3::) are given names that
end with an exclamation point.  This is similar to the convention of
designating predicates by names that end with a question mark.

   (3) We have already used `begin' implicitly in our programs, because
in Scheme the body of a procedure can be a sequence of expressions.
Also, the <CONSEQUENT> part of each clause in a `cond' expression can
be a sequence of expressions rather than a single expression.

   (4) In programming-language jargon, the variable `balance' is said
to be "encapsulated" within the `new-withdraw' procedure.
Encapsulation reflects the general system-design principle known as the "hiding
principle": One can make a system more modular and robust by protecting
parts of the system from each other; that is, by providing information
access only to those parts of the system that have a "need to know."

   (5) In contrast with `new-withdraw' above, we do not have to use
`let' to make `balance' a local variable, since formal parameters are
already local.  This will be clearer after the discussion of the
environment model of evaluation in section *Note 3-2::.  (See also
*Note Exercise 3-10::.)


File: sicp.info,  Node: 3-1-2,  Next: 3-1-3,  Prev: 3-1-1,  Up: 3-1

3.1.2 The Benefits of Introducing Assignment
--------------------------------------------

As we shall see, introducing assignment into our programming language
leads us into a thicket of difficult conceptual issues.  Nevertheless,
viewing systems as collections of objects with local state is a
powerful technique for maintaining a modular design.  As a simple
example, consider the design of a procedure `rand' that, whenever it is
called, returns an integer chosen at random.

   It is not at all clear what is meant by "chosen at random."  What we
presumably want is for successive calls to `rand' to produce a sequence
of numbers that has statistical properties of uniform distribution.  We
will not discuss methods for generating suitable sequences here.
Rather, let us assume that we have a procedure `rand-update' that has
the property that if we start with a given number x_1 and form

     x_2 = (rand-update x_1)
     x_3 = (rand-update x_2)

then the sequence of values x_1, x_2, x_3, ..., will have the desired
statistical properties.(1)

   We can implement `rand' as a procedure with a local state variable
`x' that is initialized to some fixed value `random-init'.  Each call
to `rand' computes `rand-update' of the current value of `x', returns
this as the random number, and also stores this as the new value of `x'.

     (define rand
       (let ((x random-init))
         (lambda ()
           (set! x (rand-update x))
           x)))

   Of course, we could generate the same sequence of random numbers
without using assignment by simply calling `rand-update' directly.
However, this would mean that any part of our program that used random
numbers would have to explicitly remember the current value of `x' to
be passed as an argument to `rand-update'.  To realize what an
annoyance this would be, consider using random numbers to implement a
technique called simulation "Monte Carlo simulation".

   The Monte Carlo method consists of choosing sample experiments at
random from a large set and then making deductions on the basis of the
probabilities estimated from tabulating the results of those
experiments.  For example, we can approximate [pi] using the fact that
6/[pi]^2 is the probability that two integers chosen at random will
have no factors in common; that is, that their greatest common divisor
will be 1.(2) To obtain the approximation to [pi], we perform a large
number of experiments.  In each experiment we choose two integers at
random and perform a test to see if their GCD is 1.  The fraction of
times that the test is passed gives us our estimate of 6/[pi]^2, and
from this we obtain our approximation to [pi].

   The heart of our program is a procedure `monte-carlo', which takes as
arguments the number of times to try an experiment, together with the
experiment, represented as a no-argument procedure that will return
either true or false each time it is run.  `Monte-carlo' runs the
experiment for the designated number of trials and returns a number
telling the fraction of the trials in which the experiment was found to
be true.

     (define (estimate-pi trials)
       (sqrt (/ 6 (monte-carlo trials cesaro-test))))

     (define (cesaro-test)
        (= (gcd (rand) (rand)) 1))

     (define (monte-carlo trials experiment)
       (define (iter trials-remaining trials-passed)
         (cond ((= trials-remaining 0)
                (/ trials-passed trials))
               ((experiment)
                (iter (- trials-remaining 1) (+ trials-passed 1)))
               (else
                (iter (- trials-remaining 1) trials-passed))))
       (iter trials 0))

   Now let us try the same computation using `rand-update' directly
rather than `rand', the way we would be forced to proceed if we did not
use assignment to model local state:

     (define (estimate-pi trials)
       (sqrt (/ 6 (random-gcd-test trials random-init))))

     (define (random-gcd-test trials initial-x)
       (define (iter trials-remaining trials-passed x)
         (let ((x1 (rand-update x)))
           (let ((x2 (rand-update x1)))
             (cond ((= trials-remaining 0)
                    (/ trials-passed trials))
                   ((= (gcd x1 x2) 1)
                    (iter (- trials-remaining 1)
                          (+ trials-passed 1)
                          x2))
                   (else
                    (iter (- trials-remaining 1)
                          trials-passed
                          x2))))))
       (iter trials 0 initial-x))

   While the program is still simple, it betrays some painful breaches
of modularity.  In our first version of the program, using `rand', we
can express the Monte Carlo method directly as a general `monte-carlo'
procedure that takes as an argument an arbitrary `experiment' procedure.
In our second version of the program, with no local state for the
random-number generator, `random-gcd-test' must explicitly manipulate
the random numbers `x1' and `x2' and recycle `x2' through the iterative
loop as the new input to `rand-update'.  This explicit handling of the
random numbers intertwines the structure of accumulating test results
with the fact that our particular experiment uses two random numbers,
whereas other Monte Carlo experiments might use one random number or
three.  Even the top-level procedure `estimate-pi' has to be concerned
with supplying an initial random number.  The fact that the
random-number generator's insides are leaking out into other parts of
the program makes it difficult for us to isolate the Monte Carlo idea
so that it can be applied to other tasks.  In the first version of the
program, assignment encapsulates the state of the random-number
generator within the `rand' procedure, so that the details of
random-number generation remain independent of the rest of the program.

   The general phenomenon illustrated by the Monte Carlo example is
this: From the point of view of one part of a complex process, the
other parts appear to change with time.  They have hidden time-varying
local state.  If we wish to write computer programs whose structure
reflects this decomposition, we make computational objects (such as
bank accounts and random-number generators) whose behavior changes with
time.  We model state with local state variables, and we model the
changes of state with assignments to those variables.

   It is tempting to conclude this discussion by saying that, by
introducing assignment and the technique of hiding state in local
variables, we are able to structure systems in a more modular fashion
than if all state had to be manipulated explicitly, by passing
additional parameters.  Unfortunately, as we shall see, the story is
not so simple.

     *Exercise 3.5:* "Monte Carlo integration" is a method of
     estimating definite integrals by means of Monte Carlo simulation.
     Consider computing the area of a region of space described by a
     predicate P(x, y) that is true for points (x, y) in the region and
     false for points not in the region.  For example, the region
     contained within a circle of radius 3 centered at (5, 7) is
     described by the predicate that tests whether (x - 5)^2 + (y -
     7)^2 <= 3^2.  To estimate the area of the region described by such
     a predicate, begin by choosing a rectangle that contains the
     region.  For example, a rectangle with diagonally opposite corners
     at (2, 4) and (8, 10) contains the circle above.  The desired
     integral is the area of that portion of the rectangle that lies in
     the region.  We can estimate the integral by picking, at random,
     points (x,y) that lie in the rectangle, and testing P(x, y) for
     each point to determine whether the point lies in the region.  If
     we try this with many points, then the fraction of points that
     fall in the region should give an estimate of the proportion of
     the rectangle that lies in the region.  Hence, multiplying this
     fraction by the area of the entire rectangle should produce an
     estimate of the integral.

     Implement Monte Carlo integration as a procedure
     `estimate-integral' that takes as arguments a predicate `P', upper
     and lower bounds `x1', `x2', `y1', and `y2' for the rectangle, and
     the number of trials to perform in order to produce the estimate.
     Your procedure should use the same `monte-carlo' procedure that
     was used above to estimate [pi].  Use your `estimate-integral' to
     produce an estimate of [pi] by measuring the area of a unit circle.

     You will find it useful to have a procedure that returns a number
     chosen at random from a given range.  The following
     `random-in-range' procedure implements this in terms of the
     `random' procedure used in section *Note 1-2-6::, which returns a
     nonnegative number less than its input.(3)

          (define (random-in-range low high)
            (let ((range (- high low)))
              (+ low (random range))))

     *Exercise 3.6:* It is useful to be able to reset a random-number
     generator to produce a sequence starting from a given value.
     Design a new `rand' procedure that is called with an argument that
     is either the symbol `generate' or the symbol `reset' and behaves
     as follows: `(rand 'generate)' produces a new random number;
     `((rand 'reset) <NEW-VALUE>)' resets the internal state variable
     to the designated <NEW-VALUE>.  Thus, by resetting the state, one
     can generate repeatable sequences.  These are very handy to have
     when testing and debugging programs that use random numbers.

   ---------- Footnotes ----------

   (1) One common way to implement `rand-update' is to use the rule
that x is updated to ax + b modulo m, where a, b, and m are
appropriately chosen integers.  Chapter 3 of Knuth 1981 includes an
extensive discussion of techniques for generating sequences of random
numbers and establishing their statistical properties.  Notice that the
`rand-update' procedure computes a mathematical function: Given the
same input twice, it produces the same output.  Therefore, the number
sequence produced by `rand-update' certainly is not "random," if by
"random" we insist that each number in the sequence is unrelated to the
preceding number.  The relation between "real randomness" and so-called "pseudo-random"
sequences, which are produced by well-determined computations and yet
have suitable statistical properties, is a complex question involving
difficult issues in mathematics and philosophy.  Kolmogorov,
Solomonoff, and Chaitin have made great progress in clarifying these
issues; a discussion can be found in Chaitin 1975.

   (2) This theorem is due to E. Cesa`ro.  See section 4.5.2 of Knuth
1981 for a discussion and a proof.

   (3) MIT Scheme provides such a procedure.  If `random' is given an
exact integer (as in section *Note 1-2-6::) it returns an exact
integer, but if it is given a decimal value (as in this exercise) it
returns a decimal value.


File: sicp.info,  Node: 3-1-3,  Prev: 3-1-2,  Up: 3-1

3.1.3 The Costs of Introducing Assignment
-----------------------------------------

As we have seen, the `set!' operation enables us to model objects that
have local state.  However, this advantage comes at a price.  Our
programming language can no longer be interpreted in terms of the
substitution model of procedure application that we introduced in
section *Note 1-1-5::.  Moreover, no simple model with "nice"
mathematical properties can be an adequate framework for dealing with
objects and assignment in programming languages.

   So long as we do not use assignments, two evaluations of the same
procedure with the same arguments will produce the same result, so that
procedures can be viewed as computing mathematical functions.
Programming without any use of assignments, as we did throughout the
first two chapters of this book, is accordingly known as "functional
programming".

   To understand how assignment complicates matters, consider a
simplified version of the `make-withdraw' procedure of section *Note
3-1-1:: that does not bother to check for an insufficient amount:

     (define (make-simplified-withdraw balance)
       (lambda (amount)
         (set! balance (- balance amount))
         balance))

     (define W (make-simplified-withdraw 25))

     (W 20)
     5

     (W 10)
      - 5

   Compare this procedure with the following `make-decrementer'
procedure, which does not use `set!':

     (define (make-decrementer balance)
       (lambda (amount)
         (- balance amount)))

   `Make-decrementer' returns a procedure that subtracts its input from
a designated amount `balance', but there is no accumulated effect over
successive calls, as with `make-simplified-withdraw':

     (define D (make-decrementer 25))

     (D 20)
     5

     (D 10)
     15

   We can use the substitution model to explain how `make-decrementer'
works.  For instance, let us analyze the evaluation of the expression

     ((make-decrementer 25) 20)

   We first simplify the operator of the combination by substituting 25
for `balance' in the body of `make-decrementer'.  This reduces the
expression to

     ((lambda (amount) (- 25 amount)) 20)

   Now we apply the operator by substituting 20 for `amount' in the
body of the `lambda' expression:

     (- 25 20)

   The final answer is 5.

   Observe, however, what happens if we attempt a similar substitution
analysis with `make-simplified-withdraw':

     ((make-simplified-withdraw 25) 20)

   We first simplify the operator by substituting 25 for `balance' in
the body of `make-simplified-withdraw'.  This reduces the expression
to(1)

     ((lambda (amount) (set! balance (- 25 amount)) 25) 20)

   Now we apply the operator by substituting 20 for `amount' in the
body of the `lambda' expression:

     (set! balance (- 25 20)) 25

   If we adhered to the substitution model, we would have to say that
the meaning of the procedure application is to first set `balance' to 5
and then return 25 as the value of the expression.  This gets the wrong
answer.  In order to get the correct answer, we would have to somehow
distinguish the first occurrence of `balance' (before the effect of the
`set!')  from the second occurrence of `balance' (after the effect of
the `set!'), and the substitution model cannot do this.

   The trouble here is that substitution is based ultimately on the
notion that the symbols in our language are essentially names for
values.  But as soon as we introduce `set!' and the idea that the value
of a variable can change, a variable can no longer be simply a name.
Now a variable somehow refers to a place where a value can be stored,
and the value stored at this place can change.  In section *Note 3-2::
we will see how environments play this role of "place" in our
computational model.

Sameness and change
...................

The issue surfacing here is more profound than the mere breakdown of a
particular model of computation.  As soon as we introduce change into
our computational models, many notions that were previously
straightforward become problematical.  Consider the concept of two
things being "the same."

   Suppose we call `make-decrementer' twice with the same argument to
create two procedures:

     (define D1 (make-decrementer 25))

     (define D2 (make-decrementer 25))

   Are `D1' and `D2' the same?  An acceptable answer is yes, because
`D1' and `D2' have the same computational behavior--each is a procedure
that subtracts its input from 25.  In fact, `D1' could be substituted
for `D2' in any computation without changing the result.

   Contrast this with making two calls to `make-simplified-withdraw':

     (define W1 (make-simplified-withdraw 25))

     (define W2 (make-simplified-withdraw 25))

   Are `W1' and `W2' the same?  Surely not, because calls to `W1' and
`W2' have distinct effects, as shown by the following sequence of
interactions:

     (W1 20)
     5

     (W1 20)
      - 15

     (W2 20)
     5

   Even though `W1' and `W2' are "equal" in the sense that they are
both created by evaluating the same expression,
`(make-simplified-withdraw 25)', it is not true that `W1' could be
substituted for `W2' in any expression without changing the result of
evaluating the expression.

   A language that supports the concept that "equals can be substituted
for equals" in an expresssion without changing the value of the
expression is said to be "referentially transparent".  Referential
transparency is violated when we include `set!' in our computer
language.  This makes it tricky to determine when we can simplify
expressions by substituting equivalent expressions.  Consequently,
reasoning about programs that use assignment becomes drastically more
difficult.

   Once we forgo referential transparency, the notion of what it means
for computational objects to be "the same" becomes difficult to capture
in a formal way.  Indeed, the meaning of "same" in the real world that
our programs model is hardly clear in itself.  In general, we can
determine that two apparently identical objects are indeed "the same
one" only by modifying one object and then observing whether the other
object has changed in the same way.  But how can we tell if an object
has "changed" other than by observing the "same" object twice and
seeing whether some property of the object differs from one observation
to the next?  Thus, we cannot determine "change" without some _a
priori_ notion of "sameness," and we cannot determine sameness without
observing the effects of change.

   As an example of how this issue arises in programming, consider the
situation where Peter and Paul have a bank account with $100 in it.
There is a substantial difference between modeling this as

     (define peter-acc (make-account 100))
     (define paul-acc (make-account 100))

and modeling it as

     (define peter-acc (make-account 100))
     (define paul-acc peter-acc)

   In the first situation, the two bank accounts are distinct.
Transactions made by Peter will not affect Paul's account, and vice
versa.  In the second situation, however, we have defined `paul-acc' to
be _the same thing_ as `peter-acc'.  In effect, Peter and Paul now have
a joint bank account, and if Peter makes a withdrawal from `peter-acc'
Paul will observe less money in `paul-acc'.  These two similar but
distinct situations can cause confusion in building computational
models.  With the shared account, in particular, it can be especially
confusing that there is one object (the bank account) that has two
different names (`peter-acc' and `paul-acc'); if we are searching for
all the places in our program where `paul-acc' can be changed, we must
remember to look also at things that change `peter-acc'.(2)

   With reference to the above remarks on "sameness" and "change,"
observe that if Peter and Paul could only examine their bank balances,
and could not perform operations that changed the balance, then the
issue of whether the two accounts are distinct would be moot.  In
general, so long as we never modify data objects, we can regard a
compound data object to be precisely the totality of its pieces.  For
example, a rational number is determined by giving its numerator and
its denominator.  But this view is no longer valid in the presence of
change, where a compound data object has an "identity" that is
something different from the pieces of which it is composed.  A bank
account is still "the same" bank account even if we change the balance
by making a withdrawal; conversely, we could have two different bank
accounts with the same state information.  This complication is a
consequence, not of our programming language, but of our perception of
a bank account as an object.  We do not, for example, ordinarily regard
a rational number as a changeable object with identity, such that we
could change the numerator and still have "the same" rational number.

Pitfalls of imperative programming
..................................

In contrast to functional programming, programming that makes extensive
use of assignment is known as "imperative programming".  In addition to
raising complications about computational models, programs written in
imperative style are susceptible to bugs that cannot occur in functional
programs.  For example, recall the iterative factorial program from
section *Note 1-2-1:::

     (define (factorial n)
       (define (iter product counter)
         (if (> counter n)
             product
             (iter (* counter product)
                   (+ counter 1))))
       (iter 1 1))

   Instead of passing arguments in the internal iterative loop, we
could adopt a more imperative style by using explicit assignment to
update the values of the variables `product' and `counter':

     (define (factorial n)
       (let ((product 1)
             (counter 1))
         (define (iter)
           (if (> counter n)
               product
               (begin (set! product (* counter product))
                      (set! counter (+ counter 1))
                      (iter))))
         (iter)))

   This does not change the results produced by the program, but it
does introduce a subtle trap.  How do we decide the order of the
assignments?  As it happens, the program is correct as written.  But
writing the assignments in the opposite order

     (set! counter (+ counter 1))
     (set! product (* counter product))

would have produced a different, incorrect result.  In general,
programming with assignment forces us to carefully consider the
relative orders of the assignments to make sure that each statement is
using the correct version of the variables that have been changed.
This issue simply does not arise in functional programs.(3)

   The complexity of imperative programs becomes even worse if we
consider applications in which several processes execute concurrently.
We will return to this in section *Note 3-4::.  First, however, we will
address the issue of providing a computational model for expressions
that involve assignment, and explore the uses of objects with local
state in designing simulations.

     *Exercise 3.7:* Consider the bank account objects created by
     `make-account', with the password modification described in *Note
     Exercise 3-3::.  Suppose that our banking system requires the
     ability to make joint accounts.  Define a procedure `make-joint'
     that accomplishes this.  `Make-joint' should take three arguments.
     The first is a password-protected account.  The second argument
     must match the password with which the account was defined in
     order for the `make-joint' operation to proceed.  The third
     argument is a new password.  `Make-joint' is to create an
     additional access to the original account using the new password.
     For example, if `peter-acc' is a bank account with password
     `open-sesame', then

          (define paul-acc
            (make-joint peter-acc 'open-sesame 'rosebud))

     will allow one to make transactions on `peter-acc' using the name
     `paul-acc' and the password `rosebud'.  You may wish to modify your
     solution to *Note Exercise 3-3:: to accommodate this new feature

     *Exercise 3.8:* When we defined the evaluation model in section
     *Note 1-1-3::, we said that the first step in evaluating an
     expression is to evaluate its subexpressions.  But we never
     specified the order in which the subexpressions should be
     evaluated (e.g., left to right or right to left).  When we
     introduce assignment, the order in which the arguments to a
     procedure are evaluated can make a difference to the result.
     Define a simple procedure `f' such that evaluating `(+ (f 0) (f
     1))' will return 0 if the arguments to `+' are evaluated from left
     to right but will return 1 if the arguments are evaluated from
     right to left.

   ---------- Footnotes ----------

   (1) We don't substitute for the occurrence of `balance' in the
`set!' expression because the <NAME> in a `set!' is not evaluated.  If
we did substitute for it, we would get `(set! 25 (- 25 amount))', which
makes no sense.

   (2) The phenomenon of a single computational object being accessed
by more than one name is known as "aliasing".  The joint bank account
situation illustrates a very simple example of an alias.  In section
*Note 3-3:: we will see much more complex examples, such as "distinct"
compound data structures that share parts.  Bugs can occur in our
programs if we forget that a change to an object may also, as a "side
effect," change a "different" object because the two "different"
objects are actually a single object appearing under different aliases.
These so-called "side-effect bugs" are so difficult to locate and to
analyze that some people have proposed that programming languages be
designed in such a way as to not allow side effects or aliasing
(Lampson et al. 1981; Morris, Schmidt, and Wadler 1980).

   (3) In view of this, it is ironic that introductory programming is
most often taught in a highly imperative style.  This may be a vestige
of a belief, common throughout the 1960s and 1970s, that programs that
call procedures must inherently be less efficient than programs that
perform assignments.  (Steele (1977) debunks this argument.)
Alternatively it may reflect a view that step-by-step assignment is
easier for beginners to visualize than procedure call.  Whatever the
reason, it often saddles beginning programmers with "should I set this
variable before or after that one" concerns that can complicate
programming and obscure the important ideas.


File: sicp.info,  Node: 3-2,  Next: 3-3,  Prev: 3-1,  Up: Chapter 3

3.2 The Environment Model of Evaluation
=======================================

When we introduced compound procedures in *Note Chapter 1::, we used the
substitution model of evaluation (section *Note 1-1-5::) to define what
is meant by applying a procedure to arguments:

   * To apply a compound procedure to arguments, evaluate the body of
     the procedure with each formal parameter replaced by the
     corresponding argument.


   Once we admit assignment into our programming language, such a
definition is no longer adequate.  In particular, section *Note 3-1-3::
argued that, in the presence of assignment, a variable can no longer be
considered to be merely a name for a value.  Rather, a variable must
somehow designate a "place" in which values can be stored.  In our new
model of evaluation, these places will be maintained in structures
called "environments".

   An environment is a sequence of "frames".  Each frame is a table
(possibly empty) of "bindings", which associate variable names with
their corresponding values.  (A single frame may contain at most one
binding for any variable.)  Each frame also has a pointer to its environment
"enclosing environment", unless, for the purposes of discussion, the
frame is considered to be "global".  The "value of a variable" with
respect to an environment is the value given by the binding of the
variable in the first frame in the environment that contains a binding
for that variable.  If no frame in the sequence specifies a binding for
the variable, then the variable is said to be "unbound" in the
environment.

     *Figure 3.1:* A simple environment structure.

                     +--------+
                     |      I |
                     | x: 3   |
                     | y: 5   |
                     +--------+
                        ^  ^
                        |  |
                      C |  | D
          +---------+   |  |   +----------+
          |      II |   |  |   |      III |
          | z: 6    +---+  +---+ m: 1     |
          | x: 7    |          | y: 2     |
          +---------+          +----------+

   *Note Figure 3-1:: shows a simple environment structure consisting
of three frames, labeled I, II, and III.  In the diagram, A, B, C, and
D are pointers to environments.  C and D point to the same environment.
The variables `z' and `x' are bound in frame II, while `y' and `x' are
bound in frame I.  The value of `x' in environment D is 3.  The value
of `x' with respect to environment B is also 3.  This is determined as
follows: We examine the first frame in the sequence (frame III) and do
not find a binding for `x', so we proceed to the enclosing environment
D and find the binding in frame I.  On the other hand, the value of `x'
in environment A is 7, because the first frame in the sequence (frame
II) contains a binding of `x' to 7.  With respect to environment A, the
binding of `x' to 7 in frame II is said to "shadow" the binding of `x'
to 3 in frame I.

   The environment is crucial to the evaluation process, because it
determines the context in which an expression should be evaluated.
Indeed, one could say that expressions in a programming language do
not, in themselves, have any meaning.  Rather, an expression acquires a
meaning only with respect to some environment in which it is evaluated.
Even the interpretation of an expression as straightforward as `(+ 1
1)' depends on an understanding that one is operating in a context in
which `+' is the symbol for addition.  Thus, in our model of evaluation
we will always speak of evaluating an expression with respect to some
environment.  To describe interactions with the interpreter, we will
suppose that there is a global environment, consisting of a single frame
(with no enclosing environment) that includes values for the symbols
associated with the primitive procedures.  For example, the idea that
`+' is the symbol for addition is captured by saying that the symbol
`+' is bound in the global environment to the primitive addition
procedure.

* Menu:

* 3-2-1::            The Rules for Evaluation
* 3-2-2::            Applying Simple Procedures
* 3-2-3::            Frames as the Repository of Local State
* 3-2-4::            Internal Definitions


File: sicp.info,  Node: 3-2-1,  Next: 3-2-2,  Prev: 3-2,  Up: 3-2

3.2.1 The Rules for Evaluation
------------------------------

The overall specification of how the interpreter evaluates a combination
remains the same as when we first introduced it in section *Note
1-1-3:::

   * To evaluate a combination:


  1. Evaluate the subexpressions of the combination.(1)

  2. Apply the value of the operator subexpression to the values of the
     operand subexpressions.


   The environment model of evaluation replaces the substitution model
in specifying what it means to apply a compound procedure to arguments.

   In the environment model of evaluation, a procedure is always a pair
consisting of some code and a pointer to an environment.  Procedures
are created in one way only: by evaluating a `lambda' expression.  This
produces a procedure whose code is obtained from the text of the
`lambda' expression and whose environment is the environment in which
the `lambda' expression was evaluated to produce the procedure.  For
example, consider the procedure definition

     (define (square x)
       (* x x))

evaluated in the global environment.  The procedure definition syntax
is just syntactic sugar for an underlying implicit `lambda' expression.
It would have been equivalent to have used

     (define square
       (lambda (x) (* x x)))

which evaluates `(lambda (x) (* x x))' and binds `square' to the
resulting value, all in the global environment.

   *Note Figure 3-2:: shows the result of evaluating this `define'
expression.  The procedure object is a pair whose code specifies that
the procedure has one formal parameter, namely `x', and a procedure
body `(* x x)'.  The environment part of the procedure is a pointer to
the global environment, since that is the environment in which the
`lambda' expression was evaluated to produce the procedure. A new
binding, which associates the procedure object with the symbol
`square', has been added to the global frame.  In general, `define'
creates definitions by adding bindings to frames.

     *Figure 3.2:* Environment structure produced by evaluating
     `(define (square x) (* x x))' in the global environment.

                     +----------------------+
                     | other variables      |
          global --->|                      |
          env        | square: --+          |
                     +-----------|----------+
                                 |       ^
          (define (square x)     |       |
            (* x x))             V       |
                             .---.---.   |
                             | O | O-+---+
                             `-|-^---'
                               |
                               V
                             parameters: x
                             body: (* x x)

   Now that we have seen how procedures are created, we can describe how
procedures are applied.  The environment model specifies: To apply a
procedure to arguments, create a new environment containing a frame
that binds the parameters to the values of the arguments.  The
enclosing environment of this frame is the environment specified by the
procedure.  Now, within this new environment, evaluate the procedure
body.

   To show how this rule is followed, *Note Figure 3-3:: illustrates
the environment structure created by evaluating the expression `(square
5)' in the global environment, where `square' is the procedure
generated in *Note Figure 3-2::.  Applying the procedure results in the
creation of a new environment, labeled E1 in the figure, that begins
with a frame in which `x', the formal parameter for the procedure, is
bound to the argument 5.  The pointer leading upward from this frame
shows that the frame's enclosing environment is the global environment.
The global environment is chosen here, because this is the environment
that is indicated as part of the `square' procedure object.  Within E1,
we evaluate the body of the procedure, `(* x x)'.  Since the value of
`x' in E1 is 5, the result is `(* 5 5)', or 25.

     *Figure 3.3:* Environment created by evaluating `(square 5)' in
     the global environment.

                    +------------------------------------+
                    | other variables                    |
          global -->|                                    |
          env       | square: --+                        |
                    +-----------|---------------------+--+
                                |       ^             ^
          (square 5)            |       |             |
                                V       |             |
                            .---.---.   |         +---+--+
                            | O | O-+---+   E1 -->| x: 5 |
                            `-|-^---'             +------+
                              |
                              V
                            parameters: x
                            body: (* x x)

   The environment model of procedure application can be summarized by
two rules:

   * A procedure object is applied to a set of arguments by
     constructing a frame, binding the formal parameters of the
     procedure to the arguments of the call, and then evaluating the
     body of the procedure in the context of the new environment
     constructed.  The new frame has as its enclosing environment the
     environment part of the procedure object being applied.

   * A procedure is created by evaluating a `lambda' expression
     relative to a given environment.  The resulting procedure object
     is a pair consisting of the text of the `lambda' expression and a
     pointer to the environment in which the procedure was created.


   We also specify that defining a symbol using `define' creates a
binding in the current environment frame and assigns to the symbol the
indicated value.(2) Finally, we specify the behavior of `set!', the
operation that forced us to introduce the environment model in the
first place.  Evaluating the expression `(set! <VARIABLE> <VALUE>)' in
some environment locates the binding of the variable in the environment
and changes that binding to indicate the new value.  That is, one finds
the first frame in the environment that contains a binding for the
variable and modifies that frame.  If the variable is unbound in the
environment, then `set!' signals an error.

   These evaluation rules, though considerably more complex than the
substitution model, are still reasonably straightforward.  Moreover,
the evaluation model, though abstract, provides a correct description
of how the interpreter evaluates expressions.  In *Note Chapter 4:: we
shall see how this model can serve as a blueprint for implementing a
working interpreter.  The following sections elaborate the details of
the model by analyzing some illustrative programs.

   ---------- Footnotes ----------

   (1) ssignment introduces a subtlety into step 1 of the evaluation
rule.  As shown in *Note Exercise 3-8::, the presence of assignment
allows us to write expressions that will produce different values
depending on the order in which the subexpressions in a combination are
evaluated.  Thus, to be precise, we should specify an evaluation order
in step 1 (e.g., left to right or right to left).  However, this order
should always be considered to be an implementation detail, and one
should never write programs that depend on some particular order.  For
instance, a sophisticated compiler might optimize a program by varying
the order in which subexpressions are evaluated.

   (2) If there is already a binding for the variable in the current
frame, then the binding is changed.  This is convenient because it
allows redefinition of symbols; however, it also means that `define'
can be used to change values, and this brings up the issues of
assignment without explicitly using `set!'.  Because of this, some
people prefer redefinitions of existing symbols to signal errors or
warnings.


File: sicp.info,  Node: 3-2-2,  Next: 3-2-3,  Prev: 3-2-1,  Up: 3-2

3.2.2 Applying Simple Procedures
--------------------------------

When we introduced the substitution model in section *Note 1-1-5:: we
showed how the combination `(f 5)' evaluates to 136, given the
following procedure definitions:

     (define (square x)
       (* x x))

     (define (sum-of-squares x y)
       (+ (square x) (square y)))

     (define (f a)
       (sum-of-squares (+ a 1) (* a 2)))

   We can analyze the same example using the environment model.  *Note
Figure 3-4:: shows the three procedure objects created by evaluating
the definitions of `f', `square', and `sum-of-squares' in the global
environment.  Each procedure object consists of some code, together
with a pointer to the global environment.

     *Figure 3.4:* Procedure objects in the global frame.

                    +--------------------------------------------+
                    | sum-of-squares:                            |
          global -->| square:                                    |
          env       | f: --+                                     |
                    +------|--------------+--------------+-------+
                           |     ^        |     ^        |     ^
                           |     |        |     |        |     |
                           V     |        V     |        V     |
                       .---.---. |    .---.---. |    .---.---. |
                       | O | O-+-+    | O | O-+-+    | O | O-+-+
                       `-|-^---'      `-|-^---'      `-|-^---'
                         |              |              |
                         V              V              V
             parameters: a          parameters: x  parameters: x, y
             body: (sum-of-squares  body: (* x x)  body: (+ (square x)
                     (+ a 1)                                (square y))
                     (* a 2))

   In *Note Figure 3-5:: we see the environment structure created by
evaluating the expression `(f 5)'.  The call to `f' creates a new
environment E1 beginning with a frame in which `a', the formal
parameter of `f', is bound to the argument 5.  In E1, we evaluate the
body of `f':

     (sum-of-squares (+ a 1) (* a 2))

     *Figure 3.5:* Environments created by evaluating `(f 5)' using the
     procedures in *Note Figure 3-4::.

                    +-----------------------------------------------------+
          global -->|                                                     |
          env       +-----------------------------------------------------+
                      ^              ^                ^               ^
          (f 5)       |              |                |               |
                  +------+       +-------+        +------+        +-------+
            E1 -->| a: 5 |  E2 ->| x: 6  |  E3 -->| x: 6 |  E4 -->| x: 10 |
                  |      |       | y: 10 |        |      |        |       |
                  +------+       +-------+        +------+        +-------+
             (sum-of-squares   (+ (square x)       (* x x)         (* x x)
               (+ a 1)            (square u))
               (+ a 2))

   To evaluate this combination, we first evaluate the subexpressions.
The first subexpression, `sum-of-squares', has a value that is a
procedure object.  (Notice how this value is found: We first look in
the first frame of E1, which contains no binding for `sum-of-squares'.
Then we proceed to the enclosing environment, i.e. the global
environment, and find the binding shown in *Note Figure 3-4::.)  The
other two subexpressions are evaluated by applying the primitive
operations `+' and `*' to evaluate the two combinations `(+ a 1)' and
`(* a 2)' to obtain 6 and 10, respectively.

   Now we apply the procedure object `sum-of-squares' to the arguments
6 and 10.  This results in a new environment E2 in which the formal
parameters `x' and `y' are bound to the arguments.  Within E2 we
evaluate the combination `(+ (square x) (square y))'.  This leads us to
evaluate `(square x)', where `square' is found in the global frame and
`x' is 6.  Once again, we set up a new environment, E3, in which `x' is
bound to 6, and within this we evaluate the body of `square', which is
`(* x x)'.  Also as part of applying `sum-of-squares', we must evaluate
the subexpression `(square y)', where `y' is 10.  This second call to
`square' creates another environment, E4, in which `x', the formal
parameter of `square', is bound to 10.  And within E4 we must evaluate
`(* x x)'.

   The important point to observe is that each call to `square' creates
a new environment containing a binding for `x'.  We can see here how the
different frames serve to keep separate the different local variables
all named `x'.  Notice that each frame created by `square' points to
the global environment, since this is the environment indicated by the
`square' procedure object.

   After the subexpressions are evaluated, the results are returned.
The values generated by the two calls to `square' are added by
`sum-of-squares', and this result is returned by `f'.  Since our focus
here is on the environment structures, we will not dwell on how these
returned values are passed from call to call; however, this is also an
important aspect of the evaluation process, and we will return to it in
detail in *Note Chapter 5::.

     *Exercise 3.9:* In section *Note 1-2-1:: we used the substitution
     model to analyze two procedures for computing factorials, a
     recursive version

          (define (factorial n)
            (if (= n 1)
                1
                (* n (factorial (- n 1)))))

     and an iterative version

          (define (factorial n)
            (fact-iter 1 1 n))

          (define (fact-iter product counter max-count)
            (if (> counter max-count)
                product
                (fact-iter (* counter product)
                           (+ counter 1)
                           max-count)))

     Show the environment structures created by evaluating `(factorial
     6)' using each version of the `factorial' procedure.(1)

   ---------- Footnotes ----------

   (1) The environment model will not clarify our claim in section
*Note 1-2-1:: that the interpreter can execute a procedure such as
`fact-iter' in a constant amount of space using tail recursion.  We
will discuss tail recursion when we deal with the control structure of
the interpreter in section *Note 5-4::.


File: sicp.info,  Node: 3-2-3,  Next: 3-2-4,  Prev: 3-2-2,  Up: 3-2

3.2.3 Frames as the Repository of Local State
---------------------------------------------

We can turn to the environment model to see how procedures and
assignment can be used to represent objects with local state.  As an
example, consider the "withdrawal processor" from section *Note 3-1-1::
created by calling the procedure

     (define (make-withdraw balance)
       (lambda (amount)
         (if (>= balance amount)
             (begin (set! balance (- balance amount))
                    balance)
             "Insufficient funds")))

   Let us describe the evaluation of

     (define W1 (make-withdraw 100))

followed by

     (W1 50)
     50

   *Note Figure 3-6:: shows the result of defining the `make-withdraw'
procedure in the global environment.  This produces a procedure object
that contains a pointer to the global environment.  So far, this is no
different from the examples we have already seen, except that the body
of the procedure is itself a `lambda' expression.

     *Figure 3.6:* Result of defining `make-withdraw' in the global
     environment.

                    +---------------------------+
          global -->| make-withdraw: --+        |
          env       +------------------|--------+
                                       |      ^
                                       V      |
                                   .---.---.  |
                                   | O | O-+--+
                                   `-|-^---'
                                     |
                                     V
                   parameters: balance
                   body: (lambda (amount)
                           (if (>= balance amount)
                               (begin (set! balance
                                            (- balance amount))
                                      balance)
                               "Insufficient funds"))

   The interesting part of the computation happens when we apply the
procedure `make-withdraw' to an argument:

     (define W1 (make-withdraw 100))

   We begin, as usual, by setting up an environment E1 in which the
formal parameter `balance' is bound to the argument 100.  Within this
environment, we evaluate the body of `make-withdraw', namely the
`lambda' expression.  This constructs a new procedure object, whose code
is as specified by the `lambda' and whose environment is E1, the
environment in which the `lambda' was evaluated to produce the
procedure.  The resulting procedure object is the value returned by the
call to `make-withdraw'.  This is bound to `W1' in the global
environment, since the `define' itself is being evaluated in the global
environment.  *Note Figure 3-7:: shows the resulting environment
structure.

     *Figure 3.7:* Result of evaluating `(define W1 (make-withdraw
     100))'.

                    +-----------------------------------------------+
                    | make-withdraw: -----------------------+       |
          global -->|                                       |       |
                    | W1: --+                               |       |
                    +-------|-------------------------------|-------+
                            |                ^              |     ^
                            |                |              V     |
                            |        +-------+------+   .---.---. |
                            |  E1 -->| balance: 100 |   | O | O-+-+
                            |        +--------------+   `-|-^---'
                            V                ^            |
                        .---.---.            |            V
                      +-+-O | O-+------------+    parameters: balance
                      | `---^---'                 body: ...
                      V
              parameters: amount
              body: (if (>= balance amount)
                        (begin (set! balance (- balance amount))
                               balance)
                        "Insufficient funds")

   Now we can analyze what happens when `W1' is applied to an argument:

     (W1 50)
     50

   We begin by constructing a frame in which `amount', the formal
parameter of `W1', is bound to the argument 50.  The crucial point to
observe is that this frame has as its enclosing environment not the
global environment, but rather the environment E1, because this is the
environment that is specified by the `W1' procedure object.  Within
this new environment, we evaluate the body of the procedure:

     (if (>= balance amount)
         (begin (set! balance (- balance amount))
                balance)
         "Insufficient funds")

   The resulting environment structure is shown in *Note Figure 3-8::.
The expression being evaluated references both `amount' and `balance'.
`Amount' will be found in the first frame in the environment, while
`balance' will be found by following the enclosing-environment pointer
to E1.

     *Figure 3.8:* Environments created by applying the procedure
     object `W1'.

                    +---------------------------------------------------+
                    | make-withdraw: ...                                |
          global -->|                                                   |
          env       | W1: --+                                           |
                    +-------|-------------------------------------------+
                            |               ^
                            |               |
                            |       +-------+------+ Here is the balance
                            | E1 -->| balance: 100 | that will be changed
                            |       +--------------+ by the set!.
                            V               ^   ^
                        .---.---.           |   +----+
                        | O | O-+-----------+        |
                        `-|-^---'             +------+-----+
                          |                   | amount: 50 |
                          V                   +------------+
                parameters: amount   (if (>= balance amount)
                body: ...                (begin (set! balance
                                                      (- balance amount))
                                                balance)
                                         "Insufficient funds")

   When the `set!' is executed, the binding of `balance' in E1 is
changed.  At the completion of the call to `W1', `balance' is 50, and
the frame that contains `balance' is still pointed to by the procedure
object `W1'.  The frame that binds `amount' (in which we executed the
code that changed `balance') is no longer relevant, since the procedure
call that constructed it has terminated, and there are no pointers to
that frame from other parts of the environment.  The next time `W1' is
called, this will build a new frame that binds `amount' and whose
enclosing environment is E1.  We see that E1 serves as the "place" that
holds the local state variable for the procedure object `W1'.  *Note
Figure 3-9:: shows the situation after the call to `W1'.

     *Figure 3.9:* Environments after the call to `W1'.

                     +------------------------------------+
                     | make-withdraw: ...                 |
          global --->|                                    |
          env        | W1: --+                            |
                     +-------|----------------------------+
                             |                   ^
                             |                   |
                             |            +------+------+
                             |     E1 --->| balance: 50 |
                             |            +-------------+
                             V                   ^
                         .---.---.               |
                         | O | O-+---------------+
                         `-|-^---'
                           |
                           V
                    parameters: amount
                    body: ...

   Observe what happens when we create a second "withdraw" object by
making another call to `make-withdraw':

     (define W2 (make-withdraw 100))

   This produces the environment structure of *Note Figure 3-10::,
which shows that `W2' is a procedure object, that is, a pair with some
code and an environment.  The environment E2 for `W2' was created by
the call to `make-withdraw'.  It contains a frame with its own local
binding for `balance'.  On the other hand, `W1' and `W2' have the same
code: the code specified by the `lambda' expression in the body of
`make-withdraw'.(1) We see here why `W1' and `W2' behave as independent
objects.  Calls to `W1' reference the state variable `balance' stored
in E1, whereas calls to `W2' reference the `balance' stored in E2.
Thus, changes to the local state of one object do not affect the other
object.

     *Figure 3.10:* Using `(define W2 (make-withdraw 100))' to create a
     second object.

                   +-------------------------------------------------+
                   | make-withdraw: ...                              |
          global ->| W2: ---------------------------+                |
          env      | W1: --+                        |                |
                   +-------|------------------------|----------------+
                           |              ^         |              ^
                           |              |         |              |
                           |       +------+------+  |       +------+-------+
                           |  E1 ->| balance: 50 |  |  E2 ->| balance: 100 |
                           |       +-------------+  |       +--------------+
                           V              ^         V              ^
                       .---.---.          |     .---.---.          |
                       | O | O-+----------+     | O | O-+----------+
                       `-|-^---'                `-|-^---'
                         | +----------------------+
                         V V
                  parameters: amount
                  body: ...

     *Exercise 3.10:* In the `make-withdraw' procedure, the local
     variable `balance' is created as a parameter of `make-withdraw'.
     We could also create the local state variable explicitly, using
     `let', as follows:

          (define (make-withdraw initial-amount)
            (let ((balance initial-amount))
              (lambda (amount)
                (if (>= balance amount)
                    (begin (set! balance (- balance amount))
                           balance)
                    "Insufficient funds"))))

     Recall from section *Note 1-3-2:: that `let' is simply syntactic
     sugar for a procedure call:

          (let ((<VAR> <EXP>)) <BODY>)

     is interpreted as an alternate syntax for

          ((lambda (<VAR>) <BODY>) <EXP>)

     Use the environment model to analyze this alternate version of
     `make-withdraw', drawing figures like the ones above to illustrate
     the interactions

          (define W1 (make-withdraw 100))

          (W1 50)

          (define W2 (make-withdraw 100))

     Show that the two versions of `make-withdraw' create objects with
     the same behavior.  How do the environment structures differ for
     the two versions?

   ---------- Footnotes ----------

   (1) Whether `W1' and `W2' share the same physical code stored in the
computer, or whether they each keep a copy of the code, is a detail of
the implementation.  For the interpreter we implement in *Note Chapter
4::, the code is in fact shared.


File: sicp.info,  Node: 3-2-4,  Prev: 3-2-3,  Up: 3-2

3.2.4 Internal Definitions
--------------------------

Section *Note 1-1-8:: introduced the idea that procedures can have
internal definitions, thus leading to a block structure as in the
following procedure to compute square roots:

     (define (sqrt x)
       (define (good-enough? guess)
         (< (abs (- (square guess) x)) 0.001))
       (define (improve guess)
         (average guess (/ x guess)))
       (define (sqrt-iter guess)
         (if (good-enough? guess)
             guess
             (sqrt-iter (improve guess))))
       (sqrt-iter 1.0))

   Now we can use the environment model to see why these internal
definitions behave as desired.  *Note Figure 3-11:: shows the point in
the evaluation of the expression `(sqrt 2)' where the internal
procedure `good-enough?' has been called for the first time with
`guess' equal to 1.

     *Figure 3.11:* `Sqrt' procedure with internal definitions.

                    +--------------------------------------------------+
          global -->| sqrt: --+                                        |
          env       |         |                                        |
                    +---------|----------------------------------------+
                              V       ^                   ^
                          .---.---.   |                   |
               +----------+-O | O-+---+        +----------+------------+
               |          `---^---'            | x: 2                  |
               V                         E1 -->| good-enough?: -+      |
          parameters: x                        | improve: ...   |      |
          body: (define good-enough? ...)      | sqrt-iter: ... |      |
                (define improve ...)           +----------------|------+
                (define sqrt-iter ...)          ^  ^            |     ^
                (sqrt-iter 1.0)                 |  |            V     |
                                      +---------++ |        .---.---. |
                                E2 -->| guess: 1 | |        | O | O-+-+
                                      +----------+ |        `-|-^---'
                                call to sqrt-iter  |          |
                                                   |          V
                                         +---------++    parameters: guess
                                   E3 -->| guess: 1 |    body: (< (abs ...)
                                         +----------+             ...)
                                   call to good-enough?

   Observe the structure of the environment.  `Sqrt' is a symbol in the
global environment that is bound to a procedure object whose associated
environment is the global environment.  When `sqrt' was called, a new
environment E1 was formed, subordinate to the global environment, in
which the parameter `x' is bound to 2.  The body of `sqrt' was then
evaluated in E1.  Since the first expression in the body of `sqrt' is

     (define (good-enough? guess)
       (< (abs (- (square guess) x)) 0.001))

evaluating this expression defined the procedure `good-enough?'  in the
environment E1.  To be more precise, the symbol `good-enough?' was added
to the first frame of E1, bound to a procedure object whose associated
environment is E1.  Similarly, `improve' and `sqrt-iter' were defined
as procedures in E1.  For conciseness, *Note Figure 3-11:: shows only
the procedure object for `good-enough?'.

   After the local procedures were defined, the expression `(sqrt-iter
1.0)' was evaluated, still in environment E1.  So the procedure object
bound to `sqrt-iter' in E1 was called with 1 as an argument.  This
created an environment E2 in which `guess', the parameter of
`sqrt-iter', is bound to 1.  `Sqrt-iter' in turn called `good-enough?'
with the value of `guess' (from E2) as the argument for `good-enough?'.
This set up another environment, E3, in which `guess' (the parameter of
`good-enough?') is bound to 1.  Although `sqrt-iter' and `good-enough?'
both have a parameter named `guess', these are two distinct local
variables located in different frames.  Also, E2 and E3 both have E1 as
their enclosing environment, because the `sqrt-iter' and `good-enough?'
procedures both have E1 as their environment part.  One consequence of
this is that the symbol `x' that appears in the body of `good-enough?'
will reference the binding of `x' that appears in E1, namely the value
of `x' with which the original `sqrt' procedure was called.

   The environment model thus explains the two key properties that make
local procedure definitions a useful technique for modularizing
programs:

   * The names of the local procedures do not interfere with names
     external to the enclosing procedure, because the local procedure
     names will be bound in the frame that the procedure creates when
     it is run, rather than being bound in the global environment.

   * The local procedures can access the arguments of the enclosing
     procedure, simply by using parameter names as free variables.
     This is because the body of the local procedure is evaluated in an
     environment that is subordinate to the evaluation environment for
     the enclosing procedure.


     *Exercise 3.11:* In section *Note 3-2-3:: we saw how the
     environment model described the behavior of procedures with local
     state.  Now we have seen how internal definitions work.  A typical
     message-passing procedure contains both of these aspects.
     Consider the bank account procedure of section *Note 3-1-1:::

          (define (make-account balance)
            (define (withdraw amount)
              (if (>= balance amount)
                  (begin (set! balance (- balance amount))
                         balance)
                  "Insufficient funds"))
            (define (deposit amount)
              (set! balance (+ balance amount))
              balance)
            (define (dispatch m)
              (cond ((eq? m 'withdraw) withdraw)
                    ((eq? m 'deposit) deposit)
                    (else (error "Unknown request -- MAKE-ACCOUNT"
                                 m))))
            dispatch)

     Show the environment structure generated by the sequence of
     interactions

          (define acc (make-account 50))

          ((acc 'deposit) 40)
          90

          ((acc 'withdraw) 60)
          30

     Where is the local state for `acc' kept?  Suppose we define another
     account

          (define acc2 (make-account 100))

     How are the local states for the two accounts kept distinct?
     Which parts of the environment structure are shared between `acc'
     and `acc2'?


File: sicp.info,  Node: 3-3,  Next: 3-4,  Prev: 3-2,  Up: Chapter 3

3.3 Modeling with Mutable Data
==============================

Chapter 2 dealt with compound data as a means for constructing
computational objects that have several parts, in order to model
real-world objects that have several aspects.  In that chapter we
introduced the discipline of data abstraction, according to which data
structures are specified in terms of constructors, which create data
objects, and selectors, which access the parts of compound data
objects.  But we now know that there is another aspect of data that
*Note Chapter 2:: did not address.  The desire to model systems
composed of objects that have changing state leads us to the need to
modify compound data objects, as well as to construct and select from
them.  In order to model compound objects with changing state, we will
design data abstractions to include, in addition to selectors and
constructors, operations called "mutators", which modify data objects.
For instance, modeling a banking system requires us to change account
balances.  Thus, a data structure for representing bank accounts might
admit an operation

     (set-balance! <ACCOUNT> <NEW-VALUE>)

that changes the balance of the designated account to the designated
new value.  Data objects for which mutators are defined are known as objects
"mutable data objects".

   *Note Chapter 2:: introduced pairs as a general-purpose "glue" for
synthesizing compound data.  We begin this section by defining basic
mutators for pairs, so that pairs can serve as building blocks for
constructing mutable data objects.  These mutators greatly enhance the
representational power of pairs, enabling us to build data structures
other than the sequences and trees that we worked with in section *Note
2-2::.  We also present some examples of simulations in which complex
systems are modeled as collections of objects with local state.

* Menu:

* 3-3-1::            Mutable List Structure
* 3-3-2::            Representing Queues
* 3-3-3::            Representing Tables
* 3-3-4::            A Simulator for Digital Circuits
* 3-3-5::            Propagation of Constraints


File: sicp.info,  Node: 3-3-1,  Next: 3-3-2,  Prev: 3-3,  Up: 3-3

3.3.1 Mutable List Structure
----------------------------

The basic operations on pairs--`cons', `car', and `cdr'--can be used to
construct list structure and to select parts from list structure, but
they are incapable of modifying list structure.  The same is true of the
list operations we have used so far, such as `append' and `list', since
these can be defined in terms of `cons', `car', and `cdr'.  To modify
list structures we need new operations.

     *Figure 3.12:* Lists `x': `((a b) c d)' and `y': `(e f)'.

               +---+---+     +---+---+     +---+---+
          x -->| * | *-+---->| * | *-+---->| * | / |
               +-|-+---+     +-|-+---+     +-|-+---+
                 |             V             V
                 |           +---+         +---+
                 |           | c |         | d |
                 |           +---+         +---+
                 |           +---+---+     +---+---+
                 +---------->| * | *-+---->| * | / |
                             +-|-+---+     +-|-+---+
                               V             V
                             +---+         +---+
                             | a |         | b |
                             +---+         +---+
                             +---+---+     +---+---+
                        y -->| * | *-+---->| * | / |
                             +-|-+---+     +-|-+---+
                               V             V
                             +---+         +---+
                             | e |         | f |
                             +---+         +---+

     *Figure 3.13:* Effect of `(set-car! x y)' on the lists in *Note
     Figure 3-12::.

               +---+---+     +---+---+     +---+---+
          x -->| * | *-+---->| * | *-+---->| * | / |
               +-|-+---+     +-|-+---+     +-|-+---+
                 |             V             V
                 |           +---+         +---+
                 |           | c |         | d |
                 |           +---+         +---+
                 |           +---+---+     +---+---+
                 |           | * | *-+---->| * | / |
                 |           +-|-+---+     +-|-+---+
                 |             V             V
                 |           +---+         +---+
                 |           | a |         | b |
                 |           +---+         +---+
                 +---------->+---+---+     +---+---+
                             | * | *-+---->| * | / |
                        y -->+-|-+---+     +-|-+---+
                               V             V
                             +---+         +---+
                             | e |         | f |
                             +---+         +---+

     *Figure 3.14:* Effect of `(define z (cons y (cdr x)))' on the
     lists in *Note Figure 3-12::.

               +---+---+     +---+---+     +---+---+
          x -->| * | *-+---->| * | *-+---->| * | / |
               +-|-+---+ +-->+-|-+---+     +-|-+---+
                 |       |     V             V
                 |       |   +---+         +---+
                 |       |   | c |         | d |
                 |       |   +---+         +---+
                 |       |   +---+---+     +---+---+
                 +-------+-->| * | *-+---->| * | / |
                         |   +-|-+---+     +-|-+---+
               +---+---+ |     V             V
          z -->| * | *-+-+   +---+         +---+
               +-|-+---+     | a |         | b |
                 |           +---+         +---+
                 +---------->+---+---+     +---+---+
                             | * | *-+---->| * | / |
                        y -->+-|-+---+     +-|-+---+
                               V             V
                             +---+         +---+
                             | e |         | f |
                             +---+         +---+

     *Figure 3.15:* Effect of `(set-cdr! x y)' on the lists in *Note
     Figure 3-12::.

               +---+---+     +---+---+     +---+---+
          x -->| * | * |     | * | *-+---->| * | / |
               +-|-+-|-+     +-|-+---+     +-|-+---+
                 |   |         V             V
                 |   |       +---+         +---+
                 |   |       | c |         | d |
                 |   |       +---+         +---+
                 |   |       +---+---+     +---+---+
                 +---+------>| * | *-+---->| * | / |
                     |       +-|-+---+     +-|-+---+
                     |         V             V
                     |       +---+         +---+
                     |       | a |         | b |
                     |       +---+         +---+
                     +------>+---+---+     +---+---+
                             | * | *-+---->| * | / |
                        y -->+-|-+---+     +-|-+---+
                               V             V
                             +---+         +---+
                             | e |         | f |
                             +---+         +---+

   The primitive mutators for pairs are `set-car!' and `set-cdr!'.
`Set-car!' takes two arguments, the first of which must be a pair.  It
modifies this pair, replacing the `car' pointer by a pointer to the
second argument of `set-car!'.(1)

   As an example, suppose that `x' is bound to the list `((a b) c d)'
and `y' to the list `(e f)' as illustrated in *Note Figure 3-12::.
Evaluating the expression ` (set-car!  x y)' modifies the pair to which
`x' is bound, replacing its `car' by the value of `y'.  The result of
the operation is shown in *Note Figure 3-13::.  The structure `x' has
been modified and would now be printed as `((e f) c d)'.  The pairs
representing the list `(a b)', identified by the pointer that was
replaced, are now detached from the original structure.(2)

   Compare *Note Figure 3-13:: with *Note Figure 3-14::, which
illustrates the result of executing `(define z (cons y (cdr x)))' with
`x' and `y' bound to the original lists of *Note Figure 3-12::.  The
variable `z' is now bound to a new pair created by the `cons'
operation; the list to which `x' is bound is unchanged.

   The `set-cdr!' operation is similar to `set-car!'.  The only
difference is that the `cdr' pointer of the pair, rather than the `car'
pointer, is replaced.  The effect of executing `(set-cdr! x y)' on the
lists of *Note Figure 3-12:: is shown in *Note Figure 3-15::.  Here the
`cdr' pointer of `x' has been replaced by the pointer to `(e f)'.
Also, the list `(c d)', which used to be the `cdr' of `x', is now
detached from the structure.

   `Cons' builds new list structure by creating new pairs, while
`set-car!' and `set-cdr!' modify existing pairs.  Indeed, we could
implement `cons' in terms of the two mutators, together with a procedure
`get-new-pair', which returns a new pair that is not part of any
existing list structure.  We obtain the new pair, set its `car' and
`cdr' pointers to the designated objects, and return the new pair as
the result of the `cons'.(3)

     (define (cons x y)
       (let ((new (get-new-pair)))
         (set-car! new x)
         (set-cdr! new y)
         new))

     *Exercise 3.12:* The following procedure for appending lists was
     introduced in section *Note 2-2-1:::

          (define (append x y)
            (if (null? x)
                y
                (cons (car x) (append (cdr x) y))))

     `Append' forms a new list by successively `cons'ing the elements of
     `x' onto `y'.  The procedure `append!' is similar to `append', but
     it is a mutator rather than a constructor.  It appends the lists
     by splicing them together, modifying the final pair of `x' so that
     its `cdr' is now `y'.  (It is an error to call `append!' with an
     empty `x'.)

          (define (append! x y)
            (set-cdr! (last-pair x) y)
            x)

     Here `last-pair' is a procedure that returns the last pair in its
     argument:

          (define (last-pair x)
            (if (null? (cdr x))
                x
                (last-pair (cdr x))))

     Consider the interaction

          (define x (list 'a 'b))

          (define y (list 'c 'd))

          (define z (append x y))

          z
          (a b c d)

          (cdr x)
          <RESPONSE>

          (define w (append! x y))

          w
          (a b c d)

          (cdr x)
          <RESPONSE>

     What are the missing <RESPONSE>s?  Draw box-and-pointer diagrams to
     explain your answer.

     *Exercise 3.13:* Consider the following `make-cycle' procedure,
     which uses the `last-pair' procedure defined in *Note Exercise
     3-12:::

          (define (make-cycle x)
            (set-cdr! (last-pair x) x)
            x)

     Draw a box-and-pointer diagram that shows the structure `z'
     created by

          (define z (make-cycle (list 'a 'b 'c)))

     What happens if we try to compute `(last-pair z)'?

     *Exercise 3.14:* The following procedure is quite useful, although
     obscure:

          (define (mystery x)
            (define (loop x y)
              (if (null? x)
                  y
                  (let ((temp (cdr x)))
                    (set-cdr! x y)
                    (loop temp x))))
            (loop x '()))

     `Loop' uses the "temporary" variable `temp' to hold the old value
     of the `cdr' of `x', since the `set-cdr!'  on the next line
     destroys the `cdr'.  Explain what `mystery' does in general.
     Suppose `v' is defined by `(define v (list 'a 'b 'c 'd))'. Draw the
     box-and-pointer diagram that represents the list to which `v' is
     bound.  Suppose that we now evaluate `(define w (mystery v))'. Draw
     box-and-pointer diagrams that show the structures `v' and `w' after
     evaluating this expression.  What would be printed as the values
     of `v' and `w'?

Sharing and identity
....................

We mentioned in section *Note 3-1-3:: the theoretical issues of
"sameness" and "change" raised by the introduction of assignment.
These issues arise in practice when individual pairs are "shared" among
different data objects.  For example, consider the structure formed by

     (define x (list 'a 'b))
     (define z1 (cons x x))

   As shown in *Note Figure 3-16::, `z1' is a pair whose `car' and
`cdr' both point to the same pair `x'.  This sharing of `x' by the
`car' and `cdr' of `z1' is a consequence of the straightforward way in
which `cons' is implemented.  In general, using `cons' to construct
lists will result in an interlinked structure of pairs in which many
individual pairs are shared by many different structures.

     *Figure 3.16:* The list `z1' formed by `(cons x x)'.

                +---+---+
          z1 -->| * | * |
                +-|-+-|-+
                  V   V
                +---+---+     +---+---+
           x -->| * | *-+---->| * | / |
                +-|-+---+     +-|-+---+
                  V             V
                +---+         +---+
                | a |         | b |
                +---+         +---+

     *Figure 3.17:* The list `z2' formed by `(cons (list 'a 'b) (list
     'a 'b))'.

                +---+---+     +---+---+     +---+---+
          z2 -->| * | *-+---->| * | *-+---->| * | / |
                +-|-+---+     +-|-+---+     +-|-+---+
                  |             V             V
                  |           +---+         +---+
                  |           | a |         | b |
                  |           +---+         +---+
                  |             ^             ^
                  |             |             |
                  |           +-|-+---+     +-|-+---+
                  +---------->| * | *-+---->| * | / |
                              +---+---+     +---+---+

   In contrast to *Note Figure 3-16::, *Note Figure 3-17:: shows the
structure created by

     (define z2 (cons (list 'a 'b) (list 'a 'b)))

   In this structure, the pairs in the two `(a b)' lists are distinct,
although the actual symbols are shared.(4)

   When thought of as a list, `z1' and `z2' both represent "the same"
list, `((a b) a b)'.  In general, sharing is completely undetectable if
we operate on lists using only `cons', `car', and `cdr'.  However, if
we allow mutators on list structure, sharing becomes significant.  As an
example of the difference that sharing can make, consider the following
procedure, which modifies the `car' of the structure to which it is
applied:

     (define (set-to-wow! x)
       (set-car! (car x) 'wow)
       x)

   Even though `z1' and `z2' are "the same" structure, applying
`set-to-wow!' to them yields different results.  With `z1', altering
the `car' also changes the `cdr', because in `z1' the `car' and the
`cdr' are the same pair.  With `z2', the `car' and `cdr' are distinct,
so `set-to-wow!' modifies only the `car':

     z1
     ((a b) a b)

     (set-to-wow! z1)
     ((wow b) wow b)

     z2
     ((a b) a b)

     (set-to-wow! z2)
     ((wow b) a b)

   One way to detect sharing in list structures is to use the predicate
`eq?', which we introduced in section *Note 2-3-1:: as a way to test
whether two symbols are equal.  More generally, `(eq?  x y)' tests
whether `x' and `y' are the same object (that is, whether `x' and `y'
are equal as pointers).  Thus, with `z1' and `z2' as defined in figures
*Note Figure 3-16:: and *Note Figure 3-17::, `(eq?  (car z1) (cdr z1))'
is true and `(eq? (car z2) (cdr z2))' is false.

   As will be seen in the following sections, we can exploit sharing to
greatly extend the repertoire of data structures that can be
represented by pairs.  On the other hand, sharing can also be
dangerous, since modifications made to structures will also affect
other structures that happen to share the modified parts.  The mutation
operations `set-car!' and `set-cdr!' should be used with care; unless
we have a good understanding of how our data objects are shared,
mutation can have unanticipated results.(5)

     *Exercise 3.15:* Draw box-and-pointer diagrams to explain the
     effect of `set-to-wow!' on the structures `z1' and `z2' above.

     *Exercise 3.16:* Ben Bitdiddle decides to write a procedure to
     count the number of pairs in any list structure.  "It's easy," he
     reasons.  "The number of pairs in any structure is the number in
     the `car' plus the number in the `cdr' plus one more to count the
     current pair."  So Ben writes the following procedure:

          (define (count-pairs x)
            (if (not (pair? x))
                0
                (+ (count-pairs (car x))
                   (count-pairs (cdr x))
                   1)))

     Show that this procedure is not correct.  In particular, draw
     box-and-pointer diagrams representing list structures made up of
     exactly three pairs for which Ben's procedure would return 3;
     return 4; return 7; never return at all.

     *Exercise 3.17:* Devise a correct version of the `count-pairs'
     procedure of *Note Exercise 3-16:: that returns the number of
     distinct pairs in any structure.  (Hint: Traverse the structure,
     maintaining an auxiliary data structure that is used to keep track
     of which pairs have already been counted.)

     *Exercise 3.18:* Write a procedure that examines a list and
     determines whether it contains a cycle, that is, whether a program
     that tried to find the end of the list by taking successive `cdr's
     would go into an infinite loop.  *Note Exercise 3-13:: constructed
     such lists.

     *Exercise 3.19:* Redo *Note Exercise 3-18:: using an algorithm
     that takes only a constant amount of space.  (This requires a very
     clever idea.)

Mutation is just assignment
...........................

When we introduced compound data, we observed in section *Note 2-1-3::
that pairs can be represented purely in terms of procedures:

     (define (cons x y)
       (define (dispatch m)
         (cond ((eq? m 'car) x)
               ((eq? m 'cdr) y)
               (else (error "Undefined operation -- CONS" m))))
       dispatch)

     (define (car z) (z 'car))

     (define (cdr z) (z 'cdr))

   The same observation is true for mutable data.  We can implement
mutable data objects as procedures using assignment and local state.
For instance, we can extend the above pair implementation to handle
`set-car!' and `set-cdr!' in a manner analogous to the way we
implemented bank accounts using `make-account' in section *Note 3-1-1:::

     (define (cons x y)
       (define (set-x! v) (set! x v))
       (define (set-y! v) (set! y v))
       (define (dispatch m)
         (cond ((eq? m 'car) x)
               ((eq? m 'cdr) y)
               ((eq? m 'set-car!) set-x!)
               ((eq? m 'set-cdr!) set-y!)
               (else (error "Undefined operation -- CONS" m))))
       dispatch)

     (define (car z) (z 'car))

     (define (cdr z) (z 'cdr))

     (define (set-car! z new-value)
       ((z 'set-car!) new-value)
       z)

     (define (set-cdr! z new-value)
       ((z 'set-cdr!) new-value)
       z)

   Assignment is all that is needed, theoretically, to account for the
behavior of mutable data.  As soon as we admit `set!' to our language,
we raise all the issues, not only of assignment, but of mutable data in
general.(6)

     *Exercise 3.20:* Draw environment diagrams to illustrate the
     evaluation of the sequence of expressions

          (define x (cons 1 2))
          (define z (cons x x))
          (set-car! (cdr z) 17)

          (car x)
          17

     using the procedural implementation of pairs given above.  (Compare
     *Note Exercise 3-11::.)

   ---------- Footnotes ----------

   (1) `Set-car!' and `set-cdr!' return implementation-dependent
values.  Like `set!', they should be used only for their effect.

   (2) We see from this that mutation operations on lists can create
"garbage" that is not part of any accessible structure.  We will see in
section *Note 5-3-2:: that Lisp memory-management systems include a "garbage
collector", which identifies and recycles the memory space used by
unneeded pairs.

   (3) `Get-new-pair' is one of the operations that must be implemented
as part of the memory management required by a Lisp implementation.  We
will discuss this in section *Note 5-3-1::.

   (4) The two pairs are distinct because each call to `cons' returns a
new pair.  The symbols are shared; in Scheme there is a unique symbol
with any given name.  Since Scheme provides no way to mutate a symbol,
this sharing is undetectable.  Note also that the sharing is what
enables us to compare symbols using `eq?', which simply checks equality
of pointers.

   (5) The subtleties of dealing with sharing of mutable data objects
reflect the underlying issues of "sameness" and "change" that were
raised in section *Note 3-1-3::.  We mentioned there that admitting
change to our language requires that a compound object must have an
"identity" that is something different from the pieces from which it is
composed.  In Lisp, we consider this "identity" to be the quality that
is tested by `eq?', i.e., by equality of pointers.  Since in most Lisp
implementations a pointer is essentially a memory address, we are
"solving the problem" of defining the identity of objects by
stipulating that a data object "itself" is the information stored in
some particular set of memory locations in the computer.  This suffices
for simple Lisp programs, but is hardly a general way to resolve the
issue of "sameness" in computational models.

   (6) On the other hand, from the viewpoint of implementation,
assignment requires us to modify the environment, which is itself a
mutable data structure.  Thus, assignment and mutation are equipotent:
Each can be implemented in terms of the other.


File: sicp.info,  Node: 3-3-2,  Next: 3-3-3,  Prev: 3-3-1,  Up: 3-3

3.3.2 Representing Queues
-------------------------

The mutators `set-car!' and `set-cdr!' enable us to use pairs to
construct data structures that cannot be built with `cons', `car', and
`cdr' alone.  This section shows how to use pairs to represent a data
structure called a queue.  Section *Note 3-3-3:: will show how to
represent data structures called tables.

   A "queue" is a sequence in which items are inserted at one end
(called the "rear" of the queue) and deleted from the other end (the "front").
*Note Figure 3-18:: shows an initially empty queue in which the items
`a' and `b' are inserted.  Then `a' is removed, `c' and `d' are
inserted, and `b' is removed.  Because items are always removed in the
order in which they are inserted, a queue is sometimes called a "FIFO"
(first in, first out) buffer.

     *Figure 3.18:* Queue operations.

          Operation                Resulting Queue
          (define q (make-queue))
          (insert-queue! q 'a)     a
          (insert-queue! q 'b)     a b
          (delete-queue! q)        b
          (insert-queue! q 'c)     b c
          (insert-queue! q 'd)     b c d
          (delete-queue! q)        c d

   In terms of data abstraction, we can regard a queue as defined by
the following set of operations:

   * a constructor: `(make-queue)' returns an empty queue (a queue
     containing no items).

   * two selectors:

          (empty-queue? <QUEUE>)

     tests if the queue is empty.

          (front-queue <QUEUE>)

     returns the object at the front of the queue, signaling an error
     if the queue is empty; it does not modify the queue.

   * two mutators:

          (insert-queue! <QUEUE> <ITEM>)

     inserts the item at the rear of the queue and returns the modified
     queue as its value.

          (delete-queue! <QUEUE>)

     removes the item at the front of the queue and returns the
     modified queue as its value, signaling an error if the queue is
     empty before the deletion.


   Because a queue is a sequence of items, we could certainly represent
it as an ordinary list; the front of the queue would be the `car' of
the list, inserting an item in the queue would amount to appending a
new element at the end of the list, and deleting an item from the queue
would just be taking the `cdr' of the list.  However, this
representation is inefficient, because in order to insert an item we
must scan the list until we reach the end.  Since the only method we
have for scanning a list is by successive `cdr' operations, this
scanning requires [theta](n) steps for a list of n items.  A simple
modification to the list representation overcomes this disadvantage by
allowing the queue operations to be implemented so that they require
[theta](1) steps; that is, so that the number of steps needed is
independent of the length of the queue.

   The difficulty with the list representation arises from the need to
scan to find the end of the list.  The reason we need to scan is that,
although the standard way of representing a list as a chain of pairs
readily provides us with a pointer to the beginning of the list, it
gives us no easily accessible pointer to the end.  The modification
that avoids the drawback is to represent the queue as a list, together
with an additional pointer that indicates the final pair in the list.
That way, when we go to insert an item, we can consult the rear pointer
and so avoid scanning the list.

   A queue is represented, then, as a pair of pointers, `front-ptr' and
`rear-ptr', which indicate, respectively, the first and last pairs in an
ordinary list.  Since we would like the queue to be an identifiable
object, we can use `cons' to combine the two pointers.  Thus, the queue
itself will be the `cons' of the two pointers.  *Note Figure 3-19::
illustrates this representation.

     *Figure 3.19:* Implementation of a queue as a list with front and
     rear pointers.

                 +---+---+
            q -->| * | *-+-------------------+
                 +-|-+---+                   |
                   |                         |
                   | front-ptr               | rear-ptr
                   V                         V
               +---+---+    +---+---+    +---+---+
               | * | *-+--->| * | *-+--->| * | / |
               +-|-+---+    +-|-+---+    +-|-+---+
                 V            V            V
               +---+        +---+        +---+
               | a |        | b |        | c |
               +---+        +---+        +---+

   To define the queue operations we use the following procedures,
which enable us to select and to modify the front and rear pointers of
a queue:

     (define (front-ptr queue) (car queue))

     (define (rear-ptr queue) (cdr queue))

     (define (set-front-ptr! queue item) (set-car! queue item))

     (define (set-rear-ptr! queue item) (set-cdr! queue item))

   Now we can implement the actual queue operations.  We will consider
a queue to be empty if its front pointer is the empty list:

     (define (empty-queue? queue) (null? (front-ptr queue)))

   The `make-queue' constructor returns, as an initially empty queue, a
pair whose `car' and `cdr' are both the empty list:

     (define (make-queue) (cons '() '()))

   To select the item at the front of the queue, we return the `car' of
the pair indicated by the front pointer:

     (define (front-queue queue)
       (if (empty-queue? queue)
           (error "FRONT called with an empty queue" queue)
           (car (front-ptr queue))))

   To insert an item in a queue, we follow the method whose result is
indicated in *Note Figure 3-20::.  We first create a new pair whose
`car' is the item to be inserted and whose `cdr' is the empty list.  If
the queue was initially empty, we set the front and rear pointers of
the queue to this new pair.  Otherwise, we modify the final pair in the
queue to point to the new pair, and also set the rear pointer to the
new pair.

     *Figure 3.20:* Result of using `(insert-queue!  q 'd)' on the
     queue of *Note Figure 3-19::.

                 +---+---+
            q -->| * | *-+--------------------------------+
                 +-|-+---+                                |
                   |                                      |
                   | front-ptr                            | rear-ptr
                   V                                      V
               +---+---+    +---+---+    +---+---+    +---+---+
               | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
               +-|-+---+    +-|-+---+    +-|-+---+    +-|-+---+
                 V            V            V            V
               +---+        +---+        +---+        +---+
               | a |        | b |        | c |        | d |
               +---+        +---+        +---+        +---+

     (define (insert-queue! queue item)
       (let ((new-pair (cons item '())))
         (cond ((empty-queue? queue)
                (set-front-ptr! queue new-pair)
                (set-rear-ptr! queue new-pair)
                queue)
               (else
                (set-cdr! (rear-ptr queue) new-pair)
                (set-rear-ptr! queue new-pair)
                queue))))

   To delete the item at the front of the queue, we merely modify the
front pointer so that it now points at the second item in the queue,
which can be found by following the `cdr' pointer of the first item
(see *Note Figure 3-21::):(1)

     *Figure 3.21:* Result of using `(delete-queue!  q)' on the queue
     of *Note Figure 3-20::.

                 +---+---+
            q -->| * | *-+--------------------------------+
                 +-|-+---+                                |
                   +------------+                         |
                      front-ptr |                         | rear-ptr
                                V                         V
               +---+---+    +---+---+    +---+---+    +---+---+
               | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
               +-|-+---+    +-|-+---+    +-|-+---+    +-|-+---+
                 V            V            V            V
               +---+        +---+        +---+        +---+
               | a |        | b |        | c |        | d |
               +---+        +---+        +---+        +---+

     (define (delete-queue! queue)
       (cond ((empty-queue? queue)
              (error "DELETE! called with an empty queue" queue))
             (else
              (set-front-ptr! queue (cdr (front-ptr queue)))
              queue)))

     *Exercise 3.21:* Ben Bitdiddle decides to test the queue
     implementation described above.  He types in the procedures to the
     Lisp interpreter and proceeds to try them out:

          (define q1 (make-queue))

          (insert-queue! q1 'a)
          ((a) a)

          (insert-queue! q1 'b)
          ((a b) b)

          (delete-queue! q1)
          ((b) b)

          (delete-queue! q1)
          (() b)

     "It's all wrong!" he complains.  "The interpreter's response shows
     that the last item is inserted into the queue twice.  And when I
     delete both items, the second `b' is still there, so the queue
     isn't empty, even though it's supposed to be."  Eva Lu Ator
     suggests that Ben has misunderstood what is happening.  "It's not
     that the items are going into the queue twice," she explains.
     "It's just that the standard Lisp printer doesn't know how to make
     sense of the queue representation.  If you want to see the queue
     printed correctly, you'll have to define your own print procedure
     for queues." Explain what Eva Lu is talking about.  In particular,
     show why Ben's examples produce the printed results that they do.
     Define a procedure `print-queue' that takes a queue as input and
     prints the sequence of items in the queue.

     *Exercise 3.22:* Instead of representing a queue as a pair of
     pointers, we can build a queue as a procedure with local state.
     The local state will consist of pointers to the beginning and the
     end of an ordinary list.  Thus, the `make-queue' procedure will
     have the form

          (define (make-queue)
            (let ((front-ptr ... )
                  (rear-ptr ... ))
              <DEFINITIONS OF INTERNAL PROCEDURES>
              (define (dispatch m) ...)
              dispatch))

     Complete the definition of `make-queue' and provide
     implementations of the queue operations using this representation.

     *Exercise 3.23:* A "deque" ("double-ended queue") is a sequence in
     which items can be inserted and deleted at either the front or the
     rear.  Operations on deques are the constructor `make-deque', the
     predicate `empty-deque?', selectors `front-deque' and
     `rear-deque', and mutators `front-insert-deque!',
     `rear-insert-deque!', `front-delete-deque!', and
     `rear-delete-deque!'.  Show how to represent deques using pairs,
     and give implementations of the operations.(2)  All operations
     should be accomplished in [theta](1) steps.

   ---------- Footnotes ----------

   (1) If the first item is the final item in the queue, the front
pointer will be the empty list after the deletion, which will mark the
queue as empty; we needn't worry about updating the rear pointer, which
will still point to the deleted item, because `empty-queue?' looks only
at the front pointer.

   (2) Be careful not to make the interpreter try to print a structure
that contains cycles.  (See *Note Exercise 3-13::.)


File: sicp.info,  Node: 3-3-3,  Next: 3-3-4,  Prev: 3-3-2,  Up: 3-3

3.3.3 Representing Tables
-------------------------

When we studied various ways of representing sets in *Note Chapter 2::,
we mentioned in section *Note 2-3-3:: the task of maintaining a table
of records indexed by identifying keys.  In the implementation of
data-directed programming in section *Note 2-4-3::, we made extensive
use of two-dimensional tables, in which information is stored and
retrieved using two keys.  Here we see how to build tables as mutable
list structures.

   We first consider a one-dimensional table, in which each value is
stored under a single key.  We implement the table as a list of
records, each of which is implemented as a pair consisting of a key and
the associated value. The records are glued together to form a list by
pairs whose `car's point to successive records.  These gluing pairs are
called the "backbone" of the table.  In order to have a place that we
can change when we add a new record to the table, we build the table as
a "headed list".  A headed list has a special backbone pair at the
beginning, which holds a dummy "record"--in this case the arbitrarily
chosen symbol `*table*'.  *Note Figure 3-22:: shows the box-and-pointer
diagram for the table

     a:  1
     b:  2
     c:  3

     *Figure 3.22:* A table represented as a headed list.

           +---+---+    +---+---+    +---+---+    +---+---+
           | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
           +-|-+---+    +-|-+---+    +-|-+---+    +-|-+---+
             |            |            |            |
             V            V            V            V
          +---------+   +---+---+   +---+---+   +---+---+
          | *table* |   | * | * |   | * | * |   | * | * |
          +---------+   +-|-+-|-+   +-|-+-|-+   +-|-+-|-+
                          |   |       |   |       |   |
                          V   V       V   V       V   V
                       +---+ +---+ +---+ +---+ +---+ +---+
                       | a | | 1 | | b | | 2 | | c | | 3 |
                       +---+ +---+ +---+ +---+ +---+ +---+

   To extract information from a table we use the `lookup' procedure,
which takes a key as argument and returns the associated value (or
false if there is no value stored under that key).  `Lookup' is defined
in terms of the `assoc' operation, which expects a key and a list of
records as arguments.  Note that `assoc' never sees the dummy record.
`Assoc' returns the record that has the given key as its `car'.(1)
`Lookup' then checks to see that the resulting record returned by
`assoc' is not false, and returns the value (the `cdr') of the record.

     (define (lookup key table)
       (let ((record (assoc key (cdr table))))
         (if record
             (cdr record)
             false)))

     (define (assoc key records)
       (cond ((null? records) false)
             ((equal? key (caar records)) (car records))
             (else (assoc key (cdr records)))))

   To insert a value in a table under a specified key, we first use
`assoc' to see if there is already a record in the table with this key.
If not, we form a new record by `cons'ing the key with the value, and
insert this at the head of the table's list of records, after the dummy
record.  If there already is a record with this key, we set the `cdr'
of this record to the designated new value.  The header of the table
provides us with a fixed location to modify in order to insert the new
record.(2)

     (define (insert! key value table)
       (let ((record (assoc key (cdr table))))
         (if record
             (set-cdr! record value)
             (set-cdr! table
                       (cons (cons key value) (cdr table)))))
       'ok)

   To construct a new table, we simply create a list containing the
symbol `*table*':

     (define (make-table)
       (list '*table*))

Two-dimensional tables
......................

In a two-dimensional table, each value is indexed by two keys.  We can
construct such a table as a one-dimensional table in which each key
identifies a subtable.  *Note Figure 3-23:: shows the box-and-pointer
diagram for the table

     math:
         +:  43
         -:  45
         *:  42
     letters:
         a:  97
         b:  98

which has two subtables.  (The subtables don't need a special header
symbol, since the key that identifies the subtable serves this purpose.)

     *Figure 3.23:* A two-dimensional table.

          table
            |
            V
          +---+---+   +---+---+   +---+---+
          | * | *-+-->| * | *-+-->| * | / |
          +-|-+---+   +-|-+---+   +-|-+---+
            V           |           V
          +-------+     |         +---+---+   +---+---+   +---+---+
          |*table*|     |         | * | *-+-->| * | *-+-->| * | / |
          +-------+     |         +-|-+---+   +-|-+---+   +-|-+---+
                        |           V           V           V
                        |       +-------+     +---+---+   +---+---+
                        |       |letters|     | * | * |   | * | * |
                        |       +-------+     +-|-+-|-+   +-|-+-|-+
                        |                       V   V       V   V
                        |                    +---+ +---+ +---+ +---+
                        |                    | a | | 97| | b | | 98|
                        |                    +---+ +---+ +---+ +---+
                        V
                      +---+---+   +---+---+   +---+---+   +---+---+
                      | * | *-+-->| * | *-+-->| * | *-+-->| * | / |
                      +-|-+---+   +-|-+---+   +-|-+---+   +-|-+---+
                        V           V           V           V
                    +------+      +---+---+   +---+---+   +---+---+
                    | math |      | * | * |   | * | * |   | * | * |
                    +------+      +-|-+-|-+   +-|-+-|-+   +-|-+-|-+
                                    V   V       V   V       V   V
                                 +---+ +---+ +---+ +---+ +---+ +---+
                                 | + | | 43| | - | | 45| | * | | 42|
                                 +---+ +---+ +---+ +---+ +---+ +---+

   When we look up an item, we use the first key to identify the
correct subtable.  Then we use the second key to identify the record
within the subtable.

     (define (lookup key-1 key-2 table)
       (let ((subtable (assoc key-1 (cdr table))))
         (if subtable
             (let ((record (assoc key-2 (cdr subtable))))
               (if record
                   (cdr record)
                   false))
             false)))

   To insert a new item under a pair of keys, we use `assoc' to see if
there is a subtable stored under the first key.  If not, we build a new
subtable containing the single record (`key-2', `value') and insert it
into the table under the first key.  If a subtable already exists for
the first key, we insert the new record into this subtable, using the
insertion method for one-dimensional tables described above:

     (define (insert! key-1 key-2 value table)
       (let ((subtable (assoc key-1 (cdr table))))
         (if subtable
             (let ((record (assoc key-2 (cdr subtable))))
               (if record
                   (set-cdr! record value)
                   (set-cdr! subtable
                             (cons (cons key-2 value)
                                   (cdr subtable)))))
             (set-cdr! table
                       (cons (list key-1
                                   (cons key-2 value))
                             (cdr table)))))
       'ok)

Creating local tables
.....................

The `lookup' and `insert!' operations defined above take the table as
an argument.  This enables us to use programs that access more than one
table.  Another way to deal with multiple tables is to have separate
`lookup' and `insert!' procedures for each table.  We can do this by
representing a table procedurally, as an object that maintains an
internal table as part of its local state.  When sent an appropriate
message, this "table object" supplies the procedure with which to
operate on the internal table.  Here is a generator for two-dimensional
tables represented in this fashion:

     (define (make-table)
       (let ((local-table (list '*table*)))
         (define (lookup key-1 key-2)
           (let ((subtable (assoc key-1 (cdr local-table))))
             (if subtable
                 (let ((record (assoc key-2 (cdr subtable))))
                   (if record
                       (cdr record)
                       false))
                 false)))
         (define (insert! key-1 key-2 value)
           (let ((subtable (assoc key-1 (cdr local-table))))
             (if subtable
                 (let ((record (assoc key-2 (cdr subtable))))
                   (if record
                       (set-cdr! record value)
                       (set-cdr! subtable
                                 (cons (cons key-2 value)
                                       (cdr subtable)))))
                 (set-cdr! local-table
                           (cons (list key-1
                                       (cons key-2 value))
                                 (cdr local-table)))))
           'ok)
         (define (dispatch m)
           (cond ((eq? m 'lookup-proc) lookup)
                 ((eq? m 'insert-proc!) insert!)
                 (else (error "Unknown operation -- TABLE" m))))
         dispatch))

   Using `make-table', we could implement the `get' and `put'
operations used in section *Note 2-4-3:: for data-directed programming,
as follows:

     (define operation-table (make-table))
     (define get (operation-table 'lookup-proc))
     (define put (operation-table 'insert-proc!))

   `Get' takes as arguments two keys, and `put' takes as arguments two
keys and a value.  Both operations access the same local table, which is
encapsulated within the object created by the call to `make-table'.

     *Exercise 3.24:* In the table implementations above, the keys are
     tested for equality using `equal?' (called by `assoc').  This is
     not always the appropriate test.  For instance, we might have a
     table with numeric keys in which we don't need an exact match to
     the number we're looking up, but only a number within some
     tolerance of it.  Design a table constructor `make-table' that
     takes as an argument a `same-key?' procedure that will be used to
     test "equality" of keys.  `Make-table' should return a `dispatch'
     procedure that can be used to access appropriate `lookup' and
     `insert!' procedures for a local table.

     *Exercise 3.25:* Generalizing one- and two-dimensional tables,
     show how to implement a table in which values are stored under an
     arbitrary number of keys and different values may be stored under
     different numbers of keys.  The `lookup' and `insert!' procedures
     should take as input a list of keys used to access the table.

     *Exercise 3.26:* To search a table as implemented above, one needs
     to scan through the list of records.  This is basically the
     unordered list representation of section *Note 2-3-3::.  For large
     tables, it may be more efficient to structure the table in a
     different manner.  Describe a table implementation where the (key,
     value) records are organized using a binary tree, assuming that
     keys can be ordered in some way (e.g., numerically or
     alphabetically).  (Compare *Note Exercise 2-66:: of *Note Chapter
     2::.)

     *Exercise 3.27:* "Memoization" (also called "tabulation") is a
     technique that enables a procedure to record, in a local table,
     values that have previously been computed.  This technique can
     make a vast difference in the performance of a program.  A memoized
     procedure maintains a table in which values of previous calls are
     stored using as keys the arguments that produced the values.  When
     the memoized procedure is asked to compute a value, it first
     checks the table to see if the value is already there and, if so,
     just returns that value.  Otherwise, it computes the new value in
     the ordinary way and stores this in the table.  As an example of
     memoization, recall from section *Note 1-2-2:: the exponential
     process for computing Fibonacci numbers:

          (define (fib n)
            (cond ((= n 0) 0)
                  ((= n 1) 1)
                  (else (+ (fib (- n 1))
                           (fib (- n 2))))))

     The memoized version of the same procedure is

          (define memo-fib
            (memoize (lambda (n)
                       (cond ((= n 0) 0)
                             ((= n 1) 1)
                             (else (+ (memo-fib (- n 1))
                                      (memo-fib (- n 2))))))))

     where the memoizer is defined as

          (define (memoize f)
            (let ((table (make-table)))
              (lambda (x)
                (let ((previously-computed-result (lookup x table)))
                  (or previously-computed-result
                      (let ((result (f x)))
                        (insert! x result table)
                        result))))))

     Draw an environment diagram to analyze the computation of
     `(memo-fib 3)'.  Explain why `memo-fib' computes the nth Fibonacci
     number in a number of steps proportional to n.  Would the scheme
     still work if we had simply defined `memo-fib' to be `(memoize
     fib)'?

   ---------- Footnotes ----------

   (1) Because `assoc' uses `equal?', it can recognize keys that are
symbols, numbers, or list structure.

   (2) Thus, the first backbone pair is the object that represents the
table "itself"; that is, a pointer to the table is a pointer to this
pair.  This same backbone pair always starts the table.  If we did not
arrange things in this way, `insert!' would have to return a new value
for the start of the table when it added a new record.


File: sicp.info,  Node: 3-3-4,  Next: 3-3-5,  Prev: 3-3-3,  Up: 3-3

3.3.4 A Simulator for Digital Circuits
--------------------------------------

Designing complex digital systems, such as computers, is an important
engineering activity.  Digital systems are constructed by
interconnecting simple elements.  Although the behavior of these
individual elements is simple, networks of them can have very complex
behavior.  Computer simulation of proposed circuit designs is an
important tool used by digital systems engineers.  In this section we
design a system for performing digital logic simulations.  This system
typifies a kind of program called an "event-driven simulation", in
which actions ("events") trigger further events that happen at a later
time, which in turn trigger more events, and so so.

   Our computational model of a circuit will be composed of objects that
correspond to the elementary components from which the circuit is
constructed.  There are "wires", which carry "digital signals".  A
digital signal may at any moment have only one of two possible values,
0 and 1.  There are also various types of digital "function boxes",
which connect wires carrying input signals to other output wires.  Such
boxes produce output signals computed from their input signals.  The
output signal is delayed by a time that depends on the type of the
function box.  For example, an "inverter" is a primitive function box
that inverts its input.  If the input signal to an inverter changes to
0, then one inverter-delay later the inverter will change its output
signal to 1.  If the input signal to an inverter changes to 1, then one
inverter-delay later the inverter will change its output signal to 0.
We draw an inverter symbolically as in *Note Figure 3-24::.  An "and-gate",
also shown in *Note Figure 3-24::, is a primitive function box with two
inputs and one output.  It drives its output signal to a value that is
the "logical and" of the inputs.  That is, if both of its input signals
become 1, then one and-gate-delay time later the and-gate will force
its output signal to be 1; otherwise the output will be 0.  An "or-gate"
is a similar two-input primitive function box that drives its output
signal to a value that is the "logical or" of the inputs.  That is, the
output will become 1 if at least one of the input signals is 1;
otherwise the output will become 0.

     *Figure 3.24:* Primitive functions in the digital logic simulator.

                         __          ___
            |\        --|  \       --\  \
          --| >o--      |   )--       )  >--
            |/        --|__/       --/__/

          Inverter    And-gate     Or-gate

   We can connect primitive functions together to construct more complex
functions.  To accomplish this we wire the outputs of some function
boxes to the inputs of other function boxes.  For example, the "half-adder"
circuit shown in *Note Figure 3-25:: consists of an or-gate, two
and-gates, and an inverter.  It takes two input signals, A and B, and
has two output signals, S and C.  S will become 1 whenever precisely
one of A and B is 1, and C will become 1 whenever A and B are both 1.
We can see from the figure that, because of the delays involved, the
outputs may be generated at different times.  Many of the difficulties
in the design of digital circuits arise from this fact.

     *Figure 3.25:* A half-adder circuit.

              +--------------------------------------+
              |         ____                         |
          A --------*---\   \ D               ___    |
              |     |    >   >---------------|   \   |
              |  +--|---/___/                |    )----- S
              |  |  |              |\  E  +--|___/   |
              |  |  |           +--| >o---+          |
              |  |  |    ___    |  |/                |
              |  |  +---|   \   |                    |
              |  |      |    )--*----------------------- C
          B -----*------|___/                        |
              |                                      |
              +--------------------------------------+

   We will now build a program for modeling the digital logic circuits
we wish to study.  The program will construct computational objects
modeling the wires, which will "hold" the signals.  Function boxes will
be modeled by procedures that enforce the correct relationships among
the signals.

   One basic element of our simulation will be a procedure `make-wire',
which constructs wires.  For example, we can construct six wires as
follows:

     (define a (make-wire))
     (define b (make-wire))
     (define c (make-wire))

     (define d (make-wire))
     (define e (make-wire))
     (define s (make-wire))

   We attach a function box to a set of wires by calling a procedure
that constructs that kind of box.  The arguments to the constructor
procedure are the wires to be attached to the box.  For example, given
that we can construct and-gates, or-gates, and inverters, we can wire
together the half-adder shown in *Note Figure 3-25:::

     (or-gate a b d)
     ok

     (and-gate a b c)
     ok

     (inverter c e)
     ok

     (and-gate d e s)
     ok

   Better yet, we can explicitly name this operation by defining a
procedure `half-adder' that constructs this circuit, given the four
external wires to be attached to the half-adder:

     (define (half-adder a b s c)
       (let ((d (make-wire)) (e (make-wire)))
         (or-gate a b d)
         (and-gate a b c)
         (inverter c e)
         (and-gate d e s)
         'ok))

   The advantage of making this definition is that we can use
`half-adder' itself as a building block in creating more complex
circuits.  *Note Figure 3-26::, for example, shows a "full-adder"
composed of two half-adders and an or-gate.(1) We can construct a
full-adder as follows:

     (define (full-adder a b c-in sum c-out)
       (let ((s (make-wire))
             (c1 (make-wire))
             (c2 (make-wire)))
         (half-adder b c-in s c1)
         (half-adder a s sum c2)
         (or-gate c1 c2 c-out)
         'ok))

   Having defined `full-adder' as a procedure, we can now use it as a
building block for creating still more complex circuits.  (For example,
see *Note Exercise 3-30::.)

     *Figure 3.26:* A full-adder circuit.

              +----------------------------------+
              |              +-------+           |
          A -----------------+ half  +-------------- SUM
              |  +-------+   | adder |   ____    |
          B -----+ half  +---+       +---\   \   |
              |  | adder |   +-------+    >or >----- Cout
          C -----+       +---------------/___/   |
              |  +-------+                       |
              +----------------------------------+

   In essence, our simulator provides us with the tools to construct a
language of circuits.  If we adopt the general perspective on languages
with which we approached the study of Lisp in section *Note 1-1::, we
can say that the primitive function boxes form the primitive elements
of the language, that wiring boxes together provides a means of
combination, and that specifying wiring patterns as procedures serves
as a means of abstraction.

Primitive function boxes
........................

The primitive function boxes implement the "forces" by which a change
in the signal on one wire influences the signals on other wires.  To
build function boxes, we use the following operations on wires:

   *      (get-signal <WIRE>)

     returns the current value of the signal on the wire.

   *      (set-signal! <WIRE> <NEW VALUE>)

     changes the value of the signal on the wire to the new value.

   *      (add-action! <WIRE> <PROCEDURE OF NO ARGUMENTS>)

     asserts that the designated procedure should be run whenever the
     signal on the wire changes value.  Such procedures are the
     vehicles by which changes in the signal value on the wire are
     communicated to other wires.


   In addition, we will make use of a procedure `after-delay' that
takes a time delay and a procedure to be run and executes the given
procedure after the given delay.

   Using these procedures, we can define the primitive digital logic
functions.  To connect an input to an output through an inverter, we
use `add-action!' to associate with the input wire a procedure that
will be run whenever the signal on the input wire changes value.  The
procedure computes the `logical-not' of the input signal, and then,
after one `inverter-delay', sets the output signal to be this new value:

     (define (inverter input output)
       (define (invert-input)
         (let ((new-value (logical-not (get-signal input))))
           (after-delay inverter-delay
                        (lambda ()
                          (set-signal! output new-value)))))
       (add-action! input invert-input)
       'ok)

     (define (logical-not s)
       (cond ((= s 0) 1)
             ((= s 1) 0)
             (else (error "Invalid signal" s))))

   An and-gate is a little more complex.  The action procedure must be
run if either of the inputs to the gate changes.  It computes the
`logical-and' (using a procedure analogous to `logical-not') of the
values of the signals on the input wires and sets up a change to the
new value to occur on the output wire after one `and-gate-delay'.

     (define (and-gate a1 a2 output)
       (define (and-action-procedure)
         (let ((new-value
                (logical-and (get-signal a1) (get-signal a2))))
           (after-delay and-gate-delay
                        (lambda ()
                          (set-signal! output new-value)))))
       (add-action! a1 and-action-procedure)
       (add-action! a2 and-action-procedure)
       'ok)

     *Exercise 3.28:* Define an or-gate as a primitive function box.
     Your `or-gate' constructor should be similar to `and-gate'.

     *Exercise 3.29:* Another way to construct an or-gate is as a
     compound digital logic device, built from and-gates and inverters.
     Define a procedure `or-gate' that accomplishes this.  What is the
     delay time of the or-gate in terms of `and-gate-delay' and
     `inverter-delay'?

     *Exercise 3.30:* *Note Figure 3-27:: shows a "ripple-carry adder"
     formed by stringing together n full-adders.  This is the simplest
     form of parallel adder for adding two n-bit binary numbers.  The
     inputs A_1, A_2, A_3, ..., A_n and B_1, B_2, B_3, ..., B_n are the
     two binary numbers to be added (each A_k and B_k is a 0 or a 1).
     The circuit generates S_1, S_2, S_3, ..., S_n, the n bits of the
     sum, and C, the carry from the addition.  Write a procedure
     `ripple-carry-adder' that generates this circuit.  The procedure
     should take as arguments three lists of n wires each--the A_k, the
     B_k, and the S_k--and also another wire C.  The major drawback of
     the ripple-carry adder is the need to wait for the carry signals
     to propagate.  What is the delay needed to obtain the complete
     output from an n-bit ripple-carry adder, expressed in terms of the
     delays for and-gates, or-gates, and inverters?

     *Figure 3.27:* A ripple-carry adder for n-bit numbers.

             :                                              :   :
             : A_1 B_1   C_1   A_2 B_2   C_2   A_3 B_3   C_3:   : A_n B_n C_n=0
             :  |   |   +---+   |   |   +---+   |   |   +-----  :  |   |   +-
             |  |   |   |   |   |   |   |   |   |   |   |   :   :  |   |   |
             : ++---+---++  |  ++---+---++  |  ++---+---++  :   : ++---+---++
             : |   FA    |  |  |   FA    |  |  |   FA    |  :   : |   FA    |
             : +--+---+--+  |  +--+---+--+  |  +--+---+--+  :   : +--+---+--+
             :    |   |     |     |   |     |     |   |     :   :    |   |
          C ------+   |     +-----+   |     +-----+   |     :  ------+   |
             :        |       C_1     |       C_2     |     :   :C_(n-1) |
             :        |               |               |     :   :        |
                     S_1             S_2             S_3                S_n

Representing wires
..................

A wire in our simulation will be a computational object with two local
state variables: a `signal-value' (initially taken to be 0) and a
collection of `action-procedures' to be run when the signal changes
value.  We implement the wire, using message-passing style, as a
collection of local procedures together with a `dispatch' procedure
that selects the appropriate local operation, just as we did with the
simple bank-account object in section *Note 3-1-1:::

     (define (make-wire)
       (let ((signal-value 0) (action-procedures '()))
         (define (set-my-signal! new-value)
           (if (not (= signal-value new-value))
               (begin (set! signal-value new-value)
                      (call-each action-procedures))
               'done))

         (define (accept-action-procedure! proc)
           (set! action-procedures (cons proc action-procedures))
           (proc))

         (define (dispatch m)
           (cond ((eq? m 'get-signal) signal-value)
                 ((eq? m 'set-signal!) set-my-signal!)
                 ((eq? m 'add-action!) accept-action-procedure!)
                 (else (error "Unknown operation -- WIRE" m))))
         dispatch))

   The local procedure `set-my-signal!' tests whether the new signal
value changes the signal on the wire.  If so, it runs each of the
action procedures, using the following procedure `call-each', which
calls each of the items in a list of no-argument procedures:

     (define (call-each procedures)
       (if (null? procedures)
           'done
           (begin
             ((car procedures))
             (call-each (cdr procedures)))))

   The local procedure `accept-action-procedure!' adds the given
procedure to the list of procedures to be run, and then runs the new
procedure once.  (See *Note Exercise 3-31::.)

   With the local `dispatch' procedure set up as specified, we can
provide the following procedures to access the local operations on
wires:(2)

     (define (get-signal wire)
       (wire 'get-signal))

     (define (set-signal! wire new-value)
       ((wire 'set-signal!) new-value))

     (define (add-action! wire action-procedure)
       ((wire 'add-action!) action-procedure))

   Wires, which have time-varying signals and may be incrementally
attached to devices, are typical of mutable objects.  We have modeled
them as procedures with local state variables that are modified by
assignment.  When a new wire is created, a new set of state variables
is allocated (by the `let' expression in `make-wire') and a new
`dispatch' procedure is constructed and returned, capturing the
environment with the new state variables.

   The wires are shared among the various devices that have been
connected to them.  Thus, a change made by an interaction with one
device will affect all the other devices attached to the wire.  The
wire communicates the change to its neighbors by calling the action
procedures provided to it when the connections were established.

The agenda
..........

The only thing needed to complete the simulator is `after-delay'.  The
idea here is that we maintain a data structure, called an "agenda",
that contains a schedule of things to do.  The following operations are
defined for agendas:

   * `(make-agenda)' returns a new empty agenda.

   * `(empty-agenda? <AGENDA>)' is true if the specified agenda is
     empty.

   * `(first-agenda-item <AGENDA>)' returns the first item on the
     agenda.

   * `(remove-first-agenda-item! <AGENDA>)' modifies the agenda by
     removing the first item.

   * `(add-to-agenda! <TIME> <ACTION> <AGENDA>)' modifies the agenda by
     adding the given action procedure to be run at the specified time.

   * `(current-time <AGENDA>)' returns the current simulation time.


   The particular agenda that we use is denoted by `the-agenda'.  The
procedure `after-delay' adds new elements to `the-agenda':

     (define (after-delay delay action)
       (add-to-agenda! (+ delay (current-time the-agenda))
                       action
                       the-agenda))

   The simulation is driven by the procedure `propagate', which
operates on `the-agenda', executing each procedure on the agenda in
sequence.  In general, as the simulation runs, new items will be added
to the agenda, and `propagate' will continue the simulation as long as
there are items on the agenda:

     (define (propagate)
       (if (empty-agenda? the-agenda)
           'done
           (let ((first-item (first-agenda-item the-agenda)))
             (first-item)
             (remove-first-agenda-item! the-agenda)
             (propagate))))

A sample simulation
...................

The following procedure, which places a "probe" on a wire, shows the
simulator in action.  The probe tells the wire that, whenever its signal
changes value, it should print the new signal value, together with the
current time and a name that identifies the wire:

     (define (probe name wire)
       (add-action! wire
                    (lambda ()
                      (newline)
                      (display name)
                      (display " ")
                      (display (current-time the-agenda))
                      (display "  New-value = ")
                      (display (get-signal wire)))))

   We begin by initializing the agenda and specifying delays for the
primitive function boxes:

     (define the-agenda (make-agenda))
     (define inverter-delay 2)
     (define and-gate-delay 3)
     (define or-gate-delay 5)

   Now we define four wires, placing probes on two of them:

     (define input-1 (make-wire))
     (define input-2 (make-wire))
     (define sum (make-wire))
     (define carry (make-wire))

     (probe 'sum sum)
     sum 0  New-value = 0

     (probe 'carry carry)
     carry 0  New-value = 0

   Next we connect the wires in a half-adder circuit (as in *Note
Figure 3-25::), set the signal on `input-1' to 1, and run the
simulation:

     (half-adder input-1 input-2 sum carry)
     ok

     (set-signal! input-1 1)
     done

     (propagate)
     sum 8  New-value = 1
     done

   The `sum' signal changes to 1 at time 8.  We are now eight time
units from the beginning of the simulation.  At this point, we can set
the signal on `input-2' to 1 and allow the values to propagate:

     (set-signal! input-2 1)
     done

     (propagate)
     carry 11  New-value = 1
     sum 16  New-value = 0
     done

   The `carry' changes to 1 at time 11 and the `sum' changes to 0 at
time 16.

     *Exercise 3.31:* The internal procedure `accept-action-procedure!'
     defined in `make-wire' specifies that when a new action procedure
     is added to a wire, the procedure is immediately run.  Explain why
     this initialization is necessary.  In particular, trace through the
     half-adder example in the paragraphs above and say how the
     system's response would differ if we had defined
     `accept-action-procedure!' as

          (define (accept-action-procedure! proc)
            (set! action-procedures (cons proc action-procedures)))

Implementing the agenda
.......................

Finally, we give details of the agenda data structure, which holds the
procedures that are scheduled for future execution.

   The agenda is made up of "time segments".  Each time segment is a
pair consisting of a number (the time) and a queue (see *Note Exercise
3-32::) that holds the procedures that are scheduled to be run during
that time segment.

     (define (make-time-segment time queue)
       (cons time queue))

     (define (segment-time s) (car s))

     (define (segment-queue s) (cdr s))

   We will operate on the time-segment queues using the queue
operations described in section *Note 3-3-2::.

   The agenda itself is a one-dimensional table of time segments.  It
differs from the tables described in section *Note 3-3-3:: in that the
segments will be sorted in order of increasing time.  In addition, we
store the "current time" (i.e., the time of the last action that was
processed) at the head of the agenda.  A newly constructed agenda has
no time segments and has a current time of 0:(3)

     (define (make-agenda) (list 0))

     (define (current-time agenda) (car agenda))

     (define (set-current-time! agenda time)
       (set-car! agenda time))

     (define (segments agenda) (cdr agenda))

     (define (set-segments! agenda segments)
       (set-cdr! agenda segments))

     (define (first-segment agenda) (car (segments agenda)))

     (define (rest-segments agenda) (cdr (segments agenda)))

   An agenda is empty if it has no time segments:

     (define (empty-agenda? agenda)
       (null? (segments agenda)))

   To add an action to an agenda, we first check if the agenda is
empty.  If so, we create a time segment for the action and install this
in the agenda.  Otherwise, we scan the agenda, examining the time of
each segment.  If we find a segment for our appointed time, we add the
action to the associated queue.  If we reach a time later than the one
to which we are appointed, we insert a new time segment into the agenda
just before it.  If we reach the end of the agenda, we must create a
new time segment at the end.

     (define (add-to-agenda! time action agenda)
       (define (belongs-before? segments)
         (or (null? segments)
             (< time (segment-time (car segments)))))
       (define (make-new-time-segment time action)
         (let ((q (make-queue)))
           (insert-queue! q action)
           (make-time-segment time q)))
       (define (add-to-segments! segments)
         (if (= (segment-time (car segments)) time)
             (insert-queue! (segment-queue (car segments))
                            action)
             (let ((rest (cdr segments)))
               (if (belongs-before? rest)
                   (set-cdr!
                    segments
                    (cons (make-new-time-segment time action)
                          (cdr segments)))
                   (add-to-segments! rest)))))
       (let ((segments (segments agenda)))
         (if (belongs-before? segments)
             (set-segments!
              agenda
              (cons (make-new-time-segment time action)
                    segments))
             (add-to-segments! segments))))

   The procedure that removes the first item from the agenda deletes
the item at the front of the queue in the first time segment.  If this
deletion makes the time segment empty, we remove it from the list of
segments:(4)

     (define (remove-first-agenda-item! agenda)
       (let ((q (segment-queue (first-segment agenda))))
         (delete-queue! q)
         (if (empty-queue? q)
             (set-segments! agenda (rest-segments agenda)))))

   The first agenda item is found at the head of the queue in the first
time segment.  Whenever we extract an item, we also update the current
time:(5)

     (define (first-agenda-item agenda)
       (if (empty-agenda? agenda)
           (error "Agenda is empty -- FIRST-AGENDA-ITEM")
           (let ((first-seg (first-segment agenda)))
             (set-current-time! agenda (segment-time first-seg))
             (front-queue (segment-queue first-seg)))))

     *Exercise 3.32:* The procedures to be run during each time segment
     of the agenda are kept in a queue.  Thus, the procedures for each
     segment are called in the order in which they were added to the
     agenda (first in, first out).  Explain why this order must be
     used.  In particular, trace the behavior of an and-gate whose
     inputs change from 0,1 to 1,0 in the same segment and say how the
     behavior would differ if we stored a segment's procedures in an
     ordinary list, adding and removing procedures only at the front
     (last in, first out).

   ---------- Footnotes ----------

   (1) A full-adder is a basic circuit element used in adding two
binary numbers.  Here A and B are the bits at corresponding positions in
the two numbers to be added, and C_(in) is the carry bit from the
addition one place to the right.  The circuit generates SUM, which is
the sum bit in the corresponding position, and C_(out), which is the
carry bit to be propagated to the left.

   (2) [Footnote 27] These procedures are simply syntactic sugar that
allow us to use ordinary procedural syntax to access the local
procedures of objects.  It is striking that we can interchange the role
of "procedures" and "data" in such a simple way.  For example, if we
write `(wire 'get-signal)' we think of `wire' as a procedure that is
called with the message `get-signal' as input.  Alternatively, writing
`(get-signal wire)' encourages us to think of `wire' as a data object
that is the input to a procedure `get-signal'.  The truth of the matter
is that, in a language in which we can deal with procedures as objects,
there is no fundamental difference between "procedures" and "data," and
we can choose our syntactic sugar to allow us to program in whatever
style we choose.

   (3) The agenda is a headed list, like the tables in section *Note
3-3-3::, but since the list is headed by the time, we do not need an
additional dummy header (such as the `*table*' symbol used with tables).

   (4) Observe that the `if' expression in this procedure has no
<ALTERNATIVE> expression.  Such a "one-armed `if' statement" is used to
decide whether to do something, rather than to select between two
expressions.  An `if' expression returns an unspecified value if the
predicate is false and there is no <ALTERNATIVE>.

   (5) In this way, the current time will always be the time of the
action most recently processed.  Storing this time at the head of the
agenda ensures that it will still be available even if the associated
time segment has been deleted.


File: sicp.info,  Node: 3-3-5,  Prev: 3-3-4,  Up: 3-3

3.3.5 Propagation of Constraints
--------------------------------

Computer programs are traditionally organized as one-directional
computations, which perform operations on prespecified arguments to
produce desired outputs.  On the other hand, we often model systems in
terms of relations among quantities.  For example, a mathematical model
of a mechanical structure might include the information that the
deflection d of a metal rod is related to the force f on the rod, the
length L of the rod, the cross-sectional area A, and the elastic
modulus E via the equation

     dAE = FL

   Such an equation is not one-directional.  Given any four of the
quantities, we can use it to compute the fifth.  Yet translating the
equation into a traditional computer language would force us to choose
one of the quantities to be computed in terms of the other four.  Thus,
a procedure for computing the area A could not be used to compute the
deflection d, even though the computations of A and d arise from the
same equation.(1)

   In this section, we sketch the design of a language that enables us
to work in terms of relations themselves.  The primitive elements of
the language are "primitive constraints", which state that certain
relations hold between quantities.  For example, `(adder a b c)'
specifies that the quantities a, b, and c must be related by the
equation a + b = c, `(multiplier x y z)' expresses the constraint xy =
z, and `(constant 3.14 x)' says that the value of x must be 3.14.

   Our language provides a means of combining primitive constraints in
order to express more complex relations.  We combine constraints by
constructing "constraint networks", in which constraints are joined by "connectors".
A connector is an object that "holds" a value that may participate in
one or more constraints.  For example, we know that the relationship
between Fahrenheit and Celsius temperatures is

     9C = 5(F - 32)

   Such a constraint can be thought of as a network consisting of
primitive adder, multiplier, and constant constraints (*Note Figure
3-28::).  In the figure, we see on the left a multiplier box with three
terminals, labeled m1, m2, and p.  These connect the multiplier to the
rest of the network as follows: The m1 terminal is linked to a
connector C, which will hold the Celsius temperature.  The m2 terminal
is linked to a connector w, which is also linked to a constant box that
holds 9.  The p terminal, which the multiplier box constrains to be the
product of m1 and m2, is linked to the p terminal of another multiplier
box, whose m2 is connected to a constant 5 and whose m1 is connected to
one of the terms in a sum.

     *Figure 3.28:* The relation 9C = 5(F - 32) expressed as a
     constraint network.

                 +---------+     +---------+   v   +---------+
          C -----+ m1      |  u  |      m1 +-------+ a1      |
                 |    *  p +-----+ p  *    |       |    +  s +---- F
              +--+ m2      |     |      m2 +--+ +--+ a2      |
              |  +---------+     +---------+  | |  +---------+
            w |                              x| |y
              |    +-----+        +-----+     | |     +-----+
              +----+  9  |        |  5  +-----+ +-----+  32 |
                   +-----+        +-----+             +-----+

   Computation by such a network proceeds as follows: When a connector
is given a value (by the user or by a constraint box to which it is
linked), it awakens all of its associated constraints (except for the
constraint that just awakened it) to inform them that it has a value.
Each awakened constraint box then polls its connectors to see if there
is enough information to determine a value for a connector.  If so, the
box sets that connector, which then awakens all of its associated
constraints, and so on.  For instance, in conversion between Celsius
and Fahrenheit, w, x, and y are immediately set by the constant boxes
to 9, 5, and 32, respectively.  The connectors awaken the multipliers
and the adder, which determine that there is not enough information to
proceed.  If the user (or some other part of the network) sets C to a
value (say 25), the leftmost multiplier will be awakened, and it will
set u to 25*9 = 225.  Then u awakens the second multiplier, which sets
v to 45, and v awakens the adder, which sets f to 77.

Using the constraint system
...........................

To use the constraint system to carry out the temperature computation
outlined above, we first create two connectors, `C' and `F', by calling
the constructor `make-connector', and link `C' and `F' in an
appropriate network:

     (define C (make-connector))
     (define F (make-connector))
     (celsius-fahrenheit-converter C F)
     ok

   The procedure that creates the network is defined as follows:

     (define (celsius-fahrenheit-converter c f)
       (let ((u (make-connector))
             (v (make-connector))
             (w (make-connector))
             (x (make-connector))
             (y (make-connector)))
         (multiplier c w u)
         (multiplier v x u)
         (adder v y f)
         (constant 9 w)
         (constant 5 x)
         (constant 32 y)
         'ok))

   This procedure creates the internal connectors `u', `v', `w', `x',
and `y', and links them as shown in *Note Figure 3-28:: using the
primitive constraint constructors `adder', `multiplier', and
`constant'.  Just as with the digital-circuit simulator of section
*Note 3-3-4::, expressing these combinations of primitive elements in
terms of procedures automatically provides our language with a means of
abstraction for compound objects.

   To watch the network in action, we can place probes on the
connectors `C' and `F', using a `probe' procedure similar to the one we
used to monitor wires in section *Note 3-3-4::.  Placing a probe on a
connector will cause a message to be printed whenever the connector is
given a value:

     (probe "Celsius temp" C)
     (probe "Fahrenheit temp" F)

   Next we set the value of `C' to 25.  (The third argument to
`set-value!' tells `C' that this directive comes from the `user'.)

     (set-value! C 25 'user)
     Probe: Celsius temp = 25
     Probe: Fahrenheit temp = 77
     done

   The probe on `C' awakens and reports the value.  `C' also propagates
its value through the network as described above.  This sets `F' to 77,
which is reported by the probe on `F'.

   Now we can try to set `F' to a new value, say 212:

     (set-value! F 212 'user)
     Error! Contradiction (77 212)

   The connector complains that it has sensed a contradiction: Its
value is 77, and someone is trying to set it to 212.  If we really want
to reuse the network with new values, we can tell `C' to forget its old
value:

     (forget-value! C 'user)
     Probe: Celsius temp = ?
     Probe: Fahrenheit temp = ?
     done

   `C' finds that the `user', who set its value originally, is now
retracting that value, so `C' agrees to lose its value, as shown by the
probe, and informs the rest of the network of this fact.  This
information eventually propagates to `F', which now finds that it has
no reason for continuing to believe that its own value is 77.  Thus,
`F' also gives up its value, as shown by the probe.

   Now that `F' has no value, we are free to set it to 212:

     (set-value! F 212 'user)
     Probe: Fahrenheit temp = 212
     Probe: Celsius temp = 100
     done

   This new value, when propagated through the network, forces `C' to
have a value of 100, and this is registered by the probe on `C'.
Notice that the very same network is being used to compute `C' given
`F' and to compute `F' given `C'.  This nondirectionality of
computation is the distinguishing feature of constraint-based systems.

Implementing the constraint system
..................................

The constraint system is implemented via procedural objects with local
state, in a manner very similar to the digital-circuit simulator of
section *Note 3-3-4::.  Although the primitive objects of the
constraint system are somewhat more complex, the overall system is
simpler, since there is no concern about agendas and logic delays.

   The basic operations on connectors are the following:

   * `(has-value? <CONNECTOR>)' tells whether the connector has a value.

   * `(get-value <CONNECTOR>)' returns the connector's current value.

   * `(set-value! <CONNECTOR> <NEW-VALUE> <INFORMANT>)' indicates that
     the informant is requesting the connector to set its value to the
     new value.

   * `(forget-value! <CONNECTOR> <RETRACTOR>)' tells the connector that
     the retractor is requesting it to forget its value.

   * `(connect <CONNECTOR> <NEW-CONSTRAINT>)' tells the connector to
     participate in the new constraint.


   The connectors communicate with the constraints by means of the
procedures `inform-about-value', which tells the given constraint that
the connector has a value, and `inform-about-no-value', which tells the
constraint that the connector has lost its value.

   `Adder' constructs an adder constraint among summand connectors `a1'
and `a2' and a `sum' connector.  An adder is implemented as a procedure
with local state (the procedure `me' below):

     (define (adder a1 a2 sum)
       (define (process-new-value)
         (cond ((and (has-value? a1) (has-value? a2))
                (set-value! sum
                            (+ (get-value a1) (get-value a2))
                            me))
               ((and (has-value? a1) (has-value? sum))
                (set-value! a2
                            (- (get-value sum) (get-value a1))
                            me))
               ((and (has-value? a2) (has-value? sum))
                (set-value! a1
                            (- (get-value sum) (get-value a2))
                            me))))
       (define (process-forget-value)
         (forget-value! sum me)
         (forget-value! a1 me)
         (forget-value! a2 me)
         (process-new-value))
       (define (me request)
         (cond ((eq? request 'I-have-a-value)
                (process-new-value))
               ((eq? request 'I-lost-my-value)
                (process-forget-value))
               (else
                (error "Unknown request -- ADDER" request))))
       (connect a1 me)
       (connect a2 me)
       (connect sum me)
       me)

   `Adder' connects the new adder to the designated connectors and
returns it as its value.  The procedure `me', which represents the
adder, acts as a dispatch to the local procedures.  The following
"syntax interfaces" (see footnote *Note Footnote 27:: in section *Note
3-3-4::) are used in conjunction with the dispatch:

     (define (inform-about-value constraint)
       (constraint 'I-have-a-value))

     (define (inform-about-no-value constraint)
       (constraint 'I-lost-my-value))

   The adder's local procedure `process-new-value' is called when the
adder is informed that one of its connectors has a value. The adder
first checks to see if both `a1' and `a2' have values. If so, it tells
`sum' to set its value to the sum of the two addends.  The `informant'
argument to `set-value!' is `me', which is the adder object itself.  If
`a1' and `a2' do not both have values, then the adder checks to see if
perhaps `a1' and `sum' have values.  If so, it sets `a2' to the
difference of these two.  Finally, if `a2' and `sum' have values, this
gives the adder enough information to set `a1'.  If the adder is told
that one of its connectors has lost a value, it requests that all of its
connectors now lose their values.  (Only those values that were set by
this adder are actually lost.)  Then it runs `process-new-value'.  The
reason for this last step is that one or more connectors may still have
a value (that is, a connector may have had a value that was not
originally set by the adder), and these values may need to be
propagated back through the adder.

   A multiplier is very similar to an adder. It will set its `product'
to 0 if either of the factors is 0, even if the other factor is not
known.

     (define (multiplier m1 m2 product)
       (define (process-new-value)
         (cond ((or (and (has-value? m1) (= (get-value m1) 0))
                    (and (has-value? m2) (= (get-value m2) 0)))
                (set-value! product 0 me))
               ((and (has-value? m1) (has-value? m2))
                (set-value! product
                            (* (get-value m1) (get-value m2))
                            me))
               ((and (has-value? product) (has-value? m1))
                (set-value! m2
                            (/ (get-value product) (get-value m1))
                            me))
               ((and (has-value? product) (has-value? m2))
                (set-value! m1
                            (/ (get-value product) (get-value m2))
                            me))))
       (define (process-forget-value)
         (forget-value! product me)
         (forget-value! m1 me)
         (forget-value! m2 me)
         (process-new-value))
       (define (me request)
         (cond ((eq? request 'I-have-a-value)
                (process-new-value))
               ((eq? request 'I-lost-my-value)
                (process-forget-value))
               (else
                (error "Unknown request -- MULTIPLIER" request))))
       (connect m1 me)
       (connect m2 me)
       (connect product me)
       me)

   A `constant' constructor simply sets the value of the designated
connector.  Any `I-have-a-value' or `I-lost-my-value' message sent to
the constant box will produce an error.

     (define (constant value connector)
       (define (me request)
         (error "Unknown request -- CONSTANT" request))
       (connect connector me)
       (set-value! connector value me)
       me)

   Finally, a probe prints a message about the setting or unsetting of
the designated connector:

     (define (probe name connector)
       (define (print-probe value)
         (newline)
         (display "Probe: ")
         (display name)
         (display " = ")
         (display value))
       (define (process-new-value)
         (print-probe (get-value connector)))
       (define (process-forget-value)
         (print-probe "?"))
       (define (me request)
         (cond ((eq? request 'I-have-a-value)
                (process-new-value))
               ((eq? request 'I-lost-my-value)
                (process-forget-value))
               (else
                (error "Unknown request -- PROBE" request))))
       (connect connector me)
       me)

Representing connectors
.......................

A connector is represented as a procedural object with local state
variables `value', the current value of the connector; `informant', the
object that set the connector's value; and `constraints', a list of the
constraints in which the connector participates.

     (define (make-connector)
       (let ((value false) (informant false) (constraints '()))
         (define (set-my-value newval setter)
           (cond ((not (has-value? me))
                  (set! value newval)
                  (set! informant setter)
                  (for-each-except setter
                                   inform-about-value
                                   constraints))
                 ((not (= value newval))
                  (error "Contradiction" (list value newval)))
                 (else 'ignored)))
         (define (forget-my-value retractor)
           (if (eq? retractor informant)
               (begin (set! informant false)
                      (for-each-except retractor
                                       inform-about-no-value
                                       constraints))
               'ignored))
         (define (connect new-constraint)
           (if (not (memq new-constraint constraints))
               (set! constraints
                     (cons new-constraint constraints)))
           (if (has-value? me)
               (inform-about-value new-constraint))
           'done)
         (define (me request)
           (cond ((eq? request 'has-value?)
                  (if informant true false))
                 ((eq? request 'value) value)
                 ((eq? request 'set-value!) set-my-value)
                 ((eq? request 'forget) forget-my-value)
                 ((eq? request 'connect) connect)
                 (else (error "Unknown operation -- CONNECTOR"
                              request))))
         me))

   The connector's local procedure `set-my-value' is called when there
is a request to set the connector's value.  If the connector does not
currently have a value, it will set its value and remember as
`informant' the constraint that requested the value to be set.(2)  Then
the connector will notify all of its participating constraints except
the constraint that requested the value to be set.  This is
accomplished using the following iterator, which applies a designated
procedure to all items in a list except a given one:

     (define (for-each-except exception procedure list)
       (define (loop items)
         (cond ((null? items) 'done)
               ((eq? (car items) exception) (loop (cdr items)))
               (else (procedure (car items))
                     (loop (cdr items)))))
       (loop list))

   If a connector is asked to forget its value, it runs the local
procedure `forget-my-value', which first checks to make sure that the
request is coming from the same object that set the value originally.
If so, the connector informs its associated constraints about the loss
of the value.

   The local procedure `connect' adds the designated new constraint to
the list of constraints if it is not already in that list.  Then, if
the connector has a value, it informs the new constraint of this fact.

   The connector's procedure `me' serves as a dispatch to the other
internal procedures and also represents the connector as an object.
The following procedures provide a syntax interface for the dispatch:

     (define (has-value? connector)
       (connector 'has-value?))

     (define (get-value connector)
       (connector 'value))

     (define (set-value! connector new-value informant)
       ((connector 'set-value!) new-value informant))

     (define (forget-value! connector retractor)
       ((connector 'forget) retractor))

     (define (connect connector new-constraint)
       ((connector 'connect) new-constraint))

     *Exercise 3.33:* Using primitive multiplier, adder, and constant
     constraints, define a procedure `averager' that takes three
     connectors `a', `b', and `c' as inputs and establishes the
     constraint that the value of `c' is the average of the values of
     `a' and `b'.

     *Exercise 3.34:* Louis Reasoner wants to build a squarer, a
     constraint device with two terminals such that the value of
     connector `b' on the second terminal will always be the square of
     the value `a' on the first terminal.  He proposes the following
     simple device made from a multiplier:

          (define (squarer a b)
            (multiplier a a b))

     There is a serious flaw in this idea.  Explain.

     *Exercise 3.35:* Ben Bitdiddle tells Louis that one way to avoid
     the trouble in *Note Exercise 3-34:: is to define a squarer as a
     new primitive constraint.  Fill in the missing portions in Ben's
     outline for a procedure to implement such a constraint:

          (define (squarer a b)
            (define (process-new-value)
              (if (has-value? b)
                  (if (< (get-value b) 0)
                      (error "square less than 0 -- SQUARER" (get-value b))
                      <ALTERNATIVE1>)
                  <ALTERNATIVE2>))
            (define (process-forget-value) <BODY1>)
            (define (me request) <BODY2>)
            <REST OF DEFINITION>
            me)

     *Exercise 3.36:* Suppose we evaluate the following sequence of
     expressions in the global environment:

          (define a (make-connector))
          (define b (make-connector))
          (set-value! a 10 'user)

     At some time during evaluation of the `set-value!', the following
     expression from the connector's local procedure is evaluated:

          (for-each-except setter inform-about-value constraints)

     Draw an environment diagram showing the environment in which the
     above expression is evaluated.

     *Exercise 3.37:* The `celsius-fahrenheit-converter' procedure is
     cumbersome when compared with a more expression-oriented style of
     definition, such as

          (define (celsius-fahrenheit-converter x)
            (c+ (c* (c/ (cv 9) (cv 5))
                    x)
                (cv 32)))

          (define C (make-connector))
          (define F (celsius-fahrenheit-converter C))

     Here `c+', `c*', etc. are the "constraint" versions of the
     arithmetic operations.  For example, `c+' takes two connectors as
     arguments and returns a connector that is related to these by an
     adder constraint:

          (define (c+ x y)
            (let ((z (make-connector)))
              (adder x y z)
              z))

     Define analogous procedures `c-', `c*', `c/', and `cv' (constant
     value) that enable us to define compound constraints as in the
     converter example above.(3)

   ---------- Footnotes ----------

   (1) Constraint propagation first appeared in the incredibly
forward-looking SKETCHPAD system of Ivan Sutherland (1963).  A
beautiful constraint-propagation system based on the Smalltalk language
was developed by Alan Borning (1977) at Xerox Palo Alto Research
Center.  Sussman, Stallman, and Steele applied constraint propagation
to electrical circuit analysis (Sussman and Stallman 1975; Sussman and
Steele 1980). TK!Solver (Konopasek and Jayaraman 1984) is an extensive
modeling environment based on constraints.

   (2) The `setter' might not be a constraint.  In our temperature
example, we used `user' as the `setter'.

   (3) The expression-oriented format is convenient because it avoids
the need to name the intermediate expressions in a computation.  Our
original formulation of the constraint language is cumbersome in the
same way that many languages are cumbersome when dealing with operations
on compound data.  For example, if we wanted to compute the product (a +
b) * (c + d), where the variables represent vectors, we could work in
"imperative style," using procedures that set the values of designated
vector arguments but do not themselves return vectors as values:

     (v-sum a b temp1)
     (v-sum c d temp2)
     (v-prod temp1 temp2 answer)

   Alternatively, we could deal with expressions, using procedures that
return vectors as values, and thus avoid explicitly mentioning `temp1'
and `temp2':

     (define answer (v-prod (v-sum a b) (v-sum c d)))

   Since Lisp allows us to return compound objects as values of
procedures, we can transform our imperative-style constraint language
into an expression-oriented style as shown in this exercise.  In
languages that are impoverished in handling compound objects, such as
Algol, Basic, and Pascal (unless one explicitly uses Pascal pointer
variables), one is usually stuck with the imperative style when
manipulating compound objects.  Given the advantage of the
expression-oriented format, one might ask if there is any reason to have
implemented the system in imperative style, as we did in this section.
One reason is that the non-expression-oriented constraint language
provides a handle on constraint objects (e.g., the value of the `adder'
procedure) as well as on connector objects.  This is useful if we wish
to extend the system with new operations that communicate with
constraints directly rather than only indirectly via operations on
connectors.  Although it is easy to implement the expression-oriented
style in terms of the imperative implementation, it is very difficult
to do the converse.


File: sicp.info,  Node: 3-4,  Next: 3-5,  Prev: 3-3,  Up: Chapter 3

3.4 Concurrency: Time Is of the Essence
=======================================

We've seen the power of computational objects with local state as tools
for modeling.  Yet, as section *Note 3-1-3:: warned, this power
extracts a price: the loss of referential transparency, giving rise to
a thicket of questions about sameness and change, and the need to
abandon the substitution model of evaluation in favor of the more
intricate environment model.

   The central issue lurking beneath the complexity of state, sameness,
and change is that by introducing assignment we are forced to admit "time"
into our computational models.  Before we introduced assignment, all
our programs were timeless, in the sense that any expression that has a
value always has the same value.  In contrast, recall the example of
modeling withdrawals from a bank account and returning the resulting
balance, introduced at the beginning of section *Note 3-1-1:::

     (withdraw 25)
     75

     (withdraw 25)
     50

   Here successive evaluations of the same expression yield different
values.  This behavior arises from the fact that the execution of
assignment statements (in this case, assignments to the variable
`balance') delineates "moments in time" when values change.  The result
of evaluating an expression depends not only on the expression itself,
but also on whether the evaluation occurs before or after these
moments.  Building models in terms of computational objects with local
state forces us to confront time as an essential concept in programming.

   We can go further in structuring computational models to match our
perception of the physical world.  Objects in the world do not change
one at a time in sequence.  Rather we perceive them as acting "concurrently"--all
at once.  So it is often natural to model systems as collections of
computational processes that execute concurrently.  Just as we can make
our programs modular by organizing models in terms of objects with
separate local state, it is often appropriate to divide computational
models into parts that evolve separately and concurrently.  Even if the
programs are to be executed on a sequential computer, the practice of
writing programs as if they were to be executed concurrently forces the
programmer to avoid inessential timing constraints and thus makes
programs more modular.

   In addition to making programs more modular, concurrent computation
can provide a speed advantage over sequential computation.  Sequential
computers execute only one operation at a time, so the amount of time
it takes to perform a task is proportional to the total number of
operations performed.(1)  However, if it is possible to decompose a
problem into pieces that are relatively independent and need to
communicate only rarely, it may be possible to allocate pieces to
separate computing processors, producing a speed advantage proportional
to the number of processors available.

   Unfortunately, the complexities introduced by assignment become even
more problematic in the presence of concurrency.  The fact of
concurrent execution, either because the world operates in parallel or
because our computers do, entails additional complexity in our
understanding of time.

* Menu:

* 3-4-1::            The Nature of Time in Concurrent Systems
* 3-4-2::            Mechanisms for Controlling Concurrency

   ---------- Footnotes ----------

   (1) Most real processors actually execute a few operations at a
time, following a strategy called "pipelining".  Although this
technique greatly improves the effective utilization of the hardware,
it is used only to speed up the execution of a sequential instruction
stream, while retaining the behavior of the sequential program.


File: sicp.info,  Node: 3-4-1,  Next: 3-4-2,  Prev: 3-4,  Up: 3-4

3.4.1 The Nature of Time in Concurrent Systems
----------------------------------------------

On the surface, time seems straightforward.  It is an ordering imposed
on events.(1)  For any events A and B, either A occurs before B, A and
B are simultaneous, or A occurs after B.  For instance, returning to
the bank account example, suppose that Peter withdraws $10 and Paul
withdraws $25 from a joint account that initially contains $100, leaving
$65 in the account.  Depending on the order of the two withdrawals, the
sequence of balances in the account is either $100 -> $90 -> $65 or
$100 -> $75 -> $65.  In a computer implementation of the banking
system, this changing sequence of balances could be modeled by
successive assignments to a variable `balance'.

   In complex situations, however, such a view can be problematic.
Suppose that Peter and Paul, and other people besides, are accessing
the same bank account through a network of banking machines distributed
all over the world.  The actual sequence of balances in the account
will depend critically on the detailed timing of the accesses and the
details of the communication among the machines.

   This indeterminacy in the order of events can pose serious problems
in the design of concurrent systems.  For instance, suppose that the
withdrawals made by Peter and Paul are implemented as two separate
processes sharing a common variable `balance', each process specified
by the procedure given in section *Note 3-1-1:::

     (define (withdraw amount)
       (if (>= balance amount)
           (begin (set! balance (- balance amount))
                  balance)
           "Insufficient funds"))

   If the two processes operate independently, then Peter might test the
balance and attempt to withdraw a legitimate amount.  However, Paul
might withdraw some funds in between the time that Peter checks the
balance and the time Peter completes the withdrawal, thus invalidating
Peter's test.

   Things can be worse still.  Consider the expression

     (set! balance (- balance amount))

executed as part of each withdrawal process.  This consists of three
steps: (1) accessing the value of the `balance' variable; (2) computing
the new balance; (3) setting `balance' to this new value.  If Peter and
Paul's withdrawals execute this statement concurrently, then the two
withdrawals might interleave the order in which they access `balance'
and set it to the new value.

   The timing diagram in *Note Figure 3-29:: depicts an order of events
where `balance' starts at 100, Peter withdraws 10, Paul withdraws 25,
and yet the final value of `balance' is 75.  As shown in the diagram,
the reason for this anomaly is that Paul's assignment of 75 to
`balance' is made under the assumption that the value of `balance' to
be decremented is 100.  That assumption, however, became invalid when
Peter changed `balance' to 90.  This is a catastrophic failure for the
banking system, because the total amount of money in the system is not
conserved.  Before the transactions, the total amount of money was
$100.  Afterwards, Peter has $10, Paul has $25, and the bank has $75.(2)

   The general phenomenon illustrated here is that several processes
may share a common state variable.  What makes this complicated is that
more than one process may be trying to manipulate the shared state at
the same time.  For the bank account example, during each transaction,
each customer should be able to act as if the other customers did not
exist.  When a customer changes the balance in a way that depends on
the balance, he must be able to assume that, just before the moment of
change, the balance is still what he thought it was.

Correct behavior of concurrent programs
.......................................

The above example typifies the subtle bugs that can creep into
concurrent programs.  The root of this complexity lies in the
assignments to variables that are shared among the different processes.
We already know that we must be careful in writing programs that use
`set!', because the results of a computation depend on the order in
which the assignments occur.(3)  With concurrent processes we must be
especially careful about assignments, because we may not be able to
control the order of the assignments made by the different processes.
If several such changes might be made concurrently (as with two
depositors accessing a joint account) we need some way to ensure that
our system behaves correctly.  For example, in the case of withdrawals
from a joint bank account, we must ensure that money is conserved.  To
make concurrent programs behave correctly, we may have to place some
restrictions on concurrent execution.

     *Figure 3.29:* Timing diagram showing how interleaving the order
     of events in two banking withdrawals can lead to an incorrect
     final balance.

           |           Peter              Bank              Paul
           |                              ____
           |                             /    \
           |             .--------------| $100 |-------------.
           |             |               \____/              |
           |             V                                   V
           |  .----------------------.            .----------------------.
           |  | Access balance: $100 |            | Access balance: $100 |
           |  `----------+-----------'            `----------+-----------'
           |             V                                   V
           |  .----------------------.            .----------------------.
           |  | new value: 100-10=90 |            | new value: 100-25=75 |
           |  `----------+-----------'            `----------+-----------'
           |             V                                   |
           |  .----------------------.                       |
           |  | set! balance to $90  |                       |
           |  `----------+-----------'    ____               |
           |             |               /    \              |
           |             `------------->| $ 90 |             V
           |                             \____/   .----------------------.
           |                                      | new value: 100-25=75 |
           |                              ____    `----------+-----------'
           |                             /    \              |
           |                            | $ 90 |<------------'
           V                             \____/
          time

   One possible restriction on concurrency would stipulate that no two
operations that change any shared state variables can occur at the same
time.  This is an extremely stringent requirement.  For distributed
banking, it would require the system designer to ensure that only one
transaction could proceed at a time.  This would be both inefficient
and overly conservative.  *Note Figure 3-30:: shows Peter and Paul
sharing a bank account, where Paul has a private account as well.  The
diagram illustrates two withdrawals from the shared account (one by
Peter and one by Paul) and a deposit to Paul's private account.(4)  The
two withdrawals from the shared account must not be concurrent (since
both access and update the same account), and Paul's deposit and
withdrawal must not be concurrent (since both access and update the
amount in Paul's wallet).  But there should be no problem permitting
Paul's deposit to his private account to proceed concurrently with
Peter's withdrawal from the shared account.

     *Figure 3.30:* Concurrent deposits and withdrawals from a joint
     account in Bank1 and a private account in Bank2.

           |    Peter          Bank1          Paul           Bank2
           |    ____           ____           ____           ____
           |   /    \         /    \         /    \         /    \
           |  |  $7  |--. .--| $100 |       |  $5  |--. .--| $300 |
           |   \____/   V V   \____/         \____/   V V   \____/
           |           +---+                         +---+
           |           | W |                         | D |
           |    ____   ++-++   ____           ____   ++-++   ____
           |   /    \   | |   /    \         /    \   | |   /    \
           |  | $17  |<-' `->| $90  |--. .--|  $0  |<-' `->| $305 |
           |   \____/         \____/   V V   \____/         \____/
           |                          +---+
           |                          | W |
           |    ____           ____   ++-++   ____           ____
           |   /    \         /    \   | |   /    \         /    \
           |  | $17  |       | $65  |<-' `->| $25  |       | $305 |
           |   \____/         \____/         \____/         \____/
           V
          time

   A less stringent restriction on concurrency would ensure that a
concurrent system produces the same result as if the processes had run
sequentially in some order.  There are two important aspects to this
requirement.  First, it does not require the processes to actually run
sequentially, but only to produce results that are the same _as if_
they had run sequentially.  For the example in *Note Figure 3-30::, the
designer of the bank account system can safely allow Paul's deposit and
Peter's withdrawal to happen concurrently, because the net result will
be the same as if the two operations had happened sequentially.
Second, there may be more than one possible "correct" result produced
by a concurrent program, because we require only that the result be the
same as for _some_ sequential order.  For example, suppose that Peter
and Paul's joint account starts out with $100, and Peter deposits $40
while Paul concurrently withdraws half the money in the account.  Then
sequential execution could result in the account balance being either
$70 or $90 (see *Note Exercise 3-38::).(5)

   There are still weaker requirements for correct execution of
concurrent programs.  A program for simulating diffusion (say, the flow
of heat in an object) might consist of a large number of processes,
each one representing a small volume of space, that update their values
concurrently.  Each process repeatedly changes its value to the average
of its own value and its neighbors' values.  This algorithm converges
to the right answer independent of the order in which the operations
are done; there is no need for any restrictions on concurrent use of
the shared values.

     *Exercise 3.38:* Suppose that Peter, Paul, and Mary share a joint
     bank account that initially contains $100.  Concurrently, Peter
     deposits $10, Paul withdraws $20, and Mary withdraws half the
     money in the account, by executing the following commands:

          Peter: (set! balance (+ balance 10))
          Paul:  (set! balance (- balance 20))
          Mary:  (set! balance (- balance (/ balance 2)))

       a. List all the different possible values for `balance' after
          these three transactions have been completed, assuming that
          the banking system forces the three processes to run
          sequentially in some order.

       b. What are some other values that could be produced if the
          system allows the processes to be interleaved?  Draw timing
          diagrams like the one in *Note Figure 3-29:: to explain how
          these values can occur.


   ---------- Footnotes ----------

   (1) To quote some graffiti seen on a Cambridge building wall: "Time
is a device that was invented to keep everything from happening at
once."

   (2) An even worse failure for this system could occur if the two
`set!' operations attempt to change the balance simultaneously, in
which case the actual data appearing in memory might end up being a
random combination of the information being written by the two
processes.  Most computers have interlocks on the primitive
memory-write operations, which protect against such simultaneous
access.  Even this seemingly simple kind of protection, however, raises
implementation challenges in the design of multiprocessing computers,
where elaborate "cache-coherence" protocols are required to ensure that
the various processors will maintain a consistent view of memory
contents, despite the fact that data may be replicated ("cached") among
the different processors to increase the speed of memory access.

   (3) The factorial program in section *Note 3-1-3:: illustrates this
for a single sequential process.

   (4) The columns show the contents of Peter's wallet, the joint
account (in Bank1), Paul's wallet, and Paul's private account (in
Bank2), before and after each withdrawal (W) and deposit (D).  Peter
withdraws $10 from Bank1; Paul deposits $5 in Bank2, then withdraws $25
from Bank1.

   (5) [Footnote 39] A more formal way to express this idea is to say
that concurrent programs are inherently "nondeterministic". That is,
they are described not by single-valued functions, but by functions
whose results are sets of possible values.  In section *Note 4-3:: we
will study a language for expressing nondeterministic computations.


File: sicp.info,  Node: 3-4-2,  Prev: 3-4-1,  Up: 3-4

3.4.2 Mechanisms for Controlling Concurrency
--------------------------------------------

We've seen that the difficulty in dealing with concurrent processes is
rooted in the need to consider the interleaving of the order of events
in the different processes.  For example, suppose we have two
processes, one with three ordered events (a,b,c) and one with three
ordered events (x,y,z).  If the two processes run concurrently, with no
constraints on how their execution is interleaved, then there are 20
different possible orderings for the events that are consistent with
the individual orderings for the two processes:

     (a,b,c,x,y,z)  (a,x,b,y,c,z)  (x,a,b,c,y,z)  (x,a,y,z,b,c)
     (a,b,x,c,y,z)  (a,x,b,y,z,c)  (x,a,b,y,c,z)  (x,y,a,b,c,z)
     (a,b,x,y,c,z)  (a,x,y,b,c,z)  (x,a,b,y,z,c)  (x,y,a,b,z,c)
     (a,b,x,y,z,c)  (a,x,y,b,z,c)  (x,a,y,b,c,z)  (x,y,a,z,b,c)
     (a,x,b,c,y,z)  (a,x,y,z,b,c)  (x,a,y,b,z,c)  (x,y,z,a,b,c)

   As programmers designing this system, we would have to consider the
effects of each of these 20 orderings and check that each behavior is
acceptable.  Such an approach rapidly becomes unwieldy as the numbers
of processes and events increase.

   A more practical approach to the design of concurrent systems is to
devise general mechanisms that allow us to constrain the interleaving
of concurrent processes so that we can be sure that the program
behavior is correct.  Many mechanisms have been developed for this
purpose.  In this section, we describe one of them, the "serializer".

Serializing access to shared state
..................................

Serialization implements the following idea: Processes will execute
concurrently, but there will be certain collections of procedures that
cannot be executed concurrently.  More precisely, serialization creates
distinguished sets of procedures such that only one execution of a
procedure in each serialized set is permitted to happen at a time.  If
some procedure in the set is being executed, then a process that
attempts to execute any procedure in the set will be forced to wait
until the first execution has finished.

   We can use serialization to control access to shared variables.  For
example, if we want to update a shared variable based on the previous
value of that variable, we put the access to the previous value of the
variable and the assignment of the new value to the variable in the
same procedure.  We then ensure that no other procedure that assigns to
the variable can run concurrently with this procedure by serializing
all of these procedures with the same serializer.  This guarantees that
the value of the variable cannot be changed between an access and the
corresponding assignment.

Serializers in Scheme
.....................

To make the above mechanism more concrete, suppose that we have
extended Scheme to include a procedure called `parallel-execute':

     (parallel-execute <P_1> <P_2> ... <P_K>)

   Each <P> must be a procedure of no arguments.  `Parallel-execute'
creates a separate process for each <P>, which applies <P> (to no
arguments).  These processes all run concurrently.(1)

   As an example of how this is used, consider

     (define x 10)

     (parallel-execute (lambda () (set! x (* x x)))
                       (lambda () (set! x (+ x 1))))

   This creates two concurrent processes--P_1, which sets `x' to `x'
times `x', and P_2, which increments `x'.  After execution is complete,
`x' will be left with one of five possible values, depending on the
interleaving of the events of P_1 and P_2:

     101: P_1 sets `x' to 100 and then P_2 increments
          `x' to 101.
     121: P_2 increments `x' to 11 and then P_1 sets
          `x' to `x' times `x'.
     110: P_2 changes `x' from 10 to 11 between the two
          times that P_1 accesses the value of `x' during
          the evaluation of `(* x x)'.
     11:  P_2 accesses `x', then P_1 sets `x' to
          100, then P_2 sets `x'.
     100: P_1 accesses `x' (twice), then P_2 sets
          `x' to 11, then P_1 sets `x'.

   We can constrain the concurrency by using serialized procedures,
which are created by "serializers". Serializers are constructed by
`make-serializer', whose implementation is given below.  A serializer
takes a procedure as argument and returns a serialized procedure that
behaves like the original procedure.  All calls to a given serializer
return serialized procedures in the same set.

   Thus, in contrast to the example above, executing

     (define x 10)

     (define s (make-serializer))

     (parallel-execute (s (lambda () (set! x (* x x))))
                       (s (lambda () (set! x (+ x 1)))))

can produce only two possible values for `x', 101 or 121.  The other
possibilities are eliminated, because the execution of P_1 and P_2
cannot be interleaved.

   Here is a version of the `make-account' procedure from section *Note
3-1-1::, where the deposits and withdrawals have been serialized:

     (define (make-account balance)
       (define (withdraw amount)
         (if (>= balance amount)
             (begin (set! balance (- balance amount))
                    balance)
             "Insufficient funds"))
       (define (deposit amount)
         (set! balance (+ balance amount))
         balance)
       (let ((protected (make-serializer)))
         (define (dispatch m)
           (cond ((eq? m 'withdraw) (protected withdraw))
                 ((eq? m 'deposit) (protected deposit))
                 ((eq? m 'balance) balance)
                 (else (error "Unknown request -- MAKE-ACCOUNT"
                              m))))
         dispatch))

   With this implementation, two processes cannot be withdrawing from or
depositing into a single account concurrently.  This eliminates the
source of the error illustrated in *Note Figure 3-29::, where Peter
changes the account balance between the times when Paul accesses the
balance to compute the new value and when Paul actually performs the
assignment.  On the other hand, each account has its own serializer, so
that deposits and withdrawals for different accounts can proceed
concurrently.

     *Exercise 3.39:* Which of the five possibilities in the parallel
     execution shown above remain if we instead serialize execution as
     follows:

          (define x 10)

          (define s (make-serializer))

          (parallel-execute (lambda () (set! x ((s (lambda () (* x x))))))
                            (s (lambda () (set! x (+ x 1)))))

     *Exercise 3.40:* Give all possible values of `x' that can result
     from executing

          (define x 10)

          (parallel-execute (lambda () (set! x (* x x)))
                            (lambda () (set! x (* x x x))))

     Which of these possibilities remain if we instead use serialized
     procedures:

          (define x 10)

          (define s (make-serializer))

          (parallel-execute (s (lambda () (set! x (* x x))))
                            (s (lambda () (set! x (* x x x)))))

     *Exercise 3.41:* Ben Bitdiddle worries that it would be better to
     implement the bank account as follows (where the commented line
     has been changed):

          (define (make-account balance)
            (define (withdraw amount)
              (if (>= balance amount)
                  (begin (set! balance (- balance amount))
                         balance)
                  "Insufficient funds"))
            (define (deposit amount)
              (set! balance (+ balance amount))
              balance)
            ;; continued on next page

            (let ((protected (make-serializer)))
              (define (dispatch m)
                (cond ((eq? m 'withdraw) (protected withdraw))
                      ((eq? m 'deposit) (protected deposit))
                      ((eq? m 'balance)
                       ((protected (lambda () balance)))) ; serialized
                      (else (error "Unknown request -- MAKE-ACCOUNT"
                                   m))))
              dispatch))

     because allowing unserialized access to the bank balance can
     result in anomalous behavior.  Do you agree?  Is there any
     scenario that demonstrates Ben's concern?

     *Exercise 3.42:* Ben Bitdiddle suggests that it's a waste of time
     to create a new serialized procedure in response to every
     `withdraw' and `deposit' message.  He says that `make-account'
     could be changed so that the calls to `protected' are done outside
     the `dispatch' procedure.  That is, an account would return the
     same serialized procedure (which was created at the same time as
     the account) each time it is asked for a withdrawal procedure.

          (define (make-account balance)
            (define (withdraw amount)
              (if (>= balance amount)
                  (begin (set! balance (- balance amount))
                         balance)
                  "Insufficient funds"))
            (define (deposit amount)
              (set! balance (+ balance amount))
              balance)
            (let ((protected (make-serializer)))
              (let ((protected-withdraw (protected withdraw))
                    (protected-deposit (protected deposit)))
                (define (dispatch m)
                  (cond ((eq? m 'withdraw) protected-withdraw)
                        ((eq? m 'deposit) protected-deposit)
                        ((eq? m 'balance) balance)
                        (else (error "Unknown request -- MAKE-ACCOUNT"
                                     m))))
                dispatch)))

     Is this a safe change to make?  In particular, is there any
     difference in what concurrency is allowed by these two versions of
     `make-account' ?

Complexity of using multiple shared resources
.............................................

Serializers provide a powerful abstraction that helps isolate the
complexities of concurrent programs so that they can be dealt with
carefully and (hopefully) correctly.  However, while using serializers
is relatively straightforward when there is only a single shared
resource (such as a single bank account), concurrent programming can be
treacherously difficult when there are multiple shared resources.

   To illustrate one of the difficulties that can arise, suppose we
wish to swap the balances in two bank accounts.  We access each account
to find the balance, compute the difference between the balances,
withdraw this difference from one account, and deposit it in the other
account.  We could implement this as follows:(2)

     (define (exchange account1 account2)
       (let ((difference (- (account1 'balance)
                            (account2 'balance))))
         ((account1 'withdraw) difference)
         ((account2 'deposit) difference)))

   This procedure works well when only a single process is trying to do
the exchange.  Suppose, however, that Peter and Paul both have access
to accounts a1, a2, and a3, and that Peter exchanges a1 and a2 while
Paul concurrently exchanges a1 and a3.  Even with account deposits and
withdrawals serialized for individual accounts (as in the `make-account'
procedure shown above in this section), `exchange' can still produce
incorrect results.  For example, Peter might compute the difference in
the balances for a1 and a2, but then Paul might change the balance in
a1 before Peter is able to complete the exchange.(3)  For correct
behavior, we must arrange for the `exchange' procedure to lock out any
other concurrent accesses to the accounts during the entire time of the
exchange.

   One way we can accomplish this is by using both accounts'
serializers to serialize the entire `exchange' procedure.  To do this,
we will arrange for access to an account's serializer.  Note that we
are deliberately breaking the modularity of the bank-account object by
exposing the serializer.  The following version of `make-account' is
identical to the original version given in section *Note 3-1-1::,
except that a serializer is provided to protect the balance variable,
and the serializer is exported via message passing:

     (define (make-account-and-serializer balance)
       (define (withdraw amount)
         (if (>= balance amount)
             (begin (set! balance (- balance amount))
                    balance)
             "Insufficient funds"))
       (define (deposit amount)
         (set! balance (+ balance amount))
         balance)
       (let ((balance-serializer (make-serializer)))
         (define (dispatch m)
           (cond ((eq? m 'withdraw) withdraw)
                 ((eq? m 'deposit) deposit)
                 ((eq? m 'balance) balance)
                 ((eq? m 'serializer) balance-serializer)
                 (else (error "Unknown request -- MAKE-ACCOUNT"
                              m))))
         dispatch))

   We can use this to do serialized deposits and withdrawals.  However,
unlike our earlier serialized account, it is now the responsibility of
each user of bank-account objects to explicitly manage the
serialization, for example as follows:(4)

     (define (deposit account amount)
       (let ((s (account 'serializer))
             (d (account 'deposit)))
         ((s d) amount)))

   Exporting the serializer in this way gives us enough flexibility to
implement a serialized exchange program.  We simply serialize the
original `exchange' procedure with the serializers for both accounts:

     (define (serialized-exchange account1 account2)
       (let ((serializer1 (account1 'serializer))
             (serializer2 (account2 'serializer)))
         ((serializer1 (serializer2 exchange))
          account1
          account2)))

     *Exercise 3.43:* Suppose that the balances in three accounts start
     out as $10, $20, and $30, and that multiple processes run,
     exchanging the balances in the accounts.  Argue that if the
     processes are run sequentially, after any number of concurrent
     exchanges, the account balances should be $10, $20, and $30 in
     some order.  Draw a timing diagram like the one in *Note Figure
     3-29:: to show how this condition can be violated if the exchanges
     are implemented using the first version of the account-exchange
     program in this section.  On the other hand, argue that even with
     this `exchange' program, the sum of the balances in the accounts
     will be preserved.  Draw a timing diagram to show how even this
     condition would be violated if we did not serialize the
     transactions on individual accounts.

     *Exercise 3.44:* Consider the problem of transferring an amount
     from one account to another.  Ben Bitdiddle claims that this can
     be accomplished with the following procedure, even if there are
     multiple people concurrently transferring money among multiple
     accounts, using any account mechanism that serializes deposit and
     withdrawal transactions, for example, the version of
     `make-account' in the text above.

          (define (transfer from-account to-account amount)
            ((from-account 'withdraw) amount)
            ((to-account 'deposit) amount))

     Louis Reasoner claims that there is a problem here, and that we
     need to use a more sophisticated method, such as the one required
     for dealing with the exchange problem.  Is Louis right?  If not,
     what is the essential difference between the transfer problem and
     the exchange problem?  (You should assume that the balance in
     `from-account' is at least `amount'.)

     *Exercise 3.45:* Louis Reasoner thinks our bank-account system is
     unnecessarily complex and error-prone now that deposits and
     withdrawals aren't automatically serialized.  He suggests that
     `make-account-and-serializer' should have exported the serializer
     (for use by such procedures as `serialized-exchange') in addition
     to (rather than instead of) using it to serialize accounts and
     deposits as `make-account' did.  He proposes to redefine accounts
     as follows:

          (define (make-account-and-serializer balance)
            (define (withdraw amount)
              (if (>= balance amount)
                  (begin (set! balance (- balance amount))
                         balance)
                  "Insufficient funds"))
            (define (deposit amount)
              (set! balance (+ balance amount))
              balance)
            (let ((balance-serializer (make-serializer)))
              (define (dispatch m)
                (cond ((eq? m 'withdraw) (balance-serializer withdraw))
                      ((eq? m 'deposit) (balance-serializer deposit))
                      ((eq? m 'balance) balance)
                      ((eq? m 'serializer) balance-serializer)
                      (else (error "Unknown request -- MAKE-ACCOUNT"
                                   m))))
              dispatch))

     Then deposits are handled as with the original `make-account':

          (define (deposit account amount)
           ((account 'deposit) amount))

     Explain what is wrong with Louis's reasoning.  In particular,
     consider what happens when `serialized-exchange' is called.

Implementing serializers
........................

We implement serializers in terms of a more primitive synchronization
mechanism called a "mutex".  A mutex is an object that supports two
operations--the mutex can be "acquired", and the mutex can be "released".
Once a mutex has been acquired, no other acquire operations on that
mutex may proceed until the mutex is released.(5) In our
implementation, each serializer has an associated mutex.  Given a
procedure `p', the serializer returns a procedure that acquires the
mutex, runs `p', and then releases the mutex.  This ensures that only
one of the procedures produced by the serializer can be running at
once, which is precisely the serialization property that we need to
guarantee.

     (define (make-serializer)
       (let ((mutex (make-mutex)))
         (lambda (p)
           (define (serialized-p . args)
             (mutex 'acquire)
             (let ((val (apply p args)))
               (mutex 'release)
               val))
           serialized-p)))

   The mutex is a mutable object (here we'll use a one-element list,
which we'll refer to as a "cell") that can hold the value true or
false.  When the value is false, the mutex is available to be acquired.
When the value is true, the mutex is unavailable, and any process that
attempts to acquire the mutex must wait.

   Our mutex constructor `make-mutex' begins by initializing the cell
contents to false.  To acquire the mutex, we test the cell.  If the
mutex is available, we set the cell contents to true and proceed.
Otherwise, we wait in a loop, attempting to acquire over and over
again, until we find that the mutex is available.(6)  To release the
mutex, we set the cell contents to false.

     (define (make-mutex)
       (let ((cell (list false)))
         (define (the-mutex m)
           (cond ((eq? m 'acquire)
                  (if (test-and-set! cell)
                      (the-mutex 'acquire))) ; retry
                 ((eq? m 'release) (clear! cell))))
         the-mutex))

     (define (clear! cell)
       (set-car! cell false))

   `Test-and-set!' tests the cell and returns the result of the test.
In addition, if the test was false, `test-and-set!' sets the cell
contents to true before returning false.  We can express this behavior
as the following procedure:

     (define (test-and-set! cell)
       (if (car cell)
           true
           (begin (set-car! cell true)
                  false)))

   However, this implementation of `test-and-set!' does not suffice as
it stands.  There is a crucial subtlety here, which is the essential
place where concurrency control enters the system: The `test-and-set!'
operation must be performed "atomically".  That is, we must guarantee
that, once a process has tested the cell and found it to be false, the
cell contents will actually be set to true before any other process can
test the cell.  If we do not make this guarantee, then the mutex can
fail in a way similar to the bank-account failure in *Note Figure
3-29::.  (See *Note Exercise 3-46::.)

   The actual implementation of `test-and-set!' depends on the details
of how our system runs concurrent processes.  For example, we might be
executing concurrent processes on a sequential processor using a
time-slicing mechanism that cycles through the processes, permitting
each process to run for a short time before interrupting it and moving
on to the next process.  In that case, `test-and-set!'  can work by
disabling time slicing during the testing and setting.(7)
Alternatively, multiprocessing computers provide instructions that
support atomic operations directly in hardware.(8)

     *Exercise 3.46:* Suppose that we implement `test-and-set!'  using
     an ordinary procedure as shown in the text, without attempting to
     make the operation atomic.  Draw a timing diagram like the one in
     *Note Figure 3-29:: to demonstrate how the mutex implementation
     can fail by allowing two processes to acquire the mutex at the
     same time.

     *Exercise 3.47:* A semaphore (of size n) is a generalization of a
     mutex.  Like a mutex, a semaphore supports acquire and release
     operations, but it is more general in that up to n processes can
     acquire it concurrently.  Additional processes that attempt to
     acquire the semaphore must wait for release operations.  Give
     implementations of semaphores

       a. in terms of mutexes

       b. in terms of atomic `test-and-set!' operations.


Deadlock
........

Now that we have seen how to implement serializers, we can see that
account exchanging still has a problem, even with the
`serialized-exchange' procedure above.  Imagine that Peter attempts to
exchange a1 with a2 while Paul concurrently attempts to exchange a2
with a1.  Suppose that Peter's process reaches the point where it has
entered a serialized procedure protecting a1 and, just after that,
Paul's process enters a serialized procedure protecting a2.  Now Peter
cannot proceed (to enter a serialized procedure protecting a2) until
Paul exits the serialized procedure protecting a2.  Similarly, Paul
cannot proceed until Peter exits the serialized procedure protecting
a1.  Each process is stalled forever, waiting for the other.  This
situation is called a "deadlock".  Deadlock is always a danger in
systems that provide concurrent access to multiple shared resources.

   One way to avoid the deadlock in this situation is to give each
account a unique identification number and rewrite
`serialized-exchange' so that a process will always attempt to enter a
procedure protecting the lowest-numbered account first.  Although this
method works well for the exchange problem, there are other situations
that require more sophisticated deadlock-avoidance techniques, or where
deadlock cannot be avoided at all.  (See *Note Exercise 3-48:: and
*Note Exercise 3-49::.)(9)

     *Exercise 3.48:* Explain in detail why the deadlock-avoidance
     method described above, (i.e., the accounts are numbered, and each
     process attempts to acquire the smaller-numbered account first)
     avoids deadlock in the exchange problem.  Rewrite
     `serialized-exchange' to incorporate this idea.  (You will also
     need to modify `make-account' so that each account is created with
     a number, which can be accessed by sending an appropriate message.)

     *Exercise 3.49:* Give a scenario where the deadlock-avoidance
     mechanism described above does not work.  (Hint: In the exchange
     problem, each process knows in advance which accounts it will need
     to get access to.  Consider a situation where a process must get
     access to some shared resources before it can know which
     additional shared resources it will require.)

Concurrency, time, and communication
....................................

We've seen how programming concurrent systems requires controlling the
ordering of events when different processes access shared state, and
we've seen how to achieve this control through judicious use of
serializers.  But the problems of concurrency lie deeper than this,
because, from a fundamental point of view, it's not always clear what
is meant by "shared state."

   Mechanisms such as `test-and-set!' require processes to examine a
global shared flag at arbitrary times.  This is problematic and
inefficient to implement in modern high-speed processors, where due to
optimization techniques such as pipelining and cached memory, the
contents of memory may not be in a consistent state at every instant.
In contemporary multiprocessing systems, therefore, the serializer
paradigm is being supplanted by new approaches to concurrency
control.(10)

   The problematic aspects of shared state also arise in large,
distributed systems.  For instance, imagine a distributed banking
system where individual branch banks maintain local values for bank
balances and periodically compare these with values maintained by other
branches.  In such a system the value of "the account balance" would be
undetermined, except right after synchronization.  If Peter deposits
money in an account he holds jointly with Paul, when should we say that
the account balance has changed--when the balance in the local branch
changes, or not until after the synchronization?  And if Paul accesses
the account from a different branch, what are the reasonable
constraints to place on the banking system such that the behavior is
"correct"?  The only thing that might matter for correctness is the
behavior observed by Peter and Paul individually and the "state" of the
account immediately after synchronization.  Questions about the "real"
account balance or the order of events between synchronizations may be
irrelevant or meaningless.(11)

   The basic phenomenon here is that synchronizing different processes,
establishing shared state, or imposing an order on events requires
communication among the processes.  In essence, any notion of time in
concurrency control must be intimately tied to communication.(12)  It
is intriguing that a similar connection between time and communication
also arises in the Theory of Relativity, where the speed of light (the
fastest signal that can be used to synchronize events) is a fundamental
constant relating time and space.  The complexities we encounter in
dealing with time and state in our computational models may in fact
mirror a fundamental complexity of the physical universe.

   ---------- Footnotes ----------

   (1) `Parallel-execute' is not part of standard Scheme, but it can be
implemented in MIT Scheme.  In our implementation, the new concurrent
processes also run concurrently with the original Scheme process.
Also, in our implementation, the value returned by `parallel-execute'
is a special control object that can be used to halt the newly created
processes.

   (2) We have simplified `exchange' by exploiting the fact that our
`deposit' message accepts negative amounts.  (This is a serious bug in
our banking system!)

   (3) If the account balances start out as $10, $20, and $30, then
after any number of concurrent exchanges, the balances should still be
$10, $20, and $30 in some order.  Serializing the deposits to
individual accounts is not sufficient to guarantee this.  See *Note
Exercise 3-43::.

   (4) *Note Exercise 3-45:: investigates why deposits and withdrawals
are no longer automatically serialized by the account.

   (5) The term "mutex" is an abbreviation for "mutual exclusion".  The
general problem of arranging a mechanism that permits concurrent
processes to safely share resources is called the mutual exclusion
problem.  Our mutex is a simple variant of the "semaphore" mechanism
(see *Note Exercise 3-47::), which was introduced in the "THE"
Multiprogramming System developed at the Technological University of
Eindhoven and named for the university's initials in Dutch (Dijkstra
1968a).  The acquire and release operations were originally called P
and V, from the Dutch words _passeren_ (to pass) and _vrijgeven_ (to
release), in reference to the semaphores used on railroad systems.
Dijkstra's classic exposition (1968b) was one of the first to clearly
present the issues of concurrency control, and showed how to use
semaphores to handle a variety of concurrency problems.

   (6) In most time-shared operating systems, processes that are
blocked by a mutex do not waste time "busy-waiting" as above.  Instead,
the system schedules another process to run while the first is waiting,
and the blocked process is awakened when the mutex becomes available.

   (7) In MIT Scheme for a single processor, which uses a time-slicing
model, `test-and-set!' can be implemented as follows:

     (define (test-and-set! cell)
       (without-interrupts
        (lambda ()
          (if (car cell)
              true
              (begin (set-car! cell true)
                     false)))))

   `Without-interrupts' disables time-slicing interrupts while its
procedure argument is being executed.

   (8) There are many variants of such instructions--including
test-and-set, test-and-clear, swap, compare-and-exchange, load-reserve,
and store-conditional--whose design must be carefully matched to the
machine's processor-memory interface.  One issue that arises here is to
determine what happens if two processes attempt to acquire the same
resource at exactly the same time by using such an instruction.  This
requires some mechanism for making a decision about which process gets
control.  Such a mechanism is called an "arbiter".  Arbiters usually
boil down to some sort of hardware device.  Unfortunately, it is
possible to prove that one cannot physically construct a fair arbiter
that works 100% of the time unless one allows the arbiter an
arbitrarily long time to make its decision.  The fundamental phenomenon
here was originally observed by the fourteenth-century French
philosopher Jean Buridan in his commentary on Aristotle's De caelo.
Buridan argued that a perfectly rational dog placed between two equally
attractive sources of food will starve to death, because it is
incapable of deciding which to go to first.

   (9) The general technique for avoiding deadlock by numbering the
shared resources and acquiring them in order is due to Havender (1968).
Situations where deadlock cannot be avoided require "deadlock-recovery"
methods, which entail having processes "back out" of the deadlocked
state and try again.  Deadlock-recovery mechanisms are widely used in
database management systems, a topic that is treated in detail in Gray
and Reuter 1993.

   (10) One such alternative to serialization is called "barrier
synchronization".  The programmer permits concurrent processes to
execute as they please, but establishes certain synchronization points
("barriers") through which no process can proceed until all the
processes have reached the barrier.  Modern processors provide machine
instructions that permit programmers to establish synchronization
points at places where consistency is required.  The PowerPC^( TM), for
example, includes for this purpose two instructions called SYNC and
EIEIO (Enforced In-order Execution of Input/Output).

   (11) This may seem like a strange point of view, but there are
systems that work this way.  International charges to credit-card
accounts, for example, are normally cleared on a per-country basis, and
the charges made in different countries are periodically reconciled.
Thus the account balance may be different in different countries.

   (12) For distributed systems, this perspective was pursued by
Lamport (1978), who showed how to use communication to establish
"global clocks" that can be used to establish orderings on events in
distributed systems.


File: sicp.info,  Node: 3-5,  Prev: 3-4,  Up: Chapter 3

3.5 Streams
===========

We've gained a good understanding of assignment as a tool in modeling,
as well as an appreciation of the complex problems that assignment
raises. It is time to ask whether we could have gone about things in a
different way, so as to avoid some of these problems.  In this section,
we explore an alternative approach to modeling state, based on data
structures called "streams".  As we shall see, streams can mitigate
some of the complexity of modeling state.

   Let's step back and review where this complexity comes from.  In an
attempt to model real-world phenomena, we made some apparently
reasonable decisions: We modeled real-world objects with local state by
computational objects with local variables.  We identified time
variation in the real world with time variation in the computer.  We
implemented the time variation of the states of the model objects in
the computer with assignments to the local variables of the model
objects.

   Is there another approach?  Can we avoid identifying time in the
computer with time in the modeled world?  Must we make the model change
with time in order to model phenomena in a changing world?  Think about
the issue in terms of mathematical functions.  We can describe the
time-varying behavior of a quantity x as a function of time x(t).  If
we concentrate on x instant by instant, we think of it as a changing
quantity.  Yet if we concentrate on the entire time history of values,
we do not emphasize change--the function itself does not change.(1)

   If time is measured in discrete steps, then we can model a time
function as a (possibly infinite) sequence.  In this section, we will
see how to model change in terms of sequences that represent the time
histories of the systems being modeled.  To accomplish this, we
introduce new data structures called "streams".  From an abstract point
of view, a stream is simply a sequence.  However, we will find that the
straightforward implementation of streams as lists (as in section *Note
2-2-1::) doesn't fully reveal the power of stream processing.  As an
alternative, we introduce the technique of "delayed evaluation", which
enables us to represent very large (even infinite) sequences as streams.

   Stream processing lets us model systems that have state without ever
using assignment or mutable data.  This has important implications,
both theoretical and practical, because we can build models that avoid
the drawbacks inherent in introducing assignment.  On the other hand,
the stream framework raises difficulties of its own, and the question
of which modeling technique leads to more modular and more easily
maintained systems remains open.

* Menu:

* 3-5-1::            Streams Are Delayed Lists
* 3-5-2::            Infinite Streams
* 3-5-3::            Exploiting the Stream Paradigm
* 3-5-4::            Streams and Delayed Evaluation
* 3-5-5::            Modularity of Functional Programs and Modularity of
                     Objects

   ---------- Footnotes ----------

   (1) Physicists sometimes adopt this view by introducing the "world
lines" of particles as a device for reasoning about motion.  We've also
already mentioned (section *Note 2-2-3::) that this is the natural way
to think about signal-processing systems.  We will explore applications
of streams to signal processing in section *Note 3-5-3::.


File: sicp.info,  Node: 3-5-1,  Next: 3-5-2,  Prev: 3-5,  Up: 3-5

3.5.1 Streams Are Delayed Lists
-------------------------------

As we saw in section *Note 2-2-3::, sequences can serve as standard
interfaces for combining program modules.  We formulated powerful
abstractions for manipulating sequences, such as `map', `filter', and
`accumulate', that capture a wide variety of operations in a manner that
is both succinct and elegant.

   Unfortunately, if we represent sequences as lists, this elegance is
bought at the price of severe inefficiency with respect to both the
time and space required by our computations.  When we represent
manipulations on sequences as transformations of lists, our programs
must construct and copy data structures (which may be huge) at every
step of a process.

   To see why this is true, let us compare two programs for computing
the sum of all the prime numbers in an interval.  The first program is
written in standard iterative style:(1)

     (define (sum-primes a b)
       (define (iter count accum)
         (cond ((> count b) accum)
               ((prime? count) (iter (+ count 1) (+ count accum)))
               (else (iter (+ count 1) accum))))
       (iter a 0))

   The second program performs the same computation using the sequence
operations of section *Note 2-2-3:::

     (define (sum-primes a b)
       (accumulate +
                   0
                   (filter prime? (enumerate-interval a b))))

   In carrying out the computation, the first program needs to store
only the sum being accumulated.  In contrast, the filter in the second
program cannot do any testing until `enumerate-interval' has
constructed a complete list of the numbers in the interval.  The filter
generates another list, which in turn is passed to `accumulate' before
being collapsed to form a sum.  Such large intermediate storage is not
needed by the first program, which we can think of as enumerating the
interval incrementally, adding each prime to the sum as it is generated.

   The inefficiency in using lists becomes painfully apparent if we use
the sequence paradigm to compute the second prime in the interval from
10,000 to 1,000,000 by evaluating the expression

     (car (cdr (filter prime?
                       (enumerate-interval 10000 1000000))))

   This expression does find the second prime, but the computational
overhead is outrageous.  We construct a list of almost a million
integers, filter this list by testing each element for primality, and
then ignore almost all of the result.  In a more traditional
programming style, we would interleave the enumeration and the
filtering, and stop when we reached the second prime.

   Streams are a clever idea that allows one to use sequence
manipulations without incurring the costs of manipulating sequences as
lists.  With streams we can achieve the best of both worlds: We can
formulate programs elegantly as sequence manipulations, while attaining
the efficiency of incremental computation.  The basic idea is to
arrange to construct a stream only partially, and to pass the partial
construction to the program that consumes the stream.  If the consumer
attempts to access a part of the stream that has not yet been
constructed, the stream will automatically construct just enough more
of itself to produce the required part, thus preserving the illusion
that the entire stream exists.  In other words, although we will write
programs as if we were processing complete sequences, we design our
stream implementation to automatically and transparently interleave the
construction of the stream with its use.

   On the surface, streams are just lists with different names for the
procedures that manipulate them.  There is a constructor,
`cons-stream', and two selectors, `stream-car' and `stream-cdr', which
satisfy the constraints

     (stream-car (cons-stream x y)) = x
     (stream-cdr (cons-stream x y)) = y

   There is a distinguishable object, `the-empty-stream', which cannot
be the result of any `cons-stream' operation, and which can be
identified with the predicate `stream-null?'.(2)  Thus we can make and
use streams, in just the same way as we can make and use lists, to
represent aggregate data arranged in a sequence.  In particular, we can
build stream analogs of the list operations from *Note Chapter 2::,
such as `list-ref', `map', and `for-each':(3)

     (define (stream-ref s n)
       (if (= n 0)
           (stream-car s)
           (stream-ref (stream-cdr s) (- n 1))))

     (define (stream-map proc s)
       (if (stream-null? s)
           the-empty-stream
           (cons-stream (proc (stream-car s))
                        (stream-map proc (stream-cdr s)))))

     (define (stream-for-each proc s)
       (if (stream-null? s)
           'done
           (begin (proc (stream-car s))
                  (stream-for-each proc (stream-cdr s)))))

   `Stream-for-each' is useful for viewing streams:

     (define (display-stream s)
       (stream-for-each display-line s))

     (define (display-line x)
       (newline)
       (display x))

   To make the stream implementation automatically and transparently
interleave the construction of a stream with its use, we will arrange
for the `cdr' of a stream to be evaluated when it is accessed by the
`stream-cdr' procedure rather than when the stream is constructed by
`cons-stream'.  This implementation choice is reminiscent of our
discussion of rational numbers in section *Note 2-1-2::, where we saw
that we can choose to implement rational numbers so that the reduction
of numerator and denominator to lowest terms is performed either at
construction time or at selection time.  The two rational-number
implementations produce the same data abstraction, but the choice has
an effect on efficiency.  There is a similar relationship between
streams and ordinary lists.  As a data abstraction, streams are the
same as lists.  The difference is the time at which the elements are
evaluated.  With ordinary lists, both the `car' and the `cdr' are
evaluated at construction time.  With streams, the `cdr' is evaluated
at selection time.

   Our implementation of streams will be based on a special form called
`delay'.  Evaluating `(delay <EXP>)' does not evaluate the expression
<EXP>, but rather returns a so-called object "delayed object", which we
can think of as a "promise" to evaluate <EXP> at some future time.  As
a companion to `delay', there is a procedure called `force' that takes
a delayed object as argument and performs the evaluation--in effect,
forcing the `delay' to fulfill its promise.  We will see below how
`delay' and `force' can be implemented, but first let us use these to
construct streams.

   `Cons-stream' is a special form defined so that

     (cons-stream <A> <B>)

is equivalent to

     (cons <A> (delay <B>))

   What this means is that we will construct streams using pairs.
However, rather than placing the value of the rest of the stream into
the `cdr' of the pair we will put there a promise to compute the rest
if it is ever requested.  `Stream-car' and `stream-cdr' can now be
defined as procedures:

     (define (stream-car stream) (car stream))

     (define (stream-cdr stream) (force (cdr stream)))

   `Stream-car' selects the `car' of the pair; `stream-cdr' selects the
`cdr' of the pair and evaluates the delayed expression found there to
obtain the rest of the stream.(4)

The stream implementation in action
...................................

To see how this implementation behaves, let us analyze the "outrageous"
prime computation we saw above, reformulated in terms of streams:

     (stream-car
      (stream-cdr
       (stream-filter prime?
                      (stream-enumerate-interval 10000 1000000))))

   We will see that it does indeed work efficiently.

   We begin by calling `stream-enumerate-interval' with the arguments
10,000 and 1,000,000.  `Stream-enumerate-interval' is the stream analog
of `enumerate-interval' (section *Note 2-2-3::):

     (define (stream-enumerate-interval low high)
       (if (> low high)
           the-empty-stream
           (cons-stream
            low
            (stream-enumerate-interval (+ low 1) high))))

and thus the result returned by `stream-enumerate-interval', formed by
the `cons-stream', is(5)

     (cons 10000
           (delay (stream-enumerate-interval 10001 1000000)))

   That is, `stream-enumerate-interval' returns a stream represented as
a pair whose `car' is 10,000 and whose `cdr' is a promise to enumerate
more of the interval if so requested.  This stream is now filtered for
primes, using the stream analog of the `filter' procedure (section
*Note 2-2-3::):

     (define (stream-filter pred stream)
       (cond ((stream-null? stream) the-empty-stream)
             ((pred (stream-car stream))
              (cons-stream (stream-car stream)
                           (stream-filter pred
                                          (stream-cdr stream))))
             (else (stream-filter pred (stream-cdr stream)))))

   `Stream-filter' tests the `stream-car' of the stream (the `car' of
the pair, which is 10,000).  Since this is not prime, `stream-filter'
examines the `stream-cdr' of its input stream.  The call to
`stream-cdr' forces evaluation of the delayed
`stream-enumerate-interval', which now returns

     (cons 10001
           (delay (stream-enumerate-interval 10002 1000000)))

   `Stream-filter' now looks at the `stream-car' of this stream, 10,001,
sees that this is not prime either, forces another `stream-cdr', and so
on, until `stream-enumerate-interval' yields the prime 10,007, whereupon
`stream-filter', according to its definition, returns

     (cons-stream (stream-car stream)
                  (stream-filter pred (stream-cdr stream)))

which in this case is

     (cons 10007
           (delay
             (stream-filter
              prime?
              (cons 10008
                    (delay
                      (stream-enumerate-interval 10009
                                                 1000000))))))

   This result is now passed to `stream-cdr' in our original expression.
This forces the delayed `stream-filter', which in turn keeps forcing the
delayed `stream-enumerate-interval' until it finds the next prime, which
is 10,009.  Finally, the result passed to `stream-car' in our original
expression is

     (cons 10009
           (delay
             (stream-filter
              prime?
              (cons 10010
                    (delay
                      (stream-enumerate-interval 10011
                                                 1000000))))))

   `Stream-car' returns 10,009, and the computation is complete.  Only
as many integers were tested for primality as were necessary to find
the second prime, and the interval was enumerated only as far as was
necessary to feed the prime filter.

   In general, we can think of delayed evaluation as "demand-driven"
programming, whereby each stage in the stream process is activated only
enough to satisfy the next stage.  What we have done is to decouple the
actual order of events in the computation from the apparent structure
of our procedures.  We write procedures as if the streams existed "all
at once" when, in reality, the computation is performed incrementally,
as in traditional programming styles.

Implementing `delay' and `force'
................................

Although `delay' and `force' may seem like mysterious operations, their
implementation is really quite straightforward.  `Delay' must package
an expression so that it can be evaluated later on demand, and we can
accomplish this simply by treating the expression as the body of a
procedure.  `Delay' can be a special form such that

     (delay <EXP>)

is syntactic sugar for

     (lambda () <EXP>)

   `Force' simply calls the procedure (of no arguments) produced by
`delay', so we can implement `force' as a procedure:

     (define (force delayed-object)
       (delayed-object))

   This implementation suffices for `delay' and `force' to work as
advertised, but there is an important optimization that we can include.
In many applications, we end up forcing the same delayed object many
times.  This can lead to serious inefficiency in recursive programs
involving streams.  (See *Note Exercise 3-57::.)  The solution is to
build delayed objects so that the first time they are forced, they
store the value that is computed.  Subsequent forcings will simply
return the stored value without repeating the computation.  In other
words, we implement `delay' as a special-purpose memoized procedure
similar to the one described in *Note Exercise 3-27::.  One way to
accomplish this is to use the following procedure, which takes as
argument a procedure (of no arguments) and returns a memoized version
of the procedure.  The first time the memoized procedure is run, it
saves the computed result.  On subsequent evaluations, it simply
returns the result.

     (define (memo-proc proc)
       (let ((already-run? false) (result false))
         (lambda ()
           (if (not already-run?)
               (begin (set! result (proc))
                      (set! already-run? true)
                      result)
               result))))

   `Delay' is then defined so that `(delay <EXP>)' is equivalent to

     (memo-proc (lambda () <EXP>))

and `force' is as defined previously.(6)

     *Exercise 3.50:* Complete the following definition, which
     generalizes `stream-map' to allow procedures that take multiple
     arguments, analogous to `map' in section *Note 2-2-3::, footnote
     *Note Footnote 12::.

          (define (stream-map proc . argstreams)
            (if (<??> (car argstreams))
                the-empty-stream
                (<??>
                 (apply proc (map <??> argstreams))
                 (apply stream-map
                        (cons proc (map <??> argstreams))))))

     *Exercise 3.51:* In order to take a closer look at delayed
     evaluation, we will use the following procedure, which simply
     returns its argument after printing it:

          (define (show x)
            (display-line x)
            x)

     What does the interpreter print in response to evaluating each
     expression in the following sequence?(7)

          (define x (stream-map show (stream-enumerate-interval 0 10)))

          (stream-ref x 5)

          (stream-ref x 7)

     *Exercise 3.52:* Consider the sequence of expressions

          (define sum 0)

          (define (accum x)
            (set! sum (+ x sum))
            sum)

          (define seq (stream-map accum (stream-enumerate-interval 1 20)))
          (define y (stream-filter even? seq))
          (define z (stream-filter (lambda (x) (= (remainder x 5) 0))
                                   seq))

          (stream-ref y 7)

          (display-stream z)

     What is the value of `sum' after each of the above expressions is
     evaluated?  What is the printed response to evaluating the
     `stream-ref' and `display-stream' expressions?  Would these
     responses differ if we had implemented `(delay <EXP>)' simply as
     `(lambda () <EXP>)' without using the optimization provided by
     `memo-proc'?  Explain

   ---------- Footnotes ----------

   (1) Assume that we have a predicate `prime?' (e.g., as in section
*Note 1-2-6::) that tests for primality.

   (2) In the MIT implementation, `the-empty-stream' is the same as the
empty list `'()', and `stream-null?' is the same as `null?'.

   (3) This should bother you.  The fact that we are defining such
similar procedures for streams and lists indicates that we are missing
some underlying abstraction.  Unfortunately, in order to exploit this
abstraction, we will need to exert finer control over the process of
evaluation than we can at present.  We will discuss this point further
at the end of section *Note 3-5-4::.  In section *Note 4-2::, we'll
develop a framework that unifies lists and streams.

   (4) Although `stream-car' and `stream-cdr' can be defined as
procedures, `cons-stream' must be a special form.  If `cons-stream'
were a procedure, then, according to our model of evaluation,
evaluating `(cons-stream <A> <B>)' would automatically cause <B> to be
evaluated, which is precisely what we do not want to happen.  For the
same reason, `delay' must be a special form, though `force' can be an
ordinary procedure.

   (5) The numbers shown here do not really appear in the delayed
expression.  What actually appears is the original expression, in an
environment in which the variables are bound to the appropriate numbers.
For example, `(+ low 1)' with `low' bound to 10,000 actually appears
where `10001' is shown.

   (6) There are many possible implementations of streams other than
the one described in this section.  Delayed evaluation, which is the
key to making streams practical, was inherent in Algol 60's "call-by-name"
parameter-passing method.  The use of this mechanism to implement
streams was first described by Landin (1965).  Delayed evaluation for
streams was introduced into Lisp by Friedman and Wise (1976). In their
implementation, `cons' always delays evaluating its arguments, so that
lists automatically behave as streams.  The memoizing optimization is
also known as "call-by-need".  The Algol community would refer to our
original delayed objects as "call-by-name thunks" and to the optimized
versions as "call-by-need thunks".

   (7) Exercises such as *Note Exercise 3-51:: and *Note Exercise
3-52:: are valuable for testing our understanding of how `delay' works.
On the other hand, intermixing delayed evaluation with printing--and,
even worse, with assignment--is extremely confusing, and instructors of
courses on computer languages have traditionally tormented their
students with examination questions such as the ones in this section.
Needless to say, writing programs that depend on such subtleties is
odious programming style.  Part of the power of stream processing is
that it lets us ignore the order in which events actually happen in our
programs.  Unfortunately, this is precisely what we cannot afford to do
in the presence of assignment, which forces us to be concerned with
time and change.


File: sicp.info,  Node: 3-5-2,  Next: 3-5-3,  Prev: 3-5-1,  Up: 3-5

3.5.2 Infinite Streams
----------------------

We have seen how to support the illusion of manipulating streams as
complete entities even though, in actuality, we compute only as much of
the stream as we need to access.  We can exploit this technique to
represent sequences efficiently as streams, even if the sequences are
very long.  What is more striking, we can use streams to represent
sequences that are infinitely long.  For instance, consider the
following definition of the stream of positive integers:

     (define (integers-starting-from n)
       (cons-stream n (integers-starting-from (+ n 1))))

     (define integers (integers-starting-from 1))

   This makes sense because `integers' will be a pair whose `car' is 1
and whose `cdr' is a promise to produce the integers beginning with 2.
This is an infinitely long stream, but in any given time we can examine
only a finite portion of it.  Thus, our programs will never know that
the entire infinite stream is not there.

   Using `integers' we can define other infinite streams, such as the
stream of integers that are not divisible by 7:

     (define (divisible? x y) (= (remainder x y) 0))

     (define no-sevens
       (stream-filter (lambda (x) (not (divisible? x 7)))
                      integers))

   Then we can find integers not divisible by 7 simply by accessing
elements of this stream:

     (stream-ref no-sevens 100)
     117

   In analogy with `integers', we can define the infinite stream of
Fibonacci numbers:

     (define (fibgen a b)
       (cons-stream a (fibgen b (+ a b))))

     (define fibs (fibgen 0 1))

   `Fibs' is a pair whose `car' is 0 and whose `cdr' is a promise to
evaluate `(fibgen 1 1)'.  When we evaluate this delayed `(fibgen 1 1)',
it will produce a pair whose `car' is 1 and whose `cdr' is a promise to
evaluate `(fibgen 1 2)', and so on.

   For a look at a more exciting infinite stream, we can generalize the
`no-sevens' example to construct the infinite stream of prime numbers,
using a method known as the Eratosthenes "sieve of Eratosthenes".(1) We
start with the integers beginning with 2, which is the first prime.  To
get the rest of the primes, we start by filtering the multiples of 2
from the rest of the integers.  This leaves a stream beginning with 3,
which is the next prime.  Now we filter the multiples of 3 from the
rest of this stream.  This leaves a stream beginning with 5, which is
the next prime, and so on.  In other words, we construct the primes by
a sieving process, described as follows: To sieve a stream `S', form a
stream whose first element is the first element of `S' and the rest of
which is obtained by filtering all multiples of the first element of
`S' out of the rest of `S' and sieving the result. This process is
readily described in terms of stream operations:

     (define (sieve stream)
       (cons-stream
        (stream-car stream)
        (sieve (stream-filter
                (lambda (x)
                  (not (divisible? x (stream-car stream))))
                (stream-cdr stream)))))

     (define primes (sieve (integers-starting-from 2)))

   Now to find a particular prime we need only ask for it:

     (stream-ref primes 50)
     233

   It is interesting to contemplate the signal-processing system set up
by `sieve', shown in the "Henderson diagram" in *Note Figure 3-31::.(2)
The input stream feeds into an "un`cons'er" that separates the first
element of the stream from the rest of the stream.  The first element
is used to construct a divisibility filter, through which the rest is
passed, and the output of the filter is fed to another sieve box.  Then
the original first element is `cons'ed onto the output of the internal
sieve to form the output stream.  Thus, not only is the stream
infinite, but the signal processor is also infinite, because the sieve
contains a sieve within it.

     *Figure 3.31:* The prime sieve viewed as a signal-processing
     system.

            +---------------------------------------------------------------+
            | sieve                                                         |
            |                                                               |
            |        __/|                                        |\__       |
            |     __/car|........................................|   \__    |
            |   _/      |           :                            |      \_  |
          ----><_       |           V                            |  cons _>---->
            |    \__    |    +------------+    +------------+    |    __/   |
            |       \cdr|--->| filter:    |    | sieve      |--->| __/      |
            |          \|    |            |--->|            |    |/         |
            |                | not        |    |            |               |
            |                | divisible? |    |            |               |
            |                +------------+    +------------+               |
            +---------------------------------------------------------------+

Defining streams implicitly
...........................

The `integers' and `fibs' streams above were defined by specifying
"generating" procedures that explicitly compute the stream elements one
by one. An alternative way to specify streams is to take advantage of
delayed evaluation to define streams implicitly.  For example, the
following expression defines the stream `ones' to be an infinite stream
of ones:

     (define ones (cons-stream 1 ones))

   This works much like the definition of a recursive procedure: `ones'
is a pair whose `car' is 1 and whose `cdr' is a promise to evaluate
`ones'.  Evaluating the `cdr' gives us again a 1 and a promise to
evaluate `ones', and so on.

   We can do more interesting things by manipulating streams with
operations such as `add-streams', which produces the elementwise sum of
two given streams:(3)

     (define (add-streams s1 s2)
       (stream-map + s1 s2))

   Now we can define the integers as follows:

     (define integers (cons-stream 1 (add-streams ones integers)))

   This defines `integers' to be a stream whose first element is 1 and
the rest of which is the sum of `ones' and `integers'.  Thus, the second
element of `integers' is 1 plus the first element of `integers', or 2;
the third element of `integers' is 1 plus the second element of
`integers', or 3; and so on.  This definition works because, at any
point, enough of the `integers' stream has been generated so that we
can feed it back into the definition to produce the next integer.

   We can define the Fibonacci numbers in the same style:

     (define fibs
       (cons-stream 0
                    (cons-stream 1
                                 (add-streams (stream-cdr fibs)
                                              fibs))))

   This definition says that `fibs' is a stream beginning with 0 and 1,
such that the rest of the stream can be generated by adding `fibs' to
itself shifted by one place:

           1  1  2  3  5  8   13  21  ... = `(stream-cdr fibs)'
           0  1  1  2  3  5   8   13  ... = `fibs'
     0  1  1  2  3  5  8  13  21  34  ... = `fibs'

   `Scale-stream' is another useful procedure in formulating such stream
definitions.  This multiplies each item in a stream by a given constant:

     (define (scale-stream stream factor)
       (stream-map (lambda (x) (* x factor)) stream))

   For example,

     (define double (cons-stream 1 (scale-stream double 2)))

produces the stream of powers of 2: 1, 2, 4, 8, 16, 32, ....

   An alternate definition of the stream of primes can be given by
starting with the integers and filtering them by testing for primality.
We will need the first prime, 2, to get started:

     (define primes
       (cons-stream
        2
        (stream-filter prime? (integers-starting-from 3))))

   This definition is not so straightforward as it appears, because we
will test whether a number n is prime by checking whether n is
divisible by a prime (not by just any integer) less than or equal to
_[sqrt]_(n):

     (define (prime? n)
       (define (iter ps)
         (cond ((> (square (stream-car ps)) n) true)
               ((divisible? n (stream-car ps)) false)
               (else (iter (stream-cdr ps)))))
       (iter primes))

   This is a recursive definition, since `primes' is defined in terms
of the `prime?' predicate, which itself uses the `primes' stream.  The
reason this procedure works is that, at any point, enough of the
`primes' stream has been generated to test the primality of the numbers
we need to check next.  That is, for every n we test for primality,
either n is not prime (in which case there is a prime already generated
that divides it) or n is prime (in which case there is a prime already
generated--i.e., a prime less than n--that is greater than
_[sqrt]_(n)).(4)

     *Exercise 3.53:* Without running the program, describe the
     elements of the stream defined by

          (define s (cons-stream 1 (add-streams s s)))

     *Exercise 3.54:* Define a procedure `mul-streams', analogous to
     `add-streams', that produces the elementwise product of its two
     input streams.  Use this together with the stream of `integers' to
     complete the following definition of the stream whose nth element
     (counting from 0) is n + 1 factorial:

          (define factorials (cons-stream 1 (mul-streams <??> <??>)))

     *Exercise 3.55:* Define a procedure `partial-sums' that takes as
     argument a stream S and returns the stream whose elements are S_0,
     S_0 + S_1, S_0 + S_1 + S_2, ....  For example, `(partial-sums
     integers)' should be the stream 1, 3, 6, 10, 15, ....

     *Exercise 3.56:* A famous problem, first raised by R. Hamming, is
     to enumerate, in ascending order with no repetitions, all positive
     integers with no prime factors other than 2, 3, or 5.  One obvious
     way to do this is to simply test each integer in turn to see
     whether it has any factors other than 2, 3, and 5.  But this is
     very inefficient, since, as the integers get larger, fewer and
     fewer of them fit the requirement.  As an alternative, let us call
     the required stream of numbers `S' and notice the following facts
     about it.

        * `S' begins with 1.

        * The elements of `(scale-stream S 2)' are also elements of `S'.

        * The same is true for `(scale-stream S 3)' and `(scale-stream
          5 S)'.

        * These are all the elements of `S'.


     Now all we have to do is combine elements from these sources.  For
     this we define a procedure `merge' that combines two ordered
     streams into one ordered result stream, eliminating repetitions:

          (define (merge s1 s2)
            (cond ((stream-null? s1) s2)
                  ((stream-null? s2) s1)
                  (else
                   (let ((s1car (stream-car s1))
                         (s2car (stream-car s2)))
                     (cond ((< s1car s2car)
                            (cons-stream s1car (merge (stream-cdr s1) s2)))
                           ((> s1car s2car)
                            (cons-stream s2car (merge s1 (stream-cdr s2))))
                           (else
                            (cons-stream s1car
                                         (merge (stream-cdr s1)
                                                (stream-cdr s2)))))))))

     Then the required stream may be constructed with `merge', as
     follows:

          (define S (cons-stream 1 (merge <??> <??>)))

     Fill in the missing expressions in the places marked <??> above.

     *Exercise 3.57:* How many additions are performed when we compute
     the nth Fibonacci number using the definition of `fibs' based on
     the `add-streams' procedure?  Show that the number of additions
     would be exponentially greater if we had implemented `(delay
     <EXP>)' simply as `(lambda () <EXP>)', without using the
     optimization provided by the `memo-proc' procedure described in
     section *Note 3-5-1::.(5)

     *Exercise 3.58:* Give an interpretation of the stream computed by
     the following procedure:

          (define (expand num den radix)
            (cons-stream
             (quotient (* num radix) den)
             (expand (remainder (* num radix) den) den radix)))

     (`Quotient' is a primitive that returns the integer quotient of two
     integers.)  What are the successive elements produced by `(expand
     1 7 10)'?  What is produced by `(expand 3 8 10)'?

     *Exercise 3.59:* In section *Note 2-5-3:: we saw how to implement
     a polynomial arithmetic system representing polynomials as lists
     of terms.  In a similar way, we can work with "power series", such
     as

                         x^2     x^3       x^4
          e^x = 1 + x + ----- + ----- + --------- + ...
                          2     3 * 2   4 * 3 * 2

                       x^2       x^4
          cos x = 1 - ----- + --------- - ...
                        2     4 * 3 * 2

                       x^3         x^5
          sin x = x - ----- + ------------- - ...
                      3 * 2   5 * 4 * 3 * 2

     represented as infinite streams.  We will represent the series a_0
     + a_1 x + a_2 x^2 + a_3 x^3 + ... as the stream whose elements are
     the coefficients a_0, a_1, a_2, a_3, ....

       a. The integral of the series a_0 + a_1 x + a_2 x^2 + a_3 x^3 +
          ... is the series

                            1             1             1
               c + a_0 x + --- x_1 r^2 + --- a_2 r^3 + --- a_3 r^4 + ...
                            2             3             4

          where c is any constant.  Define a procedure
          `integrate-series' that takes as input a stream a_0, a_1,
          a_2, ... representing a power series and returns the stream
          a_0, (1/2)a_1, (1/3)a_2, ... of coefficients of the
          non-constant terms of the integral of the series.  (Since the
          result has no constant term, it doesn't represent a power
          series; when we use `integrate-series', we will `cons' on the
          appropriate constant.)

       b. The function x |-> e^x is its own derivative.  This implies
          that e^x and the integral of e^x are the same series, except
          for the constant term, which is e^0 = 1.  Accordingly, we can
          generate the series for e^x as

               (define exp-series
                 (cons-stream 1 (integrate-series exp-series)))

          Show how to generate the series for sine and cosine, starting
          from the facts that the derivative of sine is cosine and the
          derivative of cosine is the negative of sine:

               (define cosine-series
                 (cons-stream 1 <??>))

               (define sine-series
                 (cons-stream 0 <??>))

     *Exercise 3.60:* With power series represented as streams of
     coefficients as in *Note Exercise 3-59::, adding series is
     implemented by `add-streams'.  Complete the definition of the
     following procedure for multiplying series:

          (define (mul-series s1 s2)
            (cons-stream <??> (add-streams <??> <??>)))

     You can test your procedure by verifying that sin^2 x + cos^2 x =
     1, using the series from *Note Exercise 3-59::.

     *Exercise 3.61:* Let S be a power series (*Note Exercise 3-59::)
     whose constant term is 1.  Suppose we want to find the power
     series 1/S, that is, the series X such that S * X = 1.  Write S =
     1 + S_R where S_R is the part of S after the constant term.  Then
     we can solve for X as follows:

                  S * X = 1
          (1 + S_R) * X = 1
            X + S_R * X = 1
                      X = 1 - S_R * X

     In other words, X is the power series whose constant term is 1 and
     whose higher-order terms are given by the negative of S_R times X.
     Use this idea to write a procedure `invert-unit-series' that
     computes 1/S for a power series S with constant term 1.  You will
     need to use `mul-series' from *Note Exercise 3-60::.

     *Exercise 3.62:* Use the results of *Note Exercise 3-60:: and
     *Note Exercise 3-61:: to define a procedure `div-series' that
     divides two power series.  `Div-series' should work for any two
     series, provided that the denominator series begins with a nonzero
     constant term.  (If the denominator has a zero constant term, then
     `div-series' should signal an error.)  Show how to use
     `div-series' together with the result of *Note Exercise 3-59:: to
     generate the power series for tangent.

   ---------- Footnotes ----------

   (1) Eratosthenes, a third-century B.C.  Alexandrian Greek
philosopher, is famous for giving the first accurate estimate of the
circumference of the Earth, which he computed by observing shadows cast
at noon on the day of the summer solstice.  Eratosthenes's sieve method,
although ancient, has formed the basis for special-purpose hardware
"sieves" that, until recently, were the most powerful tools in
existence for locating large primes.  Since the 70s, however, these
methods have been superseded by outgrowths of the probabilistic
techniques discussed in section *Note 1-2-6::.

   (2) We have named these figures after Peter Henderson, who was the
first person to show us diagrams of this sort as a way of thinking
about stream processing.  Each solid line represents a stream of values
being transmitted.  The dashed line from the `car' to the `cons' and
the `filter' indicates that this is a single value rather than a stream.

   (3) This uses the generalized version of `stream-map' from *Note
Exercise 3-50::.

   (4) This last point is very subtle and relies on the fact that
p_(n+1) <= p_n^2.  (Here, p_k denotes the kth prime.)  Estimates such
as these are very difficult to establish.  The ancient proof by Euclid
that there are an infinite number of primes shows that p_(n+1)<= p_1
p_2...p_n + 1, and no substantially better result was proved until
1851, when the Russian mathematician P. L. Chebyshev established that
p_(n+1)<= 2p_n for all n.  This result, originally conjectured in 1845,
is known as hypothesis "Bertrand's hypothesis".  A proof can be found
in section 22.3 of Hardy and Wright 1960.

   (5) This exercise shows how call-by-need is closely related to
ordinary memoization as described in *Note Exercise 3-27::.  In that
exercise, we used assignment to explicitly construct a local table.
Our call-by-need stream optimization effectively constructs such a
table automatically, storing values in the previously forced parts of
the stream.


File: sicp.info,  Node: 3-5-3,  Next: 3-5-4,  Prev: 3-5-2,  Up: 3-5

3.5.3 Exploiting the Stream Paradigm
------------------------------------

Streams with delayed evaluation can be a powerful modeling tool,
providing many of the benefits of local state and assignment.
Moreover, they avoid some of the theoretical tangles that accompany the
introduction of assignment into a programming language.

   The stream approach can be illuminating because it allows us to
build systems with different module boundaries than systems organized
around assignment to state variables.  For example, we can think of an
entire time series (or signal) as a focus of interest, rather than the
values of the state variables at individual moments.  This makes it
convenient to combine and compare components of state from different
moments.

Formulating iterations as stream processes
..........................................

In section *Note 1-2-1::, we introduced iterative processes, which
proceed by updating state variables.  We know now that we can represent
state as a "timeless" stream of values rather than as a set of
variables to be updated.  Let's adopt this perspective in revisiting
the square-root procedure from section *Note 1-1-7::.  Recall that the
idea is to generate a sequence of better and better guesses for the
square root of x by applying over and over again the procedure that
improves guesses:

     (define (sqrt-improve guess x)
       (average guess (/ x guess)))

   In our original `sqrt' procedure, we made these guesses be the
successive values of a state variable. Instead we can generate the
infinite stream of guesses, starting with an initial guess of 1:(1)

     (define (sqrt-stream x)
       (define guesses
         (cons-stream 1.0
                      (stream-map (lambda (guess)
                                    (sqrt-improve guess x))
                                  guesses)))
       guesses)

     (display-stream (sqrt-stream 2))
     1.
     1.5
     1.4166666666666665
     1.4142156862745097
     1.4142135623746899
     ...

   We can generate more and more terms of the stream to get better and
better guesses.  If we like, we can write a procedure that keeps
generating terms until the answer is good enough.  (See *Note Exercise
3-64::.)

   Another iteration that we can treat in the same way is to generate an
approximation to [pi], based upon the alternating series that we saw in
section *Note 1-3-1:::

     [pi]        1     1     1
     ---- = 1 - --- + --- - --- + ...
       4         3     5     7

   We first generate the stream of summands of the series (the
reciprocals of the odd integers, with alternating signs).  Then we take
the stream of sums of more and more terms (using the `partial-sums'
procedure of *Note Exercise 3-55::) and scale the result by 4:

     (define (pi-summands n)
       (cons-stream (/ 1.0 n)
                    (stream-map - (pi-summands (+ n 2)))))

     (define pi-stream
       (scale-stream (partial-sums (pi-summands 1)) 4))

     (display-stream pi-stream)
     4.
     2.666666666666667
     3.466666666666667
     2.8952380952380956
     3.3396825396825403
     2.9760461760461765
     3.2837384837384844
     3.017071817071818
     ...

   This gives us a stream of better and better approximations to [pi],
although the approximations converge rather slowly.  Eight terms of the
sequence bound the value of [pi] between 3.284 and 3.017.

   So far, our use of the stream of states approach is not much
different from updating state variables.  But streams give us an
opportunity to do some interesting tricks.  For example, we can
transform a stream with a "sequence accelerator" that converts a
sequence of approximations to a new sequence that converges to the same
value as the original, only faster.

   One such accelerator, due to the eighteenth-century Swiss
mathematician Leonhard Euler, works well with sequences that are
partial sums of alternating series (series of terms with alternating
signs).  In Euler's technique, if S_n is the nth term of the original
sum sequence, then the accelerated sequence has terms

                  (S_(n+1) - S_n)^2
     S_(n+1) - ------------------------
               S_(n-1) - 2S_n + S_(n+1)

   Thus, if the original sequence is represented as a stream of values,
the transformed sequence is given by

     (define (euler-transform s)
       (let ((s0 (stream-ref s 0))           ; S_(n-1)
             (s1 (stream-ref s 1))           ; S_n
             (s2 (stream-ref s 2)))          ; S_(n+1)
         (cons-stream (- s2 (/ (square (- s2 s1))
                               (+ s0 (* -2 s1) s2)))
                      (euler-transform (stream-cdr s)))))

   We can demonstrate Euler acceleration with our sequence of
approximations to [pi]:

     (display-stream (euler-transform pi-stream))
     3.166666666666667
     3.1333333333333337
     3.1452380952380956
     3.13968253968254
     3.1427128427128435
     3.1408813408813416
     3.142071817071818
     3.1412548236077655
     ...

   Even better, we can accelerate the accelerated sequence, and
recursively accelerate that, and so on.  Namely, we create a stream of
streams (a structure we'll call a "tableau") in which each stream is
the transform of the preceding one:

     (define (make-tableau transform s)
       (cons-stream s
                    (make-tableau transform
                                  (transform s))))

   The tableau has the form

     s_00   s_01   s_02   s_03   s_04   ...
            s_10   s_11   s_12   s_13   ...
                   s_20   s_21   s_22   ...
                                 ...

   Finally, we form a sequence by taking the first term in each row of
the tableau:

     (define (accelerated-sequence transform s)
       (stream-map stream-car
                   (make-tableau transform s)))

   We can demonstrate this kind of "super-acceleration" of the [pi]
sequence:

     (display-stream (accelerated-sequence euler-transform
                                           pi-stream))
     4.
     3.166666666666667
     3.142105263157895
     3.141599357319005
     3.1415927140337785
     3.1415926539752927
     3.1415926535911765
     3.141592653589778
     ...

   The result is impressive.  Taking eight terms of the sequence yields
the correct value of [pi] to 14 decimal places.  If we had used only the
original [pi] sequence, we would need to compute on the order of 10^13
terms (i.e., expanding the series far enough so that the individual
terms are less then 10^(-13)) to get that much accuracy!

   We could have implemented these acceleration techniques without
using streams.  But the stream formulation is particularly elegant and
convenient because the entire sequence of states is available to us as
a data structure that can be manipulated with a uniform set of
operations.

     *Exercise 3.63:* Louis Reasoner asks why the `sqrt-stream'
     procedure was not written in the following more straightforward
     way, without the local variable `guesses':

          (define (sqrt-stream x)
            (cons-stream 1.0
                         (stream-map (lambda (guess)
                                       (sqrt-improve guess x))
                                     (sqrt-stream x))))

     Alyssa P. Hacker replies that this version of the procedure is
     considerably less efficient because it performs redundant
     computation.  Explain Alyssa's answer.  Would the two versions
     still differ in efficiency if our implementation of `delay' used
     only `(lambda () <EXP>)' without using the optimization provided
     by `memo-proc' (section *Note 3-5-1::)?

     *Exercise 3.64:* Write a procedure `stream-limit' that takes as
     arguments a stream and a number (the tolerance).  It should
     examine the stream until it finds two successive elements that
     differ in absolute value by less than the tolerance, and return
     the second of the two elements.  Using this, we could compute
     square roots up to a given tolerance by

          (define (sqrt x tolerance)
            (stream-limit (sqrt-stream x) tolerance))

     *Exercise 3.65:* Use the series

                      1     1     1
          ln 2 = 1 - --- + --- - --- + ...
                      2     3     4

     to compute three sequences of approximations to the natural
     logarithm of 2, in the same way we did above for [pi].  How
     rapidly do these sequences converge?

Infinite streams of pairs
.........................

In section *Note 2-2-3::, we saw how the sequence paradigm handles
traditional nested loops as processes defined on sequences of pairs.
If we generalize this technique to infinite streams, then we can write
programs that are not easily represented as loops, because the
"looping" must range over an infinite set.

   For example, suppose we want to generalize the `prime-sum-pairs'
procedure of section *Note 2-2-3:: to produce the stream of pairs of
_all_ integers (i,j) with i <= j such that i + j is prime.  If
`int-pairs' is the sequence of all pairs of integers (i,j) with i <= j,
then our required stream is simply(2)

     (stream-filter (lambda (pair)
                      (prime? (+ (car pair) (cadr pair))))
                    int-pairs)

   Our problem, then, is to produce the stream `int-pairs'.  More
generally, suppose we have two streams S = (S_i) and T = (T_j), and
imagine the infinite rectangular array

     (S_0, T_0)  (S_0, T_1)  (S_0, T_2)  ...
     (S_1, T_0)  (S_1, T_1)  (S_1, T_2)  ...
     (S_2, T_0)  (S_2, T_1)  (S_2, T_2)  ...
        ...

   We wish to generate a stream that contains all the pairs in the
array that lie on or above the diagonal, i.e., the pairs

     (S_0, T_0)  (S_0, T_1)  (S_0, T_2)  ...
                 (S_1, T_1)  (S_1, T_2)  ...
                             (S_2, T_2)  ...
                                         ...

(If we take both S and T to be the stream of integers, then this will
be our desired stream `int-pairs'.)

   Call the general stream of pairs `(pairs S T)', and consider it to be
composed of three parts: the pair (S_0,T_0), the rest of the pairs in
the first row, and the remaining pairs:(3)

     (S_0, T_0) | (S_0, T_1)  (S_0, T_2)  ...
     -----------+-----------------------------
                | (S_1, T_1)  (S_1, T_2)  ...
                |             (S_2, T_2)  ...
                |                         ...

   Observe that the third piece in this decomposition (pairs that are
not in the first row) is (recursively) the pairs formed from
`(stream-cdr S)' and `(stream-cdr T)'.  Also note that the second piece
(the rest of the first row) is

     (stream-map (lambda (x) (list (stream-car s) x))
                 (stream-cdr t))

   Thus we can form our stream of pairs as follows:

     (define (pairs s t)
       (cons-stream
        (list (stream-car s) (stream-car t))
        (<COMBINE-IN-SOME-WAY>
            (stream-map (lambda (x) (list (stream-car s) x))
                        (stream-cdr t))
            (pairs (stream-cdr s) (stream-cdr t)))))

   In order to complete the procedure, we must choose some way to
combine the two inner streams.  One idea is to use the stream analog of
the `append' procedure from section *Note 2-2-1:::

     (define (stream-append s1 s2)
       (if (stream-null? s1)
           s2
           (cons-stream (stream-car s1)
                        (stream-append (stream-cdr s1) s2))))

   This is unsuitable for infinite streams, however, because it takes
all the elements from the first stream before incorporating the second
stream.  In particular, if we try to generate all pairs of positive
integers using

     (pairs integers integers)

our stream of results will first try to run through all pairs with the
first integer equal to 1, and hence will never produce pairs with any
other value of the first integer.

   To handle infinite streams, we need to devise an order of
combination that ensures that every element will eventually be reached
if we let our program run long enough.  An elegant way to accomplish
this is with the following `interleave' procedure:(4)

     (define (interleave s1 s2)
       (if (stream-null? s1)
           s2
           (cons-stream (stream-car s1)
                        (interleave s2 (stream-cdr s1)))))

   Since `interleave' takes elements alternately from the two streams,
every element of the second stream will eventually find its way into
the interleaved stream, even if the first stream is infinite.

   We can thus generate the required stream of pairs as

     (define (pairs s t)
       (cons-stream
        (list (stream-car s) (stream-car t))
        (interleave
         (stream-map (lambda (x) (list (stream-car s) x))
                     (stream-cdr t))
         (pairs (stream-cdr s) (stream-cdr t)))))

     *Exercise 3.66:* Examine the stream `(pairs integers integers)'.
     Can you make any general comments about the order in which the
     pairs are placed into the stream? For example, about how many
     pairs precede the pair (1,100)?  the pair (99,100)? the pair
     (100,100)? (If you can make precise mathematical statements here,
     all the better. But feel free to give more qualitative answers if
     you find yourself getting bogged down.)

     *Exercise 3.67:* Modify the `pairs' procedure so that `(pairs
     integers integers)' will produce the stream of _all_ pairs of
     integers (i,j) (without the condition i <= j).  Hint: You will
     need to mix in an additional stream.

     *Exercise 3.68:* Louis Reasoner thinks that building a stream of
     pairs from three parts is unnecessarily complicated.  Instead of
     separating the pair (S_0,T_0) from the rest of the pairs in the
     first row, he proposes to work with the whole first row, as
     follows:

          (define (pairs s t)
            (interleave
             (stream-map (lambda (x) (list (stream-car s) x))
                         t)
             (pairs (stream-cdr s) (stream-cdr t))))

     Does this work?  Consider what happens if we evaluate `(pairs
     integers integers)' using Louis's definition of `pairs'.

     *Exercise 3.69:* Write a procedure `triples' that takes three
     infinite streams, S, T, and U, and produces the stream of triples
     (S_i,T_j,U_k) such that i <= j <= k.  Use `triples' to generate
     the stream of all Pythagorean triples of positive integers, i.e.,
     the triples (i,j,k) such that i <= j and i^2 + j^2 = k^2.

     *Exercise 3.70:* It would be nice to be able to generate streams
     in which the pairs appear in some useful order, rather than in the
     order that results from an _ad hoc_ interleaving process.  We can
     use a technique similar to the `merge' procedure of *Note Exercise
     3-56::, if we define a way to say that one pair of integers is
     "less than" another.  One way to do this is to define a "weighting
     function" W(i,j) and stipulate that (i_1,j_1) is less than
     (i_2,j_2) if W(i_1,j_1) < W(i_2,j_2).  Write a procedure
     `merge-weighted' that is like `merge', except that
     `merge-weighted' takes an additional argument `weight', which is a
     procedure that computes the weight of a pair, and is used to
     determine the order in which elements should appear in the
     resulting merged stream.(5)  Using this, generalize `pairs' to a
     procedure `weighted-pairs' that takes two streams, together with a
     procedure that computes a weighting function, and generates the
     stream of pairs, ordered according to weight.  Use your procedure
     to generate

       a. the stream of all pairs of positive integers (i,j) with i <= j
          ordered according to the sum i + j

       b. the stream of all pairs of positive integers (i,j) with i <=
          j, where neither i nor j is divisible by 2, 3, or 5, and the
          pairs are ordered according to the sum 2 i + 3 j + 5 i j.


     *Exercise 3.71:* Numbers that can be expressed as the sum of two
     cubes in more than one way are sometimes called "Ramanujan
     numbers", in honor of the mathematician Srinivasa Ramanujan.(6)
     Ordered streams of pairs provide an elegant solution to the
     problem of computing these numbers.  To find a number that can be
     written as the sum of two cubes in two different ways, we need
     only generate the stream of pairs of integers (i,j) weighted
     according to the sum i^3 + j^3 (see *Note Exercise 3-70::), then
     search the stream for two consecutive pairs with the same weight.
     Write a procedure to generate the Ramanujan numbers.  The first
     such number is 1,729.  What are the next five?

     *Exercise 3.72:* In a similar way to *Note Exercise 3-71::
     generate a stream of all numbers that can be written as the sum of
     two squares in three different ways (showing how they can be so
     written).

Streams as signals
..................

We began our discussion of streams by describing them as computational
analogs of the "signals" in signal-processing systems.  In fact, we can
use streams to model signal-processing systems in a very direct way,
representing the values of a signal at successive time intervals as
consecutive elements of a stream.  For instance, we can implement an "integrator"
or "summer" that, for an input stream x = (x_i), an initial value C,
and a small increment dt, accumulates the sum

                i
               ---
     S_i = C + >   x_j dt
               ---
               j=1

and returns the stream of values S = (S_i).  The following `integral'
procedure is reminiscent of the "implicit style" definition of the
stream of integers (section *Note 3-5-2::):

     (define (integral integrand initial-value dt)
       (define int
         (cons-stream initial-value
                      (add-streams (scale-stream integrand dt)
                                   int)))
       int)

   *Note Figure 3-32:: is a picture of a signal-processing system that
corresponds to the `integral' procedure.  The input stream is scaled by
dt and passed through an adder, whose output is passed back through the
same adder.  The self-reference in the definition of `int' is reflected
in the figure by the feedback loop that connects the output of the
adder to one of the inputs.

     *Figure 3.32:* The `integral' procedure viewed as a
     signal-processing system.

                                       initial-value
                                            |
                 +-----------+              |   |\__
          input  |           |      |\__    +-->|   \_  integral
          ------>| scale: dt +----->|   \_      |cons_>--*------->
                 |           |      | add_>---->| __/    |
                 +-----------+  +-->| __/       |/       |
                                |   |/                   |
                                |                        |
                                +------------------------+

     *Exercise 3.73:* We can model electrical circuits using streams to
     represent the values of currents or voltages at a sequence of
     times.  For instance, suppose we have an "RC circuit" consisting
     of a resistor of resistance R and a capacitor of capacitance C in
     series.  The voltage response v of the circuit to an injected
     current i is determined by the formula in *Note Figure 3-33::,
     whose structure is shown by the accompanying signal-flow diagram.

     Write a procedure `RC' that models this circuit.  `RC' should take
     as inputs the values of R, C, and dt and should return a procedure
     that takes as inputs a stream representing the current i and an
     initial value for the capacitor voltage v_0 and produces as output
     the stream of voltages v.  For example, you should be able to use
     `RC' to model an RC circuit with R = 5 ohms, C = 1 farad, and a
     0.5-second time step by evaluating `(define RC1 (RC 5 1 0.5))'.
     This defines `RC1' as a procedure that takes a stream representing
     the time sequence of currents and an initial capacitor voltage and
     produces the output stream of voltages.

     *Figure 3.33:* An RC circuit and the associated signal-flow
     diagram.

            +                 -
           ->----'\/\/\,---| |---
            i                 C

                        / t
                        |  i
           v  =  v   +  |      dt + R i
                  0     |
                        / 0

                   +--------------+
               +-->|   scale: R   |---------------------+   |\_
               |   +--------------+                     |   |  \_
               |                                        +-->|    \   v
            i  |   +--------------+     +------------+      | add >--->
           ----+-->|  scale: 1/C  |---->|  integral  |----->|   _/
                   +--------------+     +------------+      | _/
                                                            |/

     *Exercise 3.74:* Alyssa P. Hacker is designing a system to process
     signals coming from physical sensors.  One important feature she
     wishes to produce is a signal that describes the "zero crossings"
     of the input signal.  That is, the resulting signal should be + 1
     whenever the input signal changes from negative to positive, - 1
     whenever the input signal changes from positive to negative, and 0
     otherwise.  (Assume that the sign of a 0 input is positive.)  For
     example, a typical input signal with its associated zero-crossing
     signal would be

          ... 1  2  1.5  1  0.5  -0.1  -2  -3  -2  -0.5  0.2  3  4 ...
          ...  0  0    0  0    0     -1  0   0   0     0    1  0  0 ...

     In Alyssa's system, the signal from the sensor is represented as a
     stream `sense-data' and the stream `zero-crossings' is the
     corresponding stream of zero crossings.  Alyssa first writes a
     procedure `sign-change-detector' that takes two values as
     arguments and compares the signs of the values to produce an
     appropriate 0, 1, or - 1.  She then constructs her zero-crossing
     stream as follows:

          (define (make-zero-crossings input-stream last-value)
            (cons-stream
             (sign-change-detector (stream-car input-stream) last-value)
             (make-zero-crossings (stream-cdr input-stream)
                                  (stream-car input-stream))))

          (define zero-crossings (make-zero-crossings sense-data 0))

     Alyssa's boss, Eva Lu Ator, walks by and suggests that this
     program is approximately equivalent to the following one, which
     uses the generalized version of `stream-map' from *Note Exercise
     3-50:::

          (define zero-crossings
            (stream-map sign-change-detector sense-data <EXPRESSION>))

     Complete the program by supplying the indicated <EXPRESSION>.

     *Exercise 3.75:* Unfortunately, Alyssa's zero-crossing detector in
     *Note Exercise 3-74:: proves to be insufficient, because the noisy
     signal from the sensor leads to spurious zero crossings.  Lem E.
     Tweakit, a hardware specialist, suggests that Alyssa smooth the
     signal to filter out the noise before extracting the zero
     crossings.  Alyssa takes his advice and decides to extract the
     zero crossings from the signal constructed by averaging each value
     of the sense data with the previous value.  She explains the
     problem to her assistant, Louis Reasoner, who attempts to
     implement the idea, altering Alyssa's program as follows:

          (define (make-zero-crossings input-stream last-value)
            (let ((avpt (/ (+ (stream-car input-stream) last-value) 2)))
              (cons-stream (sign-change-detector avpt last-value)
                           (make-zero-crossings (stream-cdr input-stream)
                                                avpt))))

     This does not correctly implement Alyssa's plan.  Find the bug
     that Louis has installed and fix it without changing the structure
     of the program.  (Hint: You will need to increase the number of
     arguments to `make-zero-crossings'.)

     *Exercise 3.76:* Eva Lu Ator has a criticism of Louis's approach
     in *Note Exercise 3-75::.  The program he wrote is not modular,
     because it intermixes the operation of smoothing with the
     zero-crossing extraction.  For example, the extractor should not
     have to be changed if Alyssa finds a better way to condition her
     input signal.  Help Louis by writing a procedure `smooth' that
     takes a stream as input and produces a stream in which each
     element is the average of two successive input stream elements.
     Then use `smooth' as a component to implement the zero-crossing
     detector in a more modular style.

   ---------- Footnotes ----------

   (1) We can't use `let' to bind the local variable `guesses', because
the value of `guesses' depends on `guesses' itself.  *Note Exercise
3-63:: addresses why we want a local variable here.

   (2) As in section *Note 2-2-3::, we represent a pair of integers as
a list rather than a Lisp pair.

   (3) See *Note Exercise 3-68:: for some insight into why we chose
this decomposition.

   (4) The precise statement of the required property on the order of
combination is as follows: There should be a function f of two
arguments such that the pair corresponding to element i of the first
stream and element j of the second stream will appear as element number
f(i,j) of the output stream.  The trick of using `interleave' to
accomplish this was shown to us by David Turner, who employed it in the
language KRC (Turner 1981).

   (5) We will require that the weighting function be such that the
weight of a pair increases as we move out along a row or down along a
column of the array of pairs.

   (6) To quote from G. H. Hardy's obituary of Ramanujan (Hardy 1921):
"It was Mr. Littlewood (I believe) who remarked that `every positive
integer was one of his friends.'  I remember once going to see him when
he was lying ill at Putney.  I had ridden in taxi-cab No. 1729, and
remarked that the number seemed to me a rather dull one, and that I
hoped it was not an unfavorable omen.  `No,' he replied, `it is a very
interesting number; it is the smallest number expressible as the sum of
two cubes in two different ways.'  " The trick of using weighted pairs
to generate the Ramanujan numbers was shown to us by Charles Leiserson.


File: sicp.info,  Node: 3-5-4,  Next: 3-5-5,  Prev: 3-5-3,  Up: 3-5

3.5.4 Streams and Delayed Evaluation
------------------------------------

The `integral' procedure at the end of the preceding section shows how
we can use streams to model signal-processing systems that contain
feedback loops.  The feedback loop for the adder shown in *Note Figure
3-32:: is modeled by the fact that `integral''s internal stream `int'
is defined in terms of itself:

     (define int
       (cons-stream initial-value
                    (add-streams (scale-stream integrand dt)
                                 int)))

   The interpreter's ability to deal with such an implicit definition
depends on the `delay' that is incorporated into `cons-stream'.
Without this `delay', the interpreter could not construct `int' before
evaluating both arguments to `cons-stream', which would require that
`int' already be defined.  In general, `delay' is crucial for using
streams to model signal-processing systems that contain loops.  Without
`delay', our models would have to be formulated so that the inputs to
any signal-processing component would be fully evaluated before the
output could be produced.  This would outlaw loops.

   Unfortunately, stream models of systems with loops may require uses
of `delay' beyond the "hidden" `delay' supplied by `cons-stream'.  For
instance, *Note Figure 3-34:: shows a signal-processing system for
solving the differential equation dy/dt = f(y) where f is a given
function.  The figure shows a mapping component, which applies f to its
input signal, linked in a feedback loop to an integrator in a manner
very similar to that of the analog computer circuits that are actually
used to solve such equations.

     *Figure 3.34:* An "analog computer circuit" that solves the
     equation dy/dt = f(y).

                                      y_0
                                       |
                                       V
              +----------+  dy   +----------+     y
          +-->|  map: f  +------>| integral +--*----->
          |   +----------+       +----------+  |
          |                                    |
          +------------------------------------+

   Assuming we are given an initial value y_0 for y, we could try to
model this system using the procedure

     (define (solve f y0 dt)
       (define y (integral dy y0 dt))
       (define dy (stream-map f y))
       y)

   This procedure does not work, because in the first line of `solve'
the call to `integral' requires that the input `dy' be defined, which
does not happen until the second line of `solve'.

   On the other hand, the intent of our definition does make sense,
because we can, in principle, begin to generate the `y' stream without
knowing `dy'.  Indeed, `integral' and many other stream operations have
properties similar to those of `cons-stream', in that we can generate
part of the answer given only partial information about the arguments.
For `integral', the first element of the output stream is the specified
`initial-value'.  Thus, we can generate the first element of the output
stream without evaluating the integrand `dy'.  Once we know the first
element of `y', the `stream-map' in the second line of `solve' can
begin working to generate the first element of `dy', which will produce
the next element of `y', and so on.

   To take advantage of this idea, we will redefine `integral' to
expect the integrand stream to be a "delayed argument".  `Integral' will
`force' the integrand to be evaluated only when it is required to
generate more than the first element of the output stream:

     (define (integral delayed-integrand initial-value dt)
       (define int
         (cons-stream initial-value
                      (let ((integrand (force delayed-integrand)))
                        (add-streams (scale-stream integrand dt)
                                     int))))
       int)

   Now we can implement our `solve' procedure by delaying the
evaluation of `dy' in the definition of `y':(1)

     (define (solve f y0 dt)
       (define y (integral (delay dy) y0 dt))
       (define dy (stream-map f y))
       y)

   In general, every caller of `integral' must now `delay' the integrand
argument.  We can demonstrate that the `solve' procedure works by
approximating eapprox 2.718 by computing the value at y = 1 of the
solution to the differential equation dy/dt = y with initial condition
y(0) = 1:

     (stream-ref (solve (lambda (y) y) 1 0.001) 1000)
     2.716924

     *Exercise 3.77:* The `integral' procedure used above was analogous
     to the "implicit" definition of the infinite stream of integers in
     section *Note 3-5-2::.  Alternatively, we can give a definition of
     `integral' that is more like `integers-starting-from' (also in
     section *Note 3-5-2::):

          (define (integral integrand initial-value dt)
            (cons-stream initial-value
                         (if (stream-null? integrand)
                             the-empty-stream
                             (integral (stream-cdr integrand)
                                       (+ (* dt (stream-car integrand))
                                          initial-value)
                                       dt))))

     When used in systems with loops, this procedure has the same
     problem as does our original version of `integral'.  Modify the
     procedure so that it expects the `integrand' as a delayed argument
     and hence can be used in the `solve' procedure shown above.

     *Figure 3.35:* Signal-flow diagram for the solution to a
     second-order linear differential equation.

                         dy_0                y_0
                          |                   |
                          V                   V
             ddy     +----------+    dy  +----------+    y
          +--------->| integral +-----*--+ integral +--*--->
          |          +----------+     |  +----------+  |
          |                           |                |
          |            +----------+   |                |
          |     __/|<--+ scale: a |<--+                |
          |   _/   |   +----------+                    |
          +--<_add |                                   |
               \__ |   +----------+                    |
                  \|<--+ scale: b |<-------------------+
                       +----------+

     *Exercise 3.78:* Consider the problem of designing a
     signal-processing system to study the homogeneous second-order
     linear differential equation

          d^2 y        d y
          -----  -  a -----  -  by  =  0
          d t^2        d t

     The output stream, modeling y, is generated by a network that
     contains a loop. This is because the value of d^2y/dt^2 depends
     upon the values of y and dy/dt and both of these are determined by
     integrating d^2y/dt^2.  The diagram we would like to encode is
     shown in *Note Figure 3-35::.  Write a procedure `solve-2nd' that
     takes as arguments the constants a, b, and dt and the initial
     values y_0 and dy_0 for y and dy/dt and generates the stream of
     successive values of y.

     *Exercise 3.79:* Generalize the `solve-2nd' procedure of *Note
     Exercise 3-78:: so that it can be used to solve general
     second-order differential equations d^2 y/dt^2 = f(dy/dt, y).

     *Exercise 3.80:* A "series RLC circuit" consists of a resistor, a
     capacitor, and an inductor connected in series, as shown in *Note
     Figure 3-36::.  If R, L, and C are the resistance, inductance, and
     capacitance, then the relations between voltage (v) and current
     (i) for the three components are described by the equations

          v_R = i_R R

                   d_(i L)
          v_L = L ---------
                     d t

                   d v_C
          i_C = C -------
                    d t

     and the circuit connections dictate the relations

          i_R = i_L = -i_C

          v_C = v_L + v_R

     Combining these equations shows that the state of the circuit
     (summarized by v_C, the voltage across the capacitor, and i_L, the
     current in the inductor) is described by the pair of differential
     equations

          d v_C        i_L
          -----  =  -  ---
           d t          C

          d i_L      1           R
          -----  =  --- v_C  -  --- i_L
           d t       L           L

     The signal-flow diagram representing this system of differential
     equations is shown in *Note Figure 3-37::.

     *Figure 3.36:* A series RLC circuit.
                   + v_R -
             i_R
          +--->----'\/\/\,--------+
          |                       |  i_L
         \|/          R          \|/
       +  |  i_C                  |_   +
         -+-                      __)
     v_C -+- C                   (_)   v_L
          |                       __)
       -  |                       |    -
          +-----------------------+

     *Figure 3.37:* A signal-flow diagram for the solution to a series
     RLC circuit.

                           +-------------+
          +----------------+  scale: l/L |<--+
          |                +-------------+   |
          |                                  |
          |                +-------------+   |  v_C
          |       dv_C +-->|   integral  +---*------>
          |            |   +-------------+
          |            |        ^
          |            |        | v_(C_0)
          |            |
          |            |   +-------------+
          |            +---+ scale: -l/C |<--+
          |                +-------------+   |
          |  |\__                            |
          +->|   \_  di_L  +-------------+   |  i_L
             | add_>------>|   integral  +---*------>
          +->| __/         +-------------+   |
          |  |/                 ^            |
          |                     | i_(L_0)    |
          |                                  |
          |                +-------------+   |
          +----------------+ scale: -R/L |<--+
                           +-------------+

   Write a procedure `RLC' that takes as arguments the parameters R, L,
and C of the circuit and the time increment dt.  In a manner similar to
that of the `RC' procedure of *Note Exercise 3-73::, `RLC' should
produce a procedure that takes the initial values of the state
variables, v_(C_0) and i_(L_0), and produces a pair (using `cons') of
the streams of states v_C and i_L.  Using `RLC', generate the pair of
streams that models the behavior of a series RLC circuit with R = 1
ohm, C = 0.2 farad, L = 1 henry, dt = 0.1 second, and initial values
i_(L_0) = 0 amps and v_(C_0) = 10 volts.

Normal-order evaluation
.......................

The examples in this section illustrate how the explicit use of `delay'
and `force' provides great programming flexibility, but the same
examples also show how this can make our programs more complex.  Our
new `integral' procedure, for instance, gives us the power to model
systems with loops, but we must now remember that `integral' should be
called with a delayed integrand, and every procedure that uses
`integral' must be aware of this.  In effect, we have created two
classes of procedures: ordinary procedures and procedures that take
delayed arguments.  In general, creating separate classes of procedures
forces us to create separate classes of higher-order procedures as
well.(2)

   One way to avoid the need for two different classes of procedures is
to make all procedures take delayed arguments.  We could adopt a model
of evaluation in which all arguments to procedures are automatically
delayed and arguments are forced only when they are actually needed
(for example, when they are required by a primitive operation).  This
would transform our language to use normal-order evaluation, which we
first described when we introduced the substitution model for
evaluation in section *Note 1-1-5::.  Converting to normal-order
evaluation provides a uniform and elegant way to simplify the use of
delayed evaluation, and this would be a natural strategy to adopt if we
were concerned only with stream processing.  In section *Note 4-2::,
after we have studied the evaluator, we will see how to transform our
language in just this way.  Unfortunately, including delays in
procedure calls wreaks havoc with our ability to design programs that
depend on the order of events, such as programs that use assignment,
mutate data, or perform input or output.  Even the single `delay' in
`cons-stream' can cause great confusion, as illustrated by *Note
Exercise 3-51:: and *Note Exercise 3-52::.  As far as anyone knows,
mutability and delayed evaluation do not mix well in programming
languages, and devising ways to deal with both of these at once is an
active area of research.

   ---------- Footnotes ----------

   (1) This procedure is not guaranteed to work in all Scheme
implementations, although for any implementation there is a simple
variation that will work.  The problem has to do with subtle
differences in the ways that Scheme implementations handle internal
definitions.  (See section *Note 4-1-6::.)

   (2) This is a small reflection, in Lisp, of the difficulties that
conventional strongly typed languages such as Pascal have in coping with
higher-order procedures.  In such languages, the programmer must
specify the data types of the arguments and the result of each
procedure: number, logical value, sequence, and so on.  Consequently,
we could not express an abstraction such as "map a given procedure
`proc' over all the elements in a sequence" by a single higher-order
procedure such as `stream-map'.  Rather, we would need a different
mapping procedure for each different combination of argument and result
data types that might be specified for a `proc'.  Maintaining a
practical notion of "data type" in the presence of higher-order
procedures raises many difficult issues.  One way of dealing with this
problem is illustrated by the language ML (Gordon, Milner, and
Wadsworth 1979), whose "polymorphic data types" include templates for
higher-order transformations between data types.  Moreover, data types
for most procedures in ML are never explicitly declared by the
programmer.  Instead, ML includes a "type-inferencing" mechanism that
uses information in the environment to deduce the data types for newly
defined procedures.


File: sicp.info,  Node: 3-5-5,  Prev: 3-5-4,  Up: 3-5

3.5.5 Modularity of Functional Programs and Modularity of Objects
-----------------------------------------------------------------

As we saw in section *Note 3-1-2::, one of the major benefits of
introducing assignment is that we can increase the modularity of our
systems by encapsulating, or "hiding," parts of the state of a large
system within local variables.  Stream models can provide an equivalent
modularity without the use of assignment.  As an illustration, we can
reimplement the Monte Carlo estimation of [pi], which we examined in
section *Note 3-1-2::, from a stream-processing point of view.

   The key modularity issue was that we wished to hide the internal
state of a random-number generator from programs that used random
numbers.  We began with a procedure `rand-update', whose successive
values furnished our supply of random numbers, and used this to produce
a random-number generator:

     (define rand
       (let ((x random-init))
         (lambda ()
           (set! x (rand-update x))
           x)))

   In the stream formulation there is no random-number generator _per
se_, just a stream of random numbers produced by successive calls to
`rand-update':

     (define random-numbers
       (cons-stream random-init
                    (stream-map rand-update random-numbers)))

   We use this to construct the stream of outcomes of the Cesa`ro
experiment performed on consecutive pairs in the `random-numbers'
stream:

     (define cesaro-stream
       (map-successive-pairs (lambda (r1 r2) (= (gcd r1 r2) 1))
                             random-numbers))

     (define (map-successive-pairs f s)
       (cons-stream
        (f (stream-car s) (stream-car (stream-cdr s)))
        (map-successive-pairs f (stream-cdr (stream-cdr s)))))

   The `cesaro-stream' is now fed to a `monte-carlo' procedure, which
produces a stream of estimates of probabilities.  The results are then
converted into a stream of estimates of [pi].  This version of the
program doesn't need a parameter telling how many trials to perform.
Better estimates of [pi] (from performing more experiments) are
obtained by looking farther into the `pi' stream:

     (define (monte-carlo experiment-stream passed failed)
       (define (next passed failed)
         (cons-stream
          (/ passed (+ passed failed))
          (monte-carlo
           (stream-cdr experiment-stream) passed failed)))
       (if (stream-car experiment-stream)
           (next (+ passed 1) failed)
           (next passed (+ failed 1))))

     (define pi
       (stream-map (lambda (p) (sqrt (/ 6 p)))
                   (monte-carlo cesaro-stream 0 0)))

   There is considerable modularity in this approach, because we still
can formulate a general `monte-carlo' procedure that can deal with
arbitrary experiments.  Yet there is no assignment or local state.

     *Exercise 3.81:* *Note Exercise 3-6:: discussed generalizing the
     random-number generator to allow one to reset the random-number
     sequence so as to produce repeatable sequences of "random"
     numbers.  Produce a stream formulation of this same generator that
     operates on an input stream of requests to `generate' a new random
     number or to `reset' the sequence to a specified value and that
     produces the desired stream of random numbers.  Don't use
     assignment in your solution.

     *Exercise 3.82:* Redo *Note Exercise 3-5:: on Monte Carlo
     integration in terms of streams.  The stream version of
     `estimate-integral' will not have an argument telling how many
     trials to perform.  Instead, it will produce a stream of estimates
     based on successively more trials.

A functional-programming view of time
.....................................

Let us now return to the issues of objects and state that were raised
at the beginning of this chapter and examine them in a new light.  We
introduced assignment and mutable objects to provide a mechanism for
modular construction of programs that model systems with state.  We
constructed computational objects with local state variables and used
assignment to modify these variables.  We modeled the temporal behavior
of the objects in the world by the temporal behavior of the
corresponding computational objects.

   Now we have seen that streams provide an alternative way to model
objects with local state.  We can model a changing quantity, such as
the local state of some object, using a stream that represents the time
history of successive states.  In essence, we represent time
explicitly, using streams, so that we decouple time in our simulated
world from the sequence of events that take place during evaluation.
Indeed, because of the presence of `delay' there may be little relation
between simulated time in the model and the order of events during the
evaluation.

   In order to contrast these two approaches to modeling, let us
reconsider the implementation of a "withdrawal processor" that monitors
the balance in a bank account.  In section *Note 3-1-3:: we implemented
a simplified version of such a processor:

     (define (make-simplified-withdraw balance)
       (lambda (amount)
         (set! balance (- balance amount))
         balance))

   Calls to `make-simplified-withdraw' produce computational objects,
each with a local state variable `balance' that is decremented by
successive calls to the object.  The object takes an `amount' as an
argument and returns the new balance.  We can imagine the user of a
bank account typing a sequence of inputs to such an object and
observing the sequence of returned values shown on a display screen.

   Alternatively, we can model a withdrawal processor as a procedure
that takes as input a balance and a stream of amounts to withdraw and
produces the stream of successive balances in the account:

     (define (stream-withdraw balance amount-stream)
       (cons-stream
        balance
        (stream-withdraw (- balance (stream-car amount-stream))
                         (stream-cdr amount-stream))))

   `Stream-withdraw' implements a well-defined mathematical function
whose output is fully determined by its input.  Suppose, however, that
the input `amount-stream' is the stream of successive values typed by
the user and that the resulting stream of balances is displayed.  Then,
from the perspective of the user who is typing values and watching
results, the stream process has the same behavior as the object created
by `make-simplified-withdraw'.  However, with the stream version, there
is no assignment, no local state variable, and consequently none of the
theoretical difficulties that we encountered in section *Note 3-1-3::.
Yet the system has state!

   This is really remarkable.  Even though `stream-withdraw' implements
a well-defined mathematical function whose behavior does not change,
the user's perception here is one of interacting with a system that has
a changing state.  One way to resolve this paradox is to realize that
it is the user's temporal existence that imposes state on the system.
If the user could step back from the interaction and think in terms of
streams of balances rather than individual transactions, the system
would appear stateless.(1)

   From the point of view of one part of a complex process, the other
parts appear to change with time.  They have hidden time-varying local
state.  If we wish to write programs that model this kind of natural
decomposition in our world (as we see it from our viewpoint as a part
of that world) with structures in our computer, we make computational
objects that are not functional--they must change with time.  We model
state with local state variables, and we model the changes of state
with assignments to those variables.  By doing this we make the time of
execution of a computation model time in the world that we are part of,
and thus we get "objects" in our computer.

   Modeling with objects is powerful and intuitive, largely because
this matches the perception of interacting with a world of which we are
part.  However, as we've seen repeatedly throughout this chapter, these
models raise thorny problems of constraining the order of events and of
synchronizing multiple processes.  The possibility of avoiding these
problems has stimulated the development of "functional programming
languages", which do not include any provision for assignment or
mutable data.  In such a language, all procedures implement
well-defined mathematical functions of their arguments, whose behavior
does not change.  The functional approach is extremely attractive for
dealing with concurrent systems.(2)

   On the other hand, if we look closely, we can see time-related
problems creeping into functional models as well.  One particularly
troublesome area arises when we wish to design interactive systems,
especially ones that model interactions between independent entities.
For instance, consider once more the implementation a banking system
that permits joint bank accounts.  In a conventional system using
assignment and objects, we would model the fact that Peter and Paul
share an account by having both Peter and Paul send their transaction
requests to the same bank-account object, as we saw in section *Note
3-1-3::.  From the stream point of view, where there are no "objects"
_per se_, we have already indicated that a bank account can be modeled
as a process that operates on a stream of transaction requests to
produce a stream of responses.  Accordingly, we could model the fact
that Peter and Paul have a joint bank account by merging Peter's stream
of transaction requests with Paul's stream of requests and feeding the
result to the bank-account stream process, as shown in *Note Figure
3-38::.

     *Figure 3.38:* A joint bank account, modeled by merging two
     streams of transaction requests.

          Peter's requests   +---------+     +---------+
          ------------------>|         |     |         |
          Paul's requests    |  merge  |---->| bank    |---->
          ------------------>|         |     | account |
                             +---------+     +---------+

   The trouble with this formulation is in the notion of "merge".  It
will not do to merge the two streams by simply taking alternately one
request from Peter and one request from Paul. Suppose Paul accesses the
account only very rarely.  We could hardly force Peter to wait for Paul
to access the account before he could issue a second transaction.
However such a merge is implemented, it must interleave the two
transaction streams in some way that is constrained by "real time" as
perceived by Peter and Paul, in the sense that, if Peter and Paul meet,
they can agree that certain transactions were processed before the
meeting, and other transactions were processed after the meeting.(3)
This is precisely the same constraint that we had to deal with in
section *Note 3-4-1::, where we found the need to introduce explicit
synchronization to ensure a "correct" order of events in concurrent
processing of objects with state.  Thus, in an attempt to support the
functional style, the need to merge inputs from different agents
reintroduces the same problems that the functional style was meant to
eliminate.

   We began this chapter with the goal of building computational models
whose structure matches our perception of the real world we are trying
to model.  We can model the world as a collection of separate,
time-bound, interacting objects with state, or we can model the world
as a single, timeless, stateless unity.  Each view has powerful
advantages, but neither view alone is completely satisfactory.  A grand
unification has yet to emerge.(4)

   ---------- Footnotes ----------

   (1) Similarly in physics, when we observe a moving particle, we say
that the position (state) of the particle is changing.  However, from
the perspective of the particle's world line in space-time there is no
change involved.

   (2) John Backus, the inventor of Fortran, gave high visibility to
functional programming when he was awarded the ACM Turing award in
1978.  His acceptance speech (Backus 1978) strongly advocated the
functional approach.  A good overview of functional programming is
given in Henderson 1980 and in Darlington, Henderson, and Turner 1982.

   (3) Observe that, for any two streams, there is in general more than
one acceptable order of interleaving.  Thus, technically, "merge" is a
relation rather than a function--the answer is not a deterministic
function of the inputs.  We already mentioned (*Note Footnote 39::)
that nondeterminism is essential when dealing with concurrency.  The
merge relation illustrates the same essential nondeterminism, from the
functional perspective.  In section *Note 4-3::, we will look at
nondeterminism from yet another point of view.

   (4) The object model approximates the world by dividing it into
separate pieces.  The functional model does not modularize along object
boundaries.  The object model is useful when the unshared state of the
"objects" is much larger than the state that they share.  An example of
a place where the object viewpoint fails is quantum mechanics, where
thinking of things as individual particles leads to paradoxes and
confusions.  Unifying the object view with the functional view may have
little to do with programming, but rather with fundamental
epistemological issues.


File: sicp.info,  Node: Chapter 4,  Next: Chapter 5,  Prev: Chapter 3,  Up: Top

4 Metalinguistic Abstraction
****************************

     ... It's in words that the magic is--Abracadabra, Open Sesame, and
     the rest--but the magic words in one story aren't magical in the
     next.  The real magic is to understand which words work, and when,
     and for what; the trick is to learn the trick.

     ... And those words are made from the letters of our alphabet: a
     couple-dozen squiggles we can draw with the pen.  This is the key!
     And the treasure, too, if we can only get our hands on it!  It's
     as if--as if the key to the treasure _is_ the treasure!

     --John Barth, `Chimera'

   In our study of program design, we have seen that expert programmers
control the complexity of their designs with the same general
techniques used by designers of all complex systems.  They combine
primitive elements to form compound objects, they abstract compound
objects to form higher-level building blocks, and they preserve
modularity by adopting appropriate large-scale views of system
structure.  In illustrating these techniques, we have used Lisp as a
language for describing processes and for constructing computational
data objects and processes to model complex phenomena in the real
world.  However, as we confront increasingly complex problems, we will
find that Lisp, or indeed any fixed programming language, is not
sufficient for our needs.  We must constantly turn to new languages in
order to express our ideas more effectively.  Establishing new
languages is a powerful strategy for controlling complexity in
engineering design; we can often enhance our ability to deal with a
complex problem by adopting a new language that enables us to describe
(and hence to think about) the problem in a different way, using
primitives, means of combination, and means of abstraction that are
particularly well suited to the problem at hand.(1)

   Programming is endowed with a multitude of languages.  There are
physical languages, such as the machine languages for particular
computers.  These languages are concerned with the representation of
data and control in terms of individual bits of storage and primitive
machine instructions.  The machine-language programmer is concerned
with using the given hardware to erect systems and utilities for the
efficient implementation of resource-limited computations.  High-level
languages, erected on a machine-language substrate, hide concerns about
the representation of data as collections of bits and the
representation of programs as sequences of primitive instructions.
These languages have means of combination and abstraction, such as
procedure definition, that are appropriate to the larger-scale
organization of systems.

   "Metalinguistic abstraction"--establishing new languages--plays an
important role in all branches of engineering design.  It is
particularly important to computer programming, because in programming
not only can we formulate new languages but we can also implement these
languages by constructing evaluators.  An "evaluator" (or "interpreter")
for a programming language is a procedure that, when applied to an
expression of the language, performs the actions required to evaluate
that expression.

   It is no exaggeration to regard this as the most fundamental idea in
programming:

     The evaluator, which determines the meaning of expressions in a
     programming language, is just another program.

   To appreciate this point is to change our images of ourselves as
programmers.  We come to see ourselves as designers of languages,
rather than only users of languages designed by others.

   In fact, we can regard almost any program as the evaluator for some
language.  For instance, the polynomial manipulation system of section
*Note 2-5-3:: embodies the rules of polynomial arithmetic and
implements them in terms of operations on list-structured data.  If we
augment this system with procedures to read and print polynomial
expressions, we have the core of a special-purpose language for dealing
with problems in symbolic mathematics.  The digital-logic simulator of
section *Note 3-3-4:: and the constraint propagator of section *Note
3-3-5:: are legitimate languages in their own right, each with its own
primitives, means of combination, and means of abstraction.  Seen from
this perspective, the technology for coping with large-scale computer
systems merges with the technology for building new computer languages,
and computer science itself becomes no more (and no less) than the
discipline of constructing appropriate descriptive languages.

   We now embark on a tour of the technology by which languages are
established in terms of other languages.  In this chapter we shall use
Lisp as a base, implementing evaluators as Lisp procedures.  Lisp is
particularly well suited to this task, because of its ability to
represent and manipulate symbolic expressions.  We will take the first
step in understanding how languages are implemented by building an
evaluator for Lisp itself.  The language implemented by our evaluator
will be a subset of the Scheme dialect of Lisp that we use in this
book.  Although the evaluator described in this chapter is written for a
particular dialect of Lisp, it contains the essential structure of an
evaluator for any expression-oriented language designed for writing
programs for a sequential machine.  (In fact, most language processors
contain, deep within them, a little "Lisp" evaluator.)  The evaluator
has been simplified for the purposes of illustration and discussion,
and some features have been left out that would be important to include
in a production-quality Lisp system.  Nevertheless, this simple
evaluator is adequate to execute most of the programs in this book.(2)

   An important advantage of making the evaluator accessible as a Lisp
program is that we can implement alternative evaluation rules by
describing these as modifications to the evaluator program.  One place
where we can use this power to good effect is to gain extra control
over the ways in which computational models embody the notion of time,
which was so central to the discussion in *Note Chapter 3::.  There, we
mitigated some of the complexities of state and assignment by using
streams to decouple the representation of time in the world from time
in the computer.  Our stream programs, however, were sometimes
cumbersome, because they were constrained by the applicative-order
evaluation of Scheme.  In section *Note 4-2::, we'll change the
underlying language to provide for a more elegant approach, by
modifying the evaluator to provide for "normal-order evaluation".

   Section *Note 4-3:: implements a more ambitious linguistic change,
whereby expressions have many values, rather than just a single value.
In this language of "nondeterministic computing", it is natural to
express processes that generate all possible values for expressions and
then search for those values that satisfy certain constraints.  In
terms of models of computation and time, this is like having time
branch into a set of "possible futures" and then searching for
appropriate time lines.  With our nondeterministic evaluator, keeping
track of multiple values and performing searches are handled
automatically by the underlying mechanism of the language.

   In section *Note 4-4:: we implement a "logic-programming" language in
which knowledge is expressed in terms of relations, rather than in
terms of computations with inputs and outputs.  Even though this makes
the language drastically different from Lisp, or indeed from any
conventional language, we will see that the logic-programming evaluator
shares the essential structure of the Lisp evaluator.

* Menu:

* 4-1::              The Metacircular Evaluator
* 4-2::              Variations on a Scheme -- Lazy Evaluation
* 4-3::              Variations on a Scheme -- Nondeterministic Computing
* 4-4::              Logic Programming

   ---------- Footnotes ----------

   (1) The same idea is pervasive throughout all of engineering.  For
example, electrical engineers use many different languages for
describing circuits.  Two of these are the language of electrical "networks"
and the language of electrical "systems".  The network language
emphasizes the physical modeling of devices in terms of discrete
electrical elements.  The primitive objects of the network language are
primitive electrical components such as resistors, capacitors,
inductors, and transistors, which are characterized in terms of
physical variables called voltage and current.  When describing
circuits in the network language, the engineer is concerned with the
physical characteristics of a design.  In contrast, the primitive
objects of the system language are signal-processing modules such as
filters and amplifiers.  Only the functional behavior of the modules is
relevant, and signals are manipulated without concern for their
physical realization as voltages and currents.  The system language is
erected on the network language, in the sense that the elements of
signal-processing systems are constructed from electrical networks.
Here, however, the concerns are with the large-scale organization of
electrical devices to solve a given application problem; the physical
feasibility of the parts is assumed.  This layered collection of
languages is another example of the stratified design technique
illustrated by the picture language of section *Note 2-2-4::.

   (2) The most important features that our evaluator leaves out are
mechanisms for handling errors and supporting debugging.  For a more
extensive discussion of evaluators, see Friedman, Wand, and Haynes
1992, which gives an exposition of programming languages that proceeds
via a sequence of evaluators written in Scheme.


File: sicp.info,  Node: 4-1,  Next: 4-2,  Prev: Chapter 4,  Up: Chapter 4

4.1 The Metacircular Evaluator
==============================

Our evaluator for Lisp will be implemented as a Lisp program.  It may
seem circular to think about evaluating Lisp programs using an
evaluator that is itself implemented in Lisp.  However, evaluation is a
process, so it is appropriate to describe the evaluation process using
Lisp, which, after all, is our tool for describing processes.(1) An
evaluator that is written in the same language that it evaluates is
said to be "metacircular".

   The metacircular evaluator is essentially a Scheme formulation of the
environment model of evaluation described in section *Note 3-2::.
Recall that the model has two basic parts:

  1. To evaluate a combination (a compound expression other than a
     special form), evaluate the subexpressions and then apply the
     value of the operator subexpression to the values of the operand
     subexpressions.

  2. To apply a compound procedure to a set of arguments, evaluate the
     body of the procedure in a new environment.  To construct this
     environment, extend the environment part of the procedure object
     by a frame in which the formal parameters of the procedure are
     bound to the arguments to which the procedure is applied.


   These two rules describe the essence of the evaluation process, a
basic cycle in which expressions to be evaluated in environments are
reduced to procedures to be applied to arguments, which in turn are
reduced to new expressions to be evaluated in new environments, and so
on, until we get down to symbols, whose values are looked up in the
environment, and to primitive procedures, which are applied directly
(see *Note Figure 4-1::).(2) This evaluation cycle will be embodied by
the interplay between the two critical procedures in the evaluator,
`eval' and `apply', which are described in section *Note 4-1-1:: (see
*Note Figure 4-1::).

   The implementation of the evaluator will depend upon procedures that
define the "syntax" of the expressions to be evaluated.  We will use
data abstraction to make the evaluator independent of the
representation of the language.  For example, rather than committing to
a choice that an assignment is to be represented by a list beginning
with the symbol `set!' we use an abstract predicate `assignment?' to
test for an assignment, and we use abstract selectors
`assignment-variable' and `assignment-value' to access the parts of an
assignment.  Implementation of expressions will be described in detail
in section *Note 4-1-2::.  There are also operations, described in
section *Note 4-1-3::, that specify the representation of procedures
and environments.  For example, `make-procedure' constructs compound
procedures, `lookup-variable-value' accesses the values of variables,
and `apply-primitive-procedure' applies a primitive procedure to a
given list of arguments.

* Menu:

* 4-1-1::            The Core of the Evaluator
* 4-1-2::            Representing Expressions
* 4-1-3::            Evaluator Data Structures
* 4-1-4::            Running the Evaluator as a Program
* 4-1-5::            Data as Programs
* 4-1-6::            Internal Definitions
* 4-1-7::            Separating Syntactic Analysis from Execution

   ---------- Footnotes ----------

   (1) Even so, there will remain important aspects of the evaluation
process that are not elucidated by our evaluator.  The most important
of these are the detailed mechanisms by which procedures call other
procedures and return values to their callers.  We will address these
issues in *Note Chapter 5::, where we take a closer look at the
evaluation process by implementing the evaluator as a simple register
machine.

   (2) If we grant ourselves the ability to apply primitives, then what
remains for us to implement in the evaluator?  The job of the evaluator
is not to specify the primitives of the language, but rather to provide
the connective tissue--the means of combination and the means of
abstraction--that binds a collection of primitives to form a language.
Specifically:

   * The evaluator enables us to deal with nested expressions.  For
     example, although simply applying primitives would suffice for
     evaluating the expression `(+ 1 6)', it is not adequate for
     handling `(+ 1 (* 2 3))'.  As far as the primitive procedure `+'
     is concerned, its arguments must be numbers, and it would choke if
     we passed it the expression `(* 2 3)' as an argument.  One
     important role of the evaluator is to choreograph procedure
     composition so that `(* 2 3)' is reduced to 6 before being passed
     as an argument to `+'.

   * The evaluator allows us to use variables.  For example, the
     primitive procedure for addition has no way to deal with
     expressions such as `(+ x 1)'.  We need an evaluator to keep track
     of variables and obtain their values before invoking the primitive
     procedures.

   * The evaluator allows us to define compound procedures.  This
     involves keeping track of procedure definitions, knowing how to
     use these definitions in evaluating expressions, and providing a
     mechanism that enables procedures to accept arguments.

   * The evaluator provides the special forms, which must be evaluated
     differently from procedure calls.



File: sicp.info,  Node: 4-1-1,  Next: 4-1-2,  Prev: 4-1,  Up: 4-1

4.1.1 The Core of the Evaluator
-------------------------------

     *Figure 4.1:* The `eval'-`apply' cycle exposes the essence of a
     computer language.

                                     .,ad88888888baa,
                            _    ,d8P"""        ""9888ba.      _
                           /  .a8"          ,ad88888888888a   |\
                         /   aP'          ,88888888888888888a   \
                        /  ,8"           ,88888888888888888888,  \
                       |  ,8'            (888888888888888888888, |
                      /  ,8'             `8888888888888888888888  \
                      |  8)               `888888888888888888888, |
          Procedure,  |  8                  "88888 Apply 8888888) | Expression
          Arguments   |  8     Eval          `888888888888888888) | Environment
                      |  8)                    "8888888888888888  |
                      \  (b                     "88888888888888'  /
                       | `8,                     8888888888888)  |
                       \  "8a                   ,888888888888)  /
                        \   V8,                 d88888888888"  /
                        _\| `8b,             ,d8888888888P' _/
                               `V8a,       ,ad8888888888P'
                                  ""88888888888888888P"
                                       """"""""""""

                                         [graphic by Normand Veillux, modified]

The evaluation process can be described as the interplay between two
procedures: `eval' and `apply'.

Eval
....

`Eval' takes as arguments an expression and an environment.  It
classifies the expression and directs its evaluation.  `Eval' is
structured as a case analysis of the syntactic type of the expression
to be evaluated.  In order to keep the procedure general, we express
the determination of the type of an expression abstractly, making no
commitment to any particular representation for the various types of
expressions.  Each type of expression has a predicate that tests for it
and an abstract means for selecting its parts.  This "abstract syntax"
makes it easy to see how we can change the syntax of the language by
using the same evaluator, but with a different collection of syntax
procedures.

Primitive expressions

   * For self-evaluating expressions, such as numbers, `eval' returns
     the expression itself.

   * `Eval' must look up variables in the environment to find their
     values.


Special forms

   * For quoted expressions, `eval' returns the expression that was
     quoted.

   * An assignment to (or a definition of) a variable must recursively
     call `eval' to compute the new value to be associated with the
     variable.  The environment must be modified to change (or create)
     the binding of the variable.

   * An `if' expression requires special processing of its parts, so as
     to evaluate the consequent if the predicate is true, and otherwise
     to evaluate the alternative.

   * A `lambda' expression must be transformed into an applicable
     procedure by packaging together the parameters and body specified
     by the `lambda' expression with the environment of the evaluation.

   * A `begin' expression requires evaluating its sequence of
     expressions in the order in which they appear.

   * A case analysis (`cond') is transformed into a nest of `if'
     expressions and then evaluated.


Combinations

   * For a procedure application, `eval' must recursively evaluate the
     operator part and the operands of the combination.  The resulting
     procedure and arguments are passed to `apply', which handles the
     actual procedure application.


   Here is the definition of `eval':

     (define (eval exp env)
       (cond ((self-evaluating? exp) exp)
             ((variable? exp) (lookup-variable-value exp env))
             ((quoted? exp) (text-of-quotation exp))
             ((assignment? exp) (eval-assignment exp env))
             ((definition? exp) (eval-definition exp env))
             ((if? exp) (eval-if exp env))
             ((lambda? exp)
              (make-procedure (lambda-parameters exp)
                              (lambda-body exp)
                              env))
             ((begin? exp)
              (eval-sequence (begin-actions exp) env))
             ((cond? exp) (eval (cond->if exp) env))
             ((application? exp)
              (apply (eval (operator exp) env)
                     (list-of-values (operands exp) env)))
             (else
              (error "Unknown expression type -- EVAL" exp))))

   For clarity, `eval' has been implemented as a case analysis using
`cond'.  The disadvantage of this is that our procedure handles only a
few distinguishable types of expressions, and no new ones can be
defined without editing the definition of `eval'.  In most Lisp
implementations, dispatching on the type of an expression is done in a
data-directed style.  This allows a user to add new types of
expressions that `eval' can distinguish, without modifying the
definition of `eval' itself.  (See *Note Exercise 4-3::.)

Apply
.....

`Apply' takes two arguments, a procedure and a list of arguments to
which the procedure should be applied.  `Apply' classifies procedures
into two kinds: It calls `apply-primitive-procedure' to apply
primitives; it applies compound procedures by sequentially evaluating
the expressions that make up the body of the procedure.  The
environment for the evaluation of the body of a compound procedure is
constructed by extending the base environment carried by the procedure
to include a frame that binds the parameters of the procedure to the
arguments to which the procedure is to be applied.  Here is the
definition of `apply':

     (define (apply procedure arguments)
       (cond ((primitive-procedure? procedure)
              (apply-primitive-procedure procedure arguments))
             ((compound-procedure? procedure)
              (eval-sequence
                (procedure-body procedure)
                (extend-environment
                  (procedure-parameters procedure)
                  arguments
                  (procedure-environment procedure))))
             (else
              (error
               "Unknown procedure type -- APPLY" procedure))))

Procedure arguments
...................

When `eval' processes a procedure application, it uses `list-of-values'
to produce the list of arguments to which the procedure is to be
applied. `List-of-values' takes as an argument the operands of the
combination.  It evaluates each operand and returns a list of the
corresponding values:(1)

     (define (list-of-values exps env)
       (if (no-operands? exps)
           '()
           (cons (eval (first-operand exps) env)
                 (list-of-values (rest-operands exps) env))))

Conditionals
............

`Eval-if' evaluates the predicate part of an `if' expression in the
given environment.  If the result is true, `eval-if' evaluates the
consequent, otherwise it evaluates the alternative:

     (define (eval-if exp env)
       (if (true? (eval (if-predicate exp) env))
           (eval (if-consequent exp) env)
           (eval (if-alternative exp) env)))

   The use of `true?' in `eval-if' highlights the issue of the
connection between an implemented language and an implementation
language.  The `if-predicate' is evaluated in the language being
implemented and thus yields a value in that language.  The interpreter
predicate `true?' translates that value into a value that can be tested
by the `if' in the implementation language: The metacircular
representation of truth might not be the same as that of the underlying
Scheme.(2)

Sequences
.........

`Eval-sequence' is used by `apply' to evaluate the sequence of
expressions in a procedure body and by `eval' to evaluate the sequence
of expressions in a `begin' expression.  It takes as arguments a
sequence of expressions and an environment, and evaluates the
expressions in the order in which they occur.  The value returned is
the value of the final expression.

     (define (eval-sequence exps env)
       (cond ((last-exp? exps) (eval (first-exp exps) env))
             (else (eval (first-exp exps) env)
                   (eval-sequence (rest-exps exps) env))))

Assignments and definitions
...........................

The following procedure handles assignments to variables.  It calls
`eval' to find the value to be assigned and transmits the variable and
the resulting value to `set-variable-value!' to be installed in the
designated environment.

     (define (eval-assignment exp env)
       (set-variable-value! (assignment-variable exp)
                            (eval (assignment-value exp) env)
                            env)
       'ok)

Definitions of variables are handled in a similar manner.(3)

     (define (eval-definition exp env)
       (define-variable! (definition-variable exp)
                         (eval (definition-value exp) env)
                         env)
       'ok)

   We have chosen here to return the symbol `ok' as the value of an
assignment or a definition.(4)

     *Exercise 4.1:* Notice that we cannot tell whether the
     metacircular evaluator evaluates operands from left to right or
     from right to left.  Its evaluation order is inherited from the
     underlying Lisp: If the arguments to `cons' in `list-of-values'
     are evaluated from left to right, then `list-of-values' will
     evaluate operands from left to right; and if the arguments to
     `cons' are evaluated from right to left, then `list-of-values'
     will evaluate operands from right to left.

     Write a version of `list-of-values' that evaluates operands from
     left to right regardless of the order of evaluation in the
     underlying Lisp.  Also write a version of `list-of-values' that
     evaluates operands from right to left.

   ---------- Footnotes ----------

   (1) We could have simplified the `application?' clause in `eval' by
using `map' (and stipulating that `operands' returns a list) rather
than writing an explicit `list-of-values' procedure.  We chose not to
use `map' here to emphasize the fact that the evaluator can be
implemented without any use of higher-order procedures (and thus could
be written in a language that doesn't have higher-order procedures),
even though the language that it supports will include higher-order
procedures.

   (2) In this case, the language being implemented and the
implementation language are the same.  Contemplation of the meaning of
`true?' here yields expansion of consciousness without the abuse of
substance.

   (3) This implementation of `define' ignores a subtle issue in the
handling of internal definitions, although it works correctly in most
cases.  We will see what the problem is and how to solve it in section
*Note 4-1-6::.

   (4) As we said when we introduced `define' and `set!', these values
are implementation-dependent in Scheme--that is, the implementor can
choose what value to return.


File: sicp.info,  Node: 4-1-2,  Next: 4-1-3,  Prev: 4-1-1,  Up: 4-1

4.1.2 Representing Expressions
------------------------------

The evaluator is reminiscent of the symbolic differentiation program
discussed in section *Note 2-3-2::.  Both programs operate on symbolic
expressions.  In both programs, the result of operating on a compound
expression is determined by operating recursively on the pieces of the
expression and combining the results in a way that depends on the type
of the expression.  In both programs we used data abstraction to
decouple the general rules of operation from the details of how
expressions are represented.  In the differentiation program this meant
that the same differentiation procedure could deal with algebraic
expressions in prefix form, in infix form, or in some other form.  For
the evaluator, this means that the syntax of the language being
evaluated is determined solely by the procedures that classify and
extract pieces of expressions.

   Here is the specification of the syntax of our language:

   * The only self-evaluating items are numbers and strings:

          (define (self-evaluating? exp)
            (cond ((number? exp) true)
                  ((string? exp) true)
                  (else false)))

   * Variables are represented by symbols:

          (define (variable? exp) (symbol? exp))

   * Quotations have the form `(quote <TEXT-OF-QUOTATION>)':(1)

          (define (quoted? exp)
            (tagged-list? exp 'quote))

          (define (text-of-quotation exp) (cadr exp))

     `Quoted?' is defined in terms of the procedure `tagged-list?',
     which identifies lists beginning with a designated symbol:

          (define (tagged-list? exp tag)
            (if (pair? exp)
                (eq? (car exp) tag)
                false))

   * Assignments have the form `(set! <VAR> <VALUE>)':

          (define (assignment? exp)
            (tagged-list? exp 'set!))

          (define (assignment-variable exp) (cadr exp))

          (define (assignment-value exp) (caddr exp))

   * Definitions have the form

          (define <VAR> <VALUE>)

     or the form

          (define (<VAR> <PARAMETER_1> ... <PARAMETER_N>)
            <BODY>)

     The latter form (standard procedure definition) is syntactic sugar
     for

          (define <VAR>
            (lambda (<PARAMETER_1> ... <PARAMETER_N>)
              <BODY>))

     The corresponding syntax procedures are the following:

          (define (definition? exp)
            (tagged-list? exp 'define))

          (define (definition-variable exp)
            (if (symbol? (cadr exp))
                (cadr exp)
                (caadr exp)))

          (define (definition-value exp)
            (if (symbol? (cadr exp))
                (caddr exp)
                (make-lambda (cdadr exp)   ; formal parameters
                             (cddr exp)))) ; body

   * `Lambda' expressions are lists that begin with the symbol `lambda':

          (define (lambda? exp) (tagged-list? exp 'lambda))

          (define (lambda-parameters exp) (cadr exp))

          (define (lambda-body exp) (cddr exp))

     We also provide a constructor for `lambda' expressions, which is
     used by `definition-value', above:

          (define (make-lambda parameters body)
            (cons 'lambda (cons parameters body)))

   * Conditionals begin with `if' and have a predicate, a consequent,
     and an (optional) alternative.  If the expression has no
     alternative part, we provide `false' as the alternative.(2)

          (define (if? exp) (tagged-list? exp 'if))

          (define (if-predicate exp) (cadr exp))

          (define (if-consequent exp) (caddr exp))

          (define (if-alternative exp)
            (if (not (null? (cdddr exp)))
                (cadddr exp)
                'false))

     We also provide a constructor for `if' expressions, to be used by
     `cond->if' to transform `cond' expressions into `if' expressions:

          (define (make-if predicate consequent alternative)
            (list 'if predicate consequent alternative))

   * `Begin' packages a sequence of expressions into a single
     expression.  We include syntax operations on `begin' expressions
     to extract the actual sequence from the `begin' expression, as
     well as selectors that return the first expression and the rest of
     the expressions in the sequence.(3)

          (define (begin? exp) (tagged-list? exp 'begin))

          (define (begin-actions exp) (cdr exp))

          (define (last-exp? seq) (null? (cdr seq)))

          (define (first-exp seq) (car seq))

          (define (rest-exps seq) (cdr seq))

     We also include a constructor `sequence->exp' (for use by
     `cond->if') that transforms a sequence into a single expression,
     using `begin' if necessary:

          (define (sequence->exp seq)
            (cond ((null? seq) seq)
                  ((last-exp? seq) (first-exp seq))
                  (else (make-begin seq))))

          (define (make-begin seq) (cons 'begin seq))

   * A procedure application is any compound expression that is not one
     of the above expression types.  The `car' of the expression is the
     operator, and the `cdr' is the list of operands:

          (define (application? exp) (pair? exp))

          (define (operator exp) (car exp))

          (define (operands exp) (cdr exp))

          (define (no-operands? ops) (null? ops))

          (define (first-operand ops) (car ops))

          (define (rest-operands ops) (cdr ops))


Derived expressions
...................

Some special forms in our language can be defined in terms of
expressions involving other special forms, rather than being
implemented directly.  One example is `cond', which can be implemented
as a nest of `if' expressions.  For example, we can reduce the problem
of evaluating the expression

     (cond ((> x 0) x)
           ((= x 0) (display 'zero) 0)
           (else (- x)))

to the problem of evaluating the following expression involving `if' and
`begin' expressions:

     (if (> x 0)
         x
         (if (= x 0)
             (begin (display 'zero)
                    0)
             (- x)))

   Implementing the evaluation of `cond' in this way simplifies the
evaluator because it reduces the number of special forms for which the
evaluation process must be explicitly specified.

   We include syntax procedures that extract the parts of a `cond'
expression, and a procedure `cond->if' that transforms `cond'
expressions into `if' expressions.  A case analysis begins with `cond'
and has a list of predicate-action clauses.  A clause is an `else'
clause if its predicate is the symbol `else'.(4)

     (define (cond? exp) (tagged-list? exp 'cond))

     (define (cond-clauses exp) (cdr exp))

     (define (cond-else-clause? clause)
       (eq? (cond-predicate clause) 'else))

     (define (cond-predicate clause) (car clause))

     (define (cond-actions clause) (cdr clause))

     (define (cond->if exp)
       (expand-clauses (cond-clauses exp)))

     (define (expand-clauses clauses)
       (if (null? clauses)
           'false                          ; no `else' clause
           (let ((first (car clauses))
                 (rest (cdr clauses)))
             (if (cond-else-clause? first)
                 (if (null? rest)
                     (sequence->exp (cond-actions first))
                     (error "ELSE clause isn't last -- COND->IF"
                            clauses))
                 (make-if (cond-predicate first)
                          (sequence->exp (cond-actions first))
                          (expand-clauses rest))))))

   Expressions (such as `cond') that we choose to implement as syntactic
transformations are called "derived expressions".  `Let' expressions
are also derived expressions (see *Note Exercise 4-6::).(5)

     *Exercise 4.2:* Louis Reasoner plans to reorder the `cond' clauses
     in `eval' so that the clause for procedure applications appears
     before the clause for assignments.  He argues that this will make
     the interpreter more efficient: Since programs usually contain more
     applications than assignments, definitions, and so on, his
     modified `eval' will usually check fewer clauses than the original
     `eval' before identifying the type of an expression.

       a. What is wrong with Louis's plan?  (Hint: What will Louis's
          evaluator do with the expression `(define x 3)'?)

       b. Louis is upset that his plan didn't work.  He is willing to
          go to any lengths to make his evaluator recognize procedure
          applications before it checks for most other kinds of
          expressions.  Help him by changing the syntax of the
          evaluated language so that procedure applications start with
          `call'.  For example, instead of `(factorial 3)' we will now
          have to write `(call factorial 3)' and instead of `(+ 1 2)'
          we will have to write `(call + 1 2)'.


     *Exercise 4.3:* Rewrite `eval' so that the dispatch is done in
     data-directed style.  Compare this with the data-directed
     differentiation procedure of *Note Exercise 2-73::.  (You may use
     the `car' of a compound expression as the type of the expression,
     as is appropriate for the syntax implemented in this section.)

     *Exercise 4.4:* Recall the definitions of the special forms `and'
     and `or' from *Note Chapter 1:::

        * `and': The expressions are evaluated from left to right.  If
          any expression evaluates to false, false is returned; any
          remaining expressions are not evaluated.  If all the
          expressions evaluate to true values, the value of the last
          expression is returned.  If there are no expressions then
          true is returned.

        * `or': The expressions are evaluated from left to right.  If
          any expression evaluates to a true value, that value is
          returned; any remaining expressions are not evaluated.  If
          all expressions evaluate to false, or if there are no
          expressions, then false is returned.


     Install `and' and `or' as new special forms for the evaluator by
     defining appropriate syntax procedures and evaluation procedures
     `eval-and' and `eval-or'.  Alternatively, show how to implement
     `and' and `or' as derived expressions.

     *Exercise 4.5:* Scheme allows an additional syntax for `cond'
     clauses, `(<TEST> => <RECIPIENT>)'.  If <TEST> evaluates to a true
     value, then <RECIPIENT> is evaluated.  Its value must be a
     procedure of one argument; this procedure is then invoked on the
     value of the <TEST>, and the result is returned as the value of
     the `cond' expression.  For example

          (cond ((assoc 'b '((a 1) (b 2))) => cadr)
                (else false))

     returns 2.  Modify the handling of `cond' so that it supports this
     extended syntax.

     *Exercise 4.6:* `Let' expressions are derived expressions, because

          (let ((<VAR_1> <EXP_1>) ... (<VAR_N> <EXP_N>))
            <BODY>)

     is equivalent to

          ((lambda (<VAR_1> ... <VAR_N>)
             <BODY>)
           <EXP_1>
           ...
           <EXP_N>)

     Implement a syntactic transformation `let->combination' that
     reduces evaluating `let' expressions to evaluating combinations of
     the type shown above, and add the appropriate clause to `eval' to
     handle `let' expressions.

     *Exercise 4.7:* `Let*' is similar to `let', except that the
     bindings of the `let' variables are performed sequentially from
     left to right, and each binding is made in an environment in which
     all of the preceding bindings are visible.  For example

          (let* ((x 3)
                 (y (+ x 2))
                 (z (+ x y 5)))
            (* x z))

     returns 39.  Explain how a `let*' expression can be rewritten as a
     set of nested `let' expressions, and write a procedure
     `let*->nested-lets' that performs this transformation.  If we have
     already implemented `let' (*Note Exercise 4-6::) and we want to
     extend the evaluator to handle `let*', is it sufficient to add a
     clause to `eval' whose action is

          (eval (let*->nested-lets exp) env)

     or must we explicitly expand `let*' in terms of non-derived
     expressions?

     *Exercise 4.8:* "Named `let'" is a variant of `let' that has the
     form

          (let <VAR> <BINDINGS> <BODY>)

     The <BINDINGS> and <BODY> are just as in ordinary `let', except
     that <VAR> is bound within <BODY> to a procedure whose body is
     <BODY> and whose parameters are the variables in the <BINDINGS>.
     Thus, one can repeatedly execute the <BODY> by invoking the
     procedure named <VAR>.  For example, the iterative Fibonacci
     procedure (section *Note 1-2-2::) can be rewritten using named
     `let' as follows:

          (define (fib n)
            (let fib-iter ((a 1)
                           (b 0)
                           (count n))
              (if (= count 0)
                  b
                  (fib-iter (+ a b) a (- count 1)))))

     Modify `let->combination' of *Note Exercise 4-6:: to also support
     named `let'.

     *Exercise 4.9:* Many languages support a variety of iteration
     constructs, such as `do', `for', `while', and `until'.  In Scheme,
     iterative processes can be expressed in terms of ordinary
     procedure calls, so special iteration constructs provide no
     essential gain in computational power.  On the other hand, such
     constructs are often convenient.  Design some iteration
     constructs, give examples of their use, and show how to implement
     them as derived expressions.

     *Exercise 4.10:* By using data abstraction, we were able to write
     an `eval' procedure that is independent of the particular syntax
     of the language to be evaluated.  To illustrate this, design and
     implement a new syntax for Scheme by modifying the procedures in
     this section, without changing `eval' or `apply'.

   ---------- Footnotes ----------

   (1) As mentioned in section *Note 2-3-1::, the evaluator sees a
quoted expression as a list beginning with `quote', even if the
expression is typed with the quotation mark.  For example, the
expression `'a' would be seen by the evaluator as `(quote a)'.  See
*Note Exercise 2-55::.

   (2) The value of an `if' expression when the predicate is false and
there is no alternative is unspecified in Scheme; we have chosen here
to make it false.  We will support the use of the variables `true' and
`false' in expressions to be evaluated by binding them in the global
environment.  See section *Note 4-1-4::.

   (3) These selectors for a list of expressions--and the corresponding
ones for a list of operands--are not intended as a data abstraction.
They are introduced as mnemonic names for the basic list operations in
order to make it easier to understand the explicit-control evaluator in
section *Note 5-4::.

   (4) The value of a `cond' expression when all the predicates are
false and there is no `else' clause is unspecified in Scheme; we have
chosen here to make it false.

   (5) Practical Lisp systems provide a mechanism that allows a user to
add new derived expressions and specify their implementation as
syntactic transformations without modifying the evaluator.  Such a
user-defined transformation is called a "macro".  Although it is easy
to add an elementary mechanism for defining macros, the resulting
language has subtle name-conflict problems.  There has been much
research on mechanisms for macro definition that do not cause these
difficulties.  See, for example, Kohlbecker 1986, Clinger and Rees
1991, and Hanson 1991.


File: sicp.info,  Node: 4-1-3,  Next: 4-1-4,  Prev: 4-1-2,  Up: 4-1

4.1.3 Evaluator Data Structures
-------------------------------

In addition to defining the external syntax of expressions, the
evaluator implementation must also define the data structures that the
evaluator manipulates internally, as part of the execution of a
program, such as the representation of procedures and environments and
the representation of true and false.

Testing of predicates
.....................

For conditionals, we accept anything to be true that is not the explicit
`false' object.

     (define (true? x)
       (not (eq? x false)))

     (define (false? x)
       (eq? x false))

Representing procedures
.......................

To handle primitives, we assume that we have available the following
procedures:

   * `(apply-primitive-procedure <PROC> <ARGS>)'

     applies the given primitive procedure to the argument values in
     the list <ARGS> and returns the result of the application.

   * `(primitive-procedure? <PROC>)'

     tests whether <PROC> is a primitive procedure.


   These mechanisms for handling primitives are further described in
section *Note 4-1-4::.

   Compound procedures are constructed from parameters, procedure
bodies, and environments using the constructor `make-procedure':

     (define (make-procedure parameters body env)
       (list 'procedure parameters body env))

     (define (compound-procedure? p)
       (tagged-list? p 'procedure))

     (define (procedure-parameters p) (cadr p))

     (define (procedure-body p) (caddr p))

     (define (procedure-environment p) (cadddr p))

Operations on Environments
..........................

The evaluator needs operations for manipulating environments.  As
explained in section *Note 3-2::, an environment is a sequence of
frames, where each frame is a table of bindings that associate
variables with their corresponding values.  We use the following
operations for manipulating environments:

   * `(lookup-variable-value <VAR> <ENV>)'

     returns the value that is bound to the symbol <VAR> in the
     environment <ENV>, or signals an error if the variable is unbound.

   * `(extend-environment <VARIABLES> <VALUES> <BASE-ENV>)'

     returns a new environment, consisting of a new frame in which the
     symbols in the list <VARIABLES> are bound to the corresponding
     elements in the list <VALUES>, where the enclosing environment is
     the environment <BASE-ENV>.

   * `(define-variable! <VAR> <VALUE> <ENV>)'

     adds to the first frame in the environment <ENV> a new binding that
     associates the variable <VAR> with the value <VALUE>.

   * `(set-variable-value! <VAR> <VALUE> <ENV>)'

     changes the binding of the variable <VAR> in the environment <ENV>
     so that the variable is now bound to the value <VALUE>, or signals
     an error if the variable is unbound.


   To implement these operations we represent an environment as a list
of frames.  The enclosing environment of an environment is the `cdr' of
the list.  The empty environment is simply the empty list.

     (define (enclosing-environment env) (cdr env))

     (define (first-frame env) (car env))

     (define the-empty-environment '())

   Each frame of an environment is represented as a pair of lists: a
list of the variables bound in that frame and a list of the associated
values.(1)

     (define (make-frame variables values)
       (cons variables values))

     (define (frame-variables frame) (car frame))

     (define (frame-values frame) (cdr frame))

     (define (add-binding-to-frame! var val frame)
       (set-car! frame (cons var (car frame)))
       (set-cdr! frame (cons val (cdr frame))))

   To extend an environment by a new frame that associates variables
with values, we make a frame consisting of the list of variables and
the list of values, and we adjoin this to the environment.  We signal
an error if the number of variables does not match the number of values.

     (define (extend-environment vars vals base-env)
       (if (= (length vars) (length vals))
           (cons (make-frame vars vals) base-env)
           (if (< (length vars) (length vals))
               (error "Too many arguments supplied" vars vals)
               (error "Too few arguments supplied" vars vals))))

   To look up a variable in an environment, we scan the list of
variables in the first frame.  If we find the desired variable, we
return the corresponding element in the list of values.  If we do not
find the variable in the current frame, we search the enclosing
environment, and so on.  If we reach the empty environment, we signal
an "unbound variable" error.

     (define (lookup-variable-value var env)
       (define (env-loop env)
         (define (scan vars vals)
           (cond ((null? vars)
                  (env-loop (enclosing-environment env)))
                 ((eq? var (car vars))
                  (car vals))
                 (else (scan (cdr vars) (cdr vals)))))
         (if (eq? env the-empty-environment)
             (error "Unbound variable" var)
             (let ((frame (first-frame env)))
               (scan (frame-variables frame)
                     (frame-values frame)))))
       (env-loop env))

   To set a variable to a new value in a specified environment, we scan
for the variable, just as in `lookup-variable-value', and change the
corresponding value when we find it.

     (define (set-variable-value! var val env)
       (define (env-loop env)
         (define (scan vars vals)
           (cond ((null? vars)
                  (env-loop (enclosing-environment env)))
                 ((eq? var (car vars))
                  (set-car! vals val))
                 (else (scan (cdr vars) (cdr vals)))))
         (if (eq? env the-empty-environment)
             (error "Unbound variable -- SET!" var)
             (let ((frame (first-frame env)))
               (scan (frame-variables frame)
                     (frame-values frame)))))
       (env-loop env))

   To define a variable, we search the first frame for a binding for
the variable, and change the binding if it exists (just as in
`set-variable-value!').  If no such binding exists, we adjoin one to
the first frame.

     (define (define-variable! var val env)
       (let ((frame (first-frame env)))
         (define (scan vars vals)
           (cond ((null? vars)
                  (add-binding-to-frame! var val frame))
                 ((eq? var (car vars))
                  (set-car! vals val))
                 (else (scan (cdr vars) (cdr vals)))))
         (scan (frame-variables frame)
               (frame-values frame))))

   The method described here is only one of many plausible ways to
represent environments.  Since we used data abstraction to isolate the
rest of the evaluator from the detailed choice of representation, we
could change the environment representation if we wanted to.  (See
*Note Exercise 4-11::.)  In a production-quality Lisp system, the speed
of the evaluator's environment operations--especially that of variable
lookup--has a major impact on the performance of the system.  The
representation described here, although conceptually simple, is not
efficient and would not ordinarily be used in a production system.(2)

     *Exercise 4.11:* Instead of representing a frame as a pair of
     lists, we can represent a frame as a list of bindings, where each
     binding is a name-value pair.  Rewrite the environment operations
     to use this alternative representation.

     *Exercise 4.12:* The procedures `set-variable-value!',
     `define-variable!', and `lookup-variable-value' can be expressed
     in terms of more abstract procedures for traversing the
     environment structure.  Define abstractions that capture the
     common patterns and redefine the three procedures in terms of these
     abstractions.

     *Exercise 4.13:* Scheme allows us to create new bindings for
     variables by means of `define', but provides no way to get rid of
     bindings.  Implement for the evaluator a special form
     `make-unbound!' that removes the binding of a given symbol from the
     environment in which the `make-unbound!' expression is evaluated.
     This problem is not completely specified.  For example, should we
     remove only the binding in the first frame of the environment?
     Complete the specification and justify any choices you make.

   ---------- Footnotes ----------

   (1) Frames are not really a data abstraction in the following code:
`Set-variable-value!' and `define-variable!' use `set-car!'  to
directly modify the values in a frame.  The purpose of the frame
procedures is to make the environment-manipulation procedures easy to
read.

   (2) The drawback of this representation (as well as the variant in
*Note Exercise 4-11::) is that the evaluator may have to search through
many frames in order to find the binding for a given variable.  (Such
an approach is referred to as "deep binding".)  One way to avoid this
inefficiency is to make use of a strategy called "lexical addressing",
which will be discussed in section *Note 5-5-6::.


File: sicp.info,  Node: 4-1-4,  Next: 4-1-5,  Prev: 4-1-3,  Up: 4-1

4.1.4 Running the Evaluator as a Program
----------------------------------------

Given the evaluator, we have in our hands a description (expressed in
Lisp) of the process by which Lisp expressions are evaluated.  One
advantage of expressing the evaluator as a program is that we can run
the program.  This gives us, running within Lisp, a working model of
how Lisp itself evaluates expressions.  This can serve as a framework
for experimenting with evaluation rules, as we shall do later in this
chapter.

   Our evaluator program reduces expressions ultimately to the
application of primitive procedures.  Therefore, all that we need to
run the evaluator is to create a mechanism that calls on the underlying
Lisp system to model the application of primitive procedures.

   There must be a binding for each primitive procedure name, so that
when `eval' evaluates the operator of an application of a primitive, it
will find an object to pass to `apply'.  We thus set up a global
environment that associates unique objects with the names of the
primitive procedures that can appear in the expressions we will be
evaluating.  The global environment also includes bindings for the
symbols `true' and `false', so that they can be used as variables in
expressions to be evaluated.

     (define (setup-environment)
       (let ((initial-env
              (extend-environment (primitive-procedure-names)
                                  (primitive-procedure-objects)
                                  the-empty-environment)))
         (define-variable! 'true true initial-env)
         (define-variable! 'false false initial-env)
         initial-env))

     (define the-global-environment (setup-environment))

   It does not matter how we represent the primitive procedure objects,
so long as `apply' can identify and apply them by using the procedures
`primitive-procedure?' and `apply-primitive-procedure'.  We have chosen
to represent a primitive procedure as a list beginning with the symbol
`primitive' and containing a procedure in the underlying Lisp that
implements that primitive.

     (define (primitive-procedure? proc)
       (tagged-list? proc 'primitive))

     (define (primitive-implementation proc) (cadr proc))

   `Setup-environment' will get the primitive names and implementation
procedures from a list:(1)

     (define primitive-procedures
       (list (list 'car car)
             (list 'cdr cdr)
             (list 'cons cons)
             (list 'null? null?)
             <MORE PRIMITIVES>
             ))

     (define (primitive-procedure-names)
       (map car
            primitive-procedures))

     (define (primitive-procedure-objects)
       (map (lambda (proc) (list 'primitive (cadr proc)))
            primitive-procedures))

   To apply a primitive procedure, we simply apply the implementation
procedure to the arguments, using the underlying Lisp system:(2)

     (define (apply-primitive-procedure proc args)
       (apply-in-underlying-scheme
        (primitive-implementation proc) args))

   For convenience in running the metacircular evaluator, we provide a "driver
loop" that models the read-eval-print loop of the underlying Lisp
system.  It prints a "prompt", reads an input expression, evaluates
this expression in the global environment, and prints the result.  We
precede each printed result by an "output prompt" so as to distinguish
the value of the expression from other output that may be printed.(3)

     (define input-prompt ";;; M-Eval input:")
     (define output-prompt ";;; M-Eval value:")

     (define (driver-loop)
       (prompt-for-input input-prompt)
       (let ((input (read)))
         (let ((output (eval input the-global-environment)))
           (announce-output output-prompt)
           (user-print output)))
       (driver-loop))

     (define (prompt-for-input string)
       (newline) (newline) (display string) (newline))

     (define (announce-output string)
       (newline) (display string) (newline))

   We use a special printing procedure, `user-print', to avoid printing
the environment part of a compound procedure, which may be a very long
list (or may even contain cycles).

     (define (user-print object)
       (if (compound-procedure? object)
           (display (list 'compound-procedure
                          (procedure-parameters object)
                          (procedure-body object)
                          '<procedure-env>))
           (display object)))

   Now all we need to do to run the evaluator is to initialize the
global environment and start the driver loop.  Here is a sample
interaction:

     (define the-global-environment (setup-environment))

     (driver-loop)

     ;;; M-Eval input:
     (define (append x y)
       (if (null? x)
           y
           (cons (car x)
                 (append (cdr x) y))))
     ;;; M-Eval value:
     ok

     ;;; M-Eval input:
     (append '(a b c) '(d e f))
     ;;; M-Eval value:
     (a b c d e f)

     *Exercise 4.14:* Eva Lu Ator and Louis Reasoner are each
     experimenting with the metacircular evaluator.  Eva types in the
     definition of `map', and runs some test programs that use it.
     They work fine.  Louis, in contrast, has installed the system
     version of `map' as a primitive for the metacircular evaluator.
     When he tries it, things go terribly wrong.  Explain why Louis's
     `map' fails even though Eva's works.

   ---------- Footnotes ----------

   (1) Any procedure defined in the underlying Lisp can be used as a
primitive for the metacircular evaluator.  The name of a primitive
installed in the evaluator need not be the same as the name of its
implementation in the underlying Lisp; the names are the same here
because the metacircular evaluator implements Scheme itself.  Thus, for
example, we could put `(list 'first car)' or `(list 'square (lambda (x)
(* x x)))' in the list of `primitive-procedures'.

   (2) `Apply-in-underlying-scheme' is the `apply' procedure we have
used in earlier chapters.  The metacircular evaluator's `apply'
procedure (section *Note 4-1-1::) models the working of this primitive.
Having two different things called `apply' leads to a technical
problem in running the metacircular evaluator, because defining the
metacircular evaluator's `apply' will mask the definition of the
primitive.  One way around this is to rename the metacircular `apply' to
avoid conflict with the name of the primitive procedure.  We have
assumed instead that we have saved a reference to the underlying
`apply' by doing

     (define apply-in-underlying-scheme apply)

before defining the metacircular `apply'.  This allows us to access the
original version of `apply' under a different name.

   (3) The primitive procedure `read' waits for input from the user,
and returns the next complete expression that is typed.  For example,
if the user types `(+ 23 x)', `read' returns a three-element list
containing the symbol `+', the number 23, and the symbol `x'.  If the
user types `'x', `read' returns a two-element list containing the
symbol `quote' and the symbol `x'.


File: sicp.info,  Node: 4-1-5,  Next: 4-1-6,  Prev: 4-1-4,  Up: 4-1

4.1.5 Data as Programs
----------------------

In thinking about a Lisp program that evaluates Lisp expressions, an
analogy might be helpful.  One operational view of the meaning of a
program is that a program is a description of an abstract (perhaps
infinitely large) machine.  For example, consider the familiar program
to compute factorials:

     (define (factorial n)
       (if (= n 1)
           1
           (* (factorial (- n 1)) n)))

   We may regard this program as the description of a machine
containing parts that decrement, multiply, and test for equality,
together with a two-position switch and another factorial machine. (The
factorial machine is infinite because it contains another factorial
machine within it.)  *Note Figure 4-2:: is a flow diagram for the
factorial machine, showing how the parts are wired together.

     *Figure 4.2:* The factorial program, viewed as an abstract machine.

              +-----------------------------------+
              | factorial                   |1    |
              |              |1             V     |
              |              |           +-----+  |
              |              V           | #   |  |
              |           +-----+        |     |  |
          6 --------*-----|  =  |------->|   #-+-----> 720
              |     |     +-----+        |  /  |  |
              |     |                    | #   |  |
              |     |                    +-----+  |
              |     |                       ^     |
              |     |                       |     |
              |     |                    +--+--+  |
              |     *------------------->|  *  |  |
              |     |                    +-----+  |
              |     V                       ^     |
              |  +-----+    +-----------+   |     |
              |  |  -  +--->| factorial +---+     |
              |  +-----+    +-----------+         |
              |     ^                             |
              |     |1                            |
              +-----------------------------------+

   In a similar way, we can regard the evaluator as a very special
machine that takes as input a description of a machine.  Given this
input, the evaluator configures itself to emulate the machine
described.  For example, if we feed our evaluator the definition of
`factorial', as shown in *Note Figure 4-3::, the evaluator will be able
to compute factorials.

     *Figure 4.3:* The evaluator emulating a factorial machine.

                             +--------+
                      6 ---->|  eval  |----> 720
                             +--------+
                                 /
                       . . .    /  . . .
                 . . .       ../. .      .
               .                           ..
              .   (define (factorial n)      . . .
             .      (if (= n 1)                   . .
              .         1                            .
              .         (* (factorial (- n 1)) n)))   .
                . .                       . .        .
                    . .  . .      . . . .     . . . .
                             . ..

   From this perspective, our evaluator is seen to be a machine
"universal machine".  It mimics other machines when these are described
as Lisp programs.(1) This is striking. Try to imagine an analogous
evaluator for electrical circuits.  This would be a circuit that takes
as input a signal encoding the plans for some other circuit, such as a
filter.  Given this input, the circuit evaluator would then behave like
a filter with the same description.  Such a universal electrical
circuit is almost unimaginably complex.  It is remarkable that the
program evaluator is a rather simple program.(2)

   Another striking aspect of the evaluator is that it acts as a bridge
between the data objects that are manipulated by our programming
language and the programming language itself.  Imagine that the
evaluator program (implemented in Lisp) is running, and that a user is
typing expressions to the evaluator and observing the results.  From
the perspective of the user, an input expression such as `(* x x)' is
an expression in the programming language, which the evaluator should
execute.  From the perspective of the evaluator, however, the
expression is simply a list (in this case, a list of three symbols: `*',
`x', and `x') that is to be manipulated according to a well-defined set
of rules.

   That the user's programs are the evaluator's data need not be a
source of confusion.  In fact, it is sometimes convenient to ignore
this distinction, and to give the user the ability to explicitly
evaluate a data object as a Lisp expression, by making `eval' available
for use in programs.  Many Lisp dialects provide a primitive `eval'
procedure that takes as arguments an expression and an environment and
evaluates the expression relative to the environment.(3) Thus,

     (eval '(* 5 5) user-initial-environment)

and

     (eval (cons '* (list 5 5)) user-initial-environment)

will both return 25.(4)

     *Exercise 4.15:* Given a one-argument procedure `p' and an object
     `a', `p' is said to "halt" on `a' if evaluating the expression `(p
     a)' returns a value (as opposed to terminating with an error
     message or running forever).  Show that it is impossible to write
     a procedure `halts?' that correctly determines whether `p' halts
     on `a' for any procedure `p' and object `a'.  Use the following
     reasoning: If you had such a procedure `halts?', you could
     implement the following program:

          (define (run-forever) (run-forever))

          (define (try p)
            (if (halts? p p)
                (run-forever)
                'halted))

     Now consider evaluating the expression `(try try)' and show that
     any possible outcome (either halting or running forever) violates
     the intended behavior of `halts?'.(5)

   ---------- Footnotes ----------

   (1) The fact that the machines are described in Lisp is inessential.
If we give our evaluator a Lisp program that behaves as an evaluator
for some other language, say C, the Lisp evaluator will emulate the C
evaluator, which in turn can emulate any machine described as a C
program.  Similarly, writing a Lisp evaluator in C produces a C program
that can execute any Lisp program.  The deep idea here is that any
evaluator can emulate any other.  Thus, the notion of "what can in
principle be computed" (ignoring practicalities of time and memory
required) is independent of the language or the computer, and instead
reflects an underlying notion of "computability".  This was first
demonstrated in a clear way by Alan M. Turing (1912-1954), whose 1936
paper laid the foundations for theoretical computer science.  In the
paper, Turing presented a simple computational model--now known as a "Turing
machine"--and argued that any "effective process" can be formulated as
a program for such a machine.  (This argument is known as the "Church-Turing
thesis".)  Turing then implemented a universal machine, i.e., a Turing
machine that behaves as an evaluator for Turing-machine programs.  He
used this framework to demonstrate that there are well-posed problems
that cannot be computed by Turing machines (see *Note Exercise 4-15::),
and so by implication cannot be formulated as "effective processes."
Turing went on to make fundamental contributions to practical computer
science as well.  For example, he invented the idea of structuring
programs using general-purpose subroutines.  See Hodges 1983 for a
biography of Turing.

   (2) Some people find it counterintuitive that an evaluator, which is
implemented by a relatively simple procedure, can emulate programs that
are more complex than the evaluator itself.  The existence of a
universal evaluator machine is a deep and wonderful property of
computation.  theory "Recursion theory", a branch of mathematical
logic, is concerned with logical limits of computation.  Douglas
Hofstadter's beautiful book `Go"del, Escher, Bach' (1979) explores some
of these ideas.

   (3) Warning: This `eval' primitive is not identical to the `eval'
procedure we implemented in section *Note 4-1-1::, because it uses
_actual_ Scheme environments rather than the sample environment
structures we built in section *Note 4-1-3::.  These actual
environments cannot be manipulated by the user as ordinary lists; they
must be accessed via `eval' or other special operations.  Similarly,
the `apply' primitive we saw earlier is not identical to the
metacircular `apply', because it uses actual Scheme procedures rather
than the procedure objects we constructed in sections *Note 4-1-3:: and
*Note 4-1-4::.

   (4) The MIT implementation of Scheme includes `eval', as well as a
symbol `user-initial-environment' that is bound to the initial
environment in which the user's input expressions are evaluated.

   (5) Although we stipulated that `halts?' is given a procedure
object, notice that this reasoning still applies even if `halts?' can
gain access to the procedure's text and its environment.  This is
Turing's celebrated "Halting Theorem", which gave the first clear
example of a "non-computable" problem, i.e., a well-posed task that
cannot be carried out as a computational procedure.


File: sicp.info,  Node: 4-1-6,  Next: 4-1-7,  Prev: 4-1-5,  Up: 4-1

4.1.6 Internal Definitions
--------------------------

Our environment model of evaluation and our metacircular evaluator
execute definitions in sequence, extending the environment frame one
definition at a time.  This is particularly convenient for interactive
program development, in which the programmer needs to freely mix the
application of procedures with the definition of new procedures.
However, if we think carefully about the internal definitions used to
implement block structure (introduced in section *Note 1-1-8::), we
will find that name-by-name extension of the environment may not be the
best way to define local variables.

   Consider a procedure with internal definitions, such as

     (define (f x)
       (define (even? n)
         (if (= n 0)
             true
             (odd? (- n 1))))
       (define (odd? n)
         (if (= n 0)
             false
             (even? (- n 1))))
       <REST OF BODY OF `F'>)

   Our intention here is that the name `odd?' in the body of the
procedure `even?' should refer to the procedure `odd?' that is defined
after `even?'.  The scope of the name `odd?' is the entire body of `f',
not just the portion of the body of `f' starting at the point where the
`define' for `odd?' occurs.  Indeed, when we consider that `odd?' is
itself defined in terms of `even?'--so that `even?' and `odd?' are
mutually recursive procedures--we see that the only satisfactory
interpretation of the two `define's is to regard them as if the names
`even?' and `odd?' were being added to the environment simultaneously.
More generally, in block structure, the scope of a local name is the
entire procedure body in which the `define' is evaluated.

   As it happens, our interpreter will evaluate calls to `f' correctly,
but for an "accidental" reason: Since the definitions of the internal
procedures come first, no calls to these procedures will be evaluated
until all of them have been defined.  Hence, `odd?'  will have been
defined by the time `even?' is executed.  In fact, our sequential
evaluation mechanism will give the same result as a mechanism that
directly implements simultaneous definition for any procedure in which
the internal definitions come first in a body and evaluation of the
value expressions for the defined variables doesn't actually use any of
the defined variables.  (For an example of a procedure that doesn't
obey these restrictions, so that sequential definition isn't equivalent
to simultaneous definition, see *Note Exercise 4-19::.)(1)

   There is, however, a simple way to treat definitions so that
internally defined names have truly simultaneous scope--just create all
local variables that will be in the current environment before
evaluating any of the value expressions.  One way to do this is by a
syntax transformation on `lambda' expressions.  Before evaluating the
body of a `lambda' expression, we "scan out" and eliminate all the
internal definitions in the body.  The internally defined variables
will be created with a `let' and then set to their values by
assignment.  For example, the procedure

     (lambda <VARS>
       (define u <E1>)
       (define v <E2>)
       <E3>)

would be transformed into

     (lambda <VARS>
       (let ((u '*unassigned*)
             (v '*unassigned*))
         (set! u <E1>)
         (set! v <E2>)
         <E3>))

where `*unassigned*' is a special symbol that causes looking up a
variable to signal an error if an attempt is made to use the value of
the not-yet-assigned variable.

   An alternative strategy for scanning out internal definitions is
shown in *Note Exercise 4-18::.  Unlike the transformation shown above,
this enforces the restriction that the defined variables' values can be
evaluated without using any of the variables' values.(2)

     *Exercise 4.16:* In this exercise we implement the method just
     described for interpreting internal definitions.  We assume that
     the evaluator supports `let' (see *Note Exercise 4-6::).

       a. Change `lookup-variable-value' (section *Note 4-1-3::) to
          signal an error if the value it finds is the symbol
          `*unassigned*'.

       b. Write a procedure `scan-out-defines' that takes a procedure
          body and returns an equivalent one that has no internal
          definitions, by making the transformation described above.

       c. Install `scan-out-defines' in the interpreter, either in
          `make-procedure' or in `procedure-body' (see section *Note
          4-1-3::).  Which place is better?  Why?


     *Exercise 4.17:* Draw diagrams of the environment in effect when
     evaluating the expression <E3> in the procedure in the text,
     comparing how this will be structured when definitions are
     interpreted sequentially with how it will be structured if
     definitions are scanned out as described.  Why is there an extra
     frame in the transformed program?  Explain why this difference in
     environment structure can never make a difference in the behavior
     of a correct program.  Design a way to make the interpreter
     implement the "simultaneous" scope rule for internal definitions
     without constructing the extra frame.

     *Exercise 4.18:* Consider an alternative strategy for scanning out
     definitions that translates the example in the text to

          (lambda <VARS>
            (let ((u '*unassigned*)
                  (v '*unassigned*))
              (let ((a <E1>)
                    (b <E2>))
                (set! u a)
                (set! v b))
              <E3>))

     Here `a' and `b' are meant to represent new variable names, created
     by the interpreter, that do not appear in the user's program.
     Consider the `solve' procedure from section *Note 3-5-4:::

          (define (solve f y0 dt)
            (define y (integral (delay dy) y0 dt))
            (define dy (stream-map f y))
            y)

     Will this procedure work if internal definitions are scanned out
     as shown in this exercise?  What if they are scanned out as shown
     in the text?  Explain.

     *Exercise 4.19:* Ben Bitdiddle, Alyssa P. Hacker, and Eva Lu Ator
     are arguing about the desired result of evaluating the expression

          (let ((a 1))
            (define (f x)
              (define b (+ a x))
              (define a 5)
              (+ a b))
            (f 10))

     Ben asserts that the result should be obtained using the
     sequential rule for `define': `b' is defined to be 11, then `a' is
     defined to be 5, so the result is 16.  Alyssa objects that mutual
     recursion requires the simultaneous scope rule for internal
     procedure definitions, and that it is unreasonable to treat
     procedure names differently from other names.  Thus, she argues
     for the mechanism implemented in *Note Exercise 4-16::.  This
     would lead to `a' being unassigned at the time that the value for
     `b' is to be computed.  Hence, in Alyssa's view the procedure
     should produce an error.  Eva has a third opinion.  She says that
     if the definitions of `a' and `b' are truly meant to be
     simultaneous, then the value 5 for `a' should be used in
     evaluating `b'.  Hence, in Eva's view `a' should be 5, `b' should
     be 15, and the result should be 20.  Which (if any) of these
     viewpoints do you support?  Can you devise a way to implement
     internal definitions so that they behave as Eva prefers?(3)

     *Exercise 4.20:* Because internal definitions look sequential but
     are actually simultaneous, some people prefer to avoid them
     entirely, and use the special form `letrec' instead.  `Letrec'
     looks like `let', so it is not surprising that the variables it
     binds are bound simultaneously and have the same scope as each
     other.  The sample procedure `f' above can be written without
     internal definitions, but with exactly the same meaning, as

          (define (f x)
            (letrec ((even?
                      (lambda (n)
                        (if (= n 0)
                            true
                            (odd? (- n 1)))))
                     (odd?
                      (lambda (n)
                        (if (= n 0)
                            false
                            (even? (- n 1))))))
              <REST OF BODY OF `F'>))

     `Letrec' expressions, which have the form

          (letrec ((<VAR_1> <EXP_1>) ... (<VAR_N> <EXP_N>))
            <BODY>)

     are a variation on `let' in which the expressions <EXP_K> that
     provide the initial values for the variables <VAR_K> are evaluated
     in an environment that includes all the `letrec' bindings.  This
     permits recursion in the bindings, such as the mutual recursion of
     `even?' and `odd?' in the example above, or the evaluation of 10
     factorial with

          (letrec ((fact
                    (lambda (n)
                      (if (= n 1)
                          1
                          (* n (fact (- n 1)))))))
            (fact 10))

       a. Implement `letrec' as a derived expression, by transforming a
          `letrec' expression into a `let' expression as shown in the
          text above or in *Note Exercise 4-18::.  That is, the
          `letrec' variables should be created with a `let' and then be
          assigned their values with `set!'.

       b. Louis Reasoner is confused by all this fuss about internal
          definitions.  The way he sees it, if you don't like to use
          `define' inside a procedure, you can just use `let'.
          Illustrate what is loose about his reasoning by drawing an
          environment diagram that shows the environment in which the
          <REST OF BODY OF `F'> is evaluated during evaluation of the
          expression `(f 5)', with `f' defined as in this exercise.
          Draw an environment diagram for the same evaluation, but with
          `let' in place of `letrec' in the definition of `f'.


     *Exercise 4.21:* Amazingly, Louis's intuition in *Note Exercise
     4-20:: is correct.  It is indeed possible to specify recursive
     procedures without using `letrec' (or even `define'), although the
     method for accomplishing this is much more subtle than Louis
     imagined.  The following expression computes 10 factorial by
     applying a recursive factorial procedure:(4)

          ((lambda (n)
             ((lambda (fact)
                (fact fact n))
              (lambda (ft k)
                (if (= k 1)
                    1
                    (* k (ft ft (- k 1)))))))
           10)

       a. Check (by evaluating the expression) that this really does
          compute factorials.  Devise an analogous expression for
          computing Fibonacci numbers.

       b. Consider the following procedure, which includes mutually
          recursive internal definitions:

               (define (f x)
                 (define (even? n)
                   (if (= n 0)
                       true
                       (odd? (- n 1))))
                 (define (odd? n)
                   (if (= n 0)
                       false
                       (even? (- n 1))))
                 (even? x))

          Fill in the missing expressions to complete an alternative
          definition of `f', which uses neither internal definitions
          nor `letrec':

               (define (f x)
                 ((lambda (even? odd?)
                    (even? even? odd? x))
                  (lambda (ev? od? n)
                    (if (= n 0) true (od? <??> <??> <??>)))
                  (lambda (ev? od? n)
                    (if (= n 0) false (ev? <??> <??> <??>)))))

   ---------- Footnotes ----------

   (1) Wanting programs to not depend on this evaluation mechanism is
the reason for the "management is not responsible" remark in *Note
Footnote 28:: of *Note Chapter 1::.  By insisting that internal
definitions come first and do not use each other while the definitions
are being evaluated, the IEEE standard for Scheme leaves implementors
some choice in the mechanism used to evaluate these definitions.  The
choice of one evaluation rule rather than another here may seem like a
small issue, affecting only the interpretation of "badly formed"
programs.  However, we will see in section *Note 5-5-6:: that moving to
a model of simultaneous scoping for internal definitions avoids some
nasty difficulties that would otherwise arise in implementing a
compiler.

   (2) The IEEE standard for Scheme allows for different implementation
strategies by specifying that it is up to the programmer to obey this
restriction, not up to the implementation to enforce it.  Some Scheme
implementations, including MIT Scheme, use the transformation shown
above.  Thus, some programs that don't obey this restriction will in
fact run in such implementations.

   (3) The MIT implementors of Scheme support Alyssa on the following
grounds: Eva is in principle correct - the definitions should be
regarded as simultaneous.  But it seems difficult to implement a
general, efficient mechanism that does what Eva requires.  In the
absence of such a mechanism, it is better to generate an error in the
difficult cases of simultaneous definitions (Alyssa's notion) than to
produce an incorrect answer (as Ben would have it).

   (4) This example illustrates a programming trick for formulating
recursive procedures without using `define'.  The most general trick of
this sort is the Y "operator", which can be used to give a "pure
[lambda]-calculus" implementation of recursion.  (See Stoy 1977 for
details on the [lambda] calculus, and Gabriel 1988 for an exposition of
the Y operator in Scheme.)


File: sicp.info,  Node: 4-1-7,  Prev: 4-1-6,  Up: 4-1

4.1.7 Separating Syntactic Analysis from Execution
--------------------------------------------------

The evaluator implemented above is simple, but it is very inefficient,
because the syntactic analysis of expressions is interleaved with their
execution.  Thus if a program is executed many times, its syntax is
analyzed many times.  Consider, for example, evaluating `(factorial 4)'
using the following definition of `factorial':

     (define (factorial n)
       (if (= n 1)
           1
           (* (factorial (- n 1)) n)))

   Each time `factorial' is called, the evaluator must determine that
the body is an `if' expression and extract the predicate.  Only then
can it evaluate the predicate and dispatch on its value.  Each time it
evaluates the expression `(* (factorial (- n 1)) n)', or the
subexpressions `(factorial (- n 1))' and `(- n 1)', the evaluator must
perform the case analysis in `eval' to determine that the expression is
an application, and must extract its operator and operands.  This
analysis is expensive.  Performing it repeatedly is wasteful.

   We can transform the evaluator to be significantly more efficient by
arranging things so that syntactic analysis is performed only once.(1)
We split `eval', which takes an expression and an environment, into two
parts.  The procedure `analyze' takes only the expression.  It performs
the syntactic analysis and returns a new procedure, the "execution
procedure", that encapsulates the work to be done in executing the
analyzed expression.  The execution procedure takes an environment as
its argument and completes the evaluation.  This saves work because
`analyze' will be called only once on an expression, while the
execution procedure may be called many times.

   With the separation into analysis and execution, `eval' now becomes

     (define (eval exp env)
       ((analyze exp) env))

   The result of calling `analyze' is the execution procedure to be
applied to the environment.  The `analyze' procedure is the same case
analysis as performed by the original `eval' of section *Note 4-1-1::,
except that the procedures to which we dispatch perform only analysis,
not full evaluation:

     (define (analyze exp)
       (cond ((self-evaluating? exp)
              (analyze-self-evaluating exp))
             ((quoted? exp) (analyze-quoted exp))
             ((variable? exp) (analyze-variable exp))
             ((assignment? exp) (analyze-assignment exp))
             ((definition? exp) (analyze-definition exp))
             ((if? exp) (analyze-if exp))
             ((lambda? exp) (analyze-lambda exp))
             ((begin? exp) (analyze-sequence (begin-actions exp)))
             ((cond? exp) (analyze (cond->if exp)))
             ((application? exp) (analyze-application exp))
             (else
              (error "Unknown expression type -- ANALYZE" exp))))

   Here is the simplest syntactic analysis procedure, which handles
self-evaluating expressions.  It returns an execution procedure that
ignores its environment argument and just returns the expression:

     (define (analyze-self-evaluating exp)
       (lambda (env) exp))

   For a quoted expression, we can gain a little efficiency by
extracting the text of the quotation only once, in the analysis phase,
rather than in the execution phase.

     (define (analyze-quoted exp)
       (let ((qval (text-of-quotation exp)))
         (lambda (env) qval)))

   Looking up a variable value must still be done in the execution
phase, since this depends upon knowing the environment.(2)

     (define (analyze-variable exp)
       (lambda (env) (lookup-variable-value exp env)))

   `Analyze-assignment' also must defer actually setting the variable
until the execution, when the environment has been supplied.  However,
the fact that the `assignment-value' expression can be analyzed
(recursively) during analysis is a major gain in efficiency, because
the `assignment-value' expression will now be analyzed only once.  The
same holds true for definitions.

     (define (analyze-assignment exp)
       (let ((var (assignment-variable exp))
             (vproc (analyze (assignment-value exp))))
         (lambda (env)
           (set-variable-value! var (vproc env) env)
           'ok)))

     (define (analyze-definition exp)
       (let ((var (definition-variable exp))
             (vproc (analyze (definition-value exp))))
         (lambda (env)
           (define-variable! var (vproc env) env)
           'ok)))

   For `if' expressions, we extract and analyze the predicate,
consequent, and alternative at analysis time.

     (define (analyze-if exp)
       (let ((pproc (analyze (if-predicate exp)))
             (cproc (analyze (if-consequent exp)))
             (aproc (analyze (if-alternative exp))))
         (lambda (env)
           (if (true? (pproc env))
               (cproc env)
               (aproc env)))))

   Analyzing a `lambda' expression also achieves a major gain in
efficiency: We analyze the `lambda' body only once, even though
procedures resulting from evaluation of the `lambda' may be applied
many times.

     (define (analyze-lambda exp)
       (let ((vars (lambda-parameters exp))
             (bproc (analyze-sequence (lambda-body exp))))
         (lambda (env) (make-procedure vars bproc env))))

   Analysis of a sequence of expressions (as in a `begin' or the body
of a `lambda' expression) is more involved.(3) Each expression in the
sequence is analyzed, yielding an execution procedure.  These execution
procedures are combined to produce an execution procedure that takes an
environment as argument and sequentially calls each individual execution
procedure with the environment as argument.

     (define (analyze-sequence exps)
       (define (sequentially proc1 proc2)
         (lambda (env) (proc1 env) (proc2 env)))
       (define (loop first-proc rest-procs)
         (if (null? rest-procs)
             first-proc
             (loop (sequentially first-proc (car rest-procs))
                   (cdr rest-procs))))
       (let ((procs (map analyze exps)))
         (if (null? procs)
             (error "Empty sequence -- ANALYZE"))
         (loop (car procs) (cdr procs))))

   To analyze an application, we analyze the operator and operands and
construct an execution procedure that calls the operator execution
procedure (to obtain the actual procedure to be applied) and the
operand execution procedures (to obtain the actual arguments).  We then
pass these to `execute-application', which is the analog of `apply' in
section *Note 4-1-1::.  `Execute-application' differs from `apply' in
that the procedure body for a compound procedure has already been
analyzed, so there is no need to do further analysis.  Instead, we just
call the execution procedure for the body on the extended environment.

     (define (analyze-application exp)
       (let ((fproc (analyze (operator exp)))
             (aprocs (map analyze (operands exp))))
         (lambda (env)
           (execute-application (fproc env)
                                (map (lambda (aproc) (aproc env))
                                     aprocs)))))

     (define (execute-application proc args)
       (cond ((primitive-procedure? proc)
              (apply-primitive-procedure proc args))
             ((compound-procedure? proc)
              ((procedure-body proc)
               (extend-environment (procedure-parameters proc)
                                   args
                                   (procedure-environment proc))))
             (else
              (error
               "Unknown procedure type -- EXECUTE-APPLICATION"
               proc))))

   Our new evaluator uses the same data structures, syntax procedures,
and run-time support procedures as in sections *Note 4-1-2::, *Note
4-1-3::, and *Note 4-1-4::.

     *Exercise 4.22:* Extend the evaluator in this section to support
     the special form `let'.  (See *Note Exercise 4-6::.)

     *Exercise 4.23:* Alyssa P. Hacker doesn't understand why
     `analyze-sequence' needs to be so complicated.  All the other
     analysis procedures are straightforward transformations of the
     corresponding evaluation procedures (or `eval' clauses) in section
     *Note 4-1-1::.  She expected `analyze-sequence' to look like this:

          (define (analyze-sequence exps)
            (define (execute-sequence procs env)
              (cond ((null? (cdr procs)) ((car procs) env))
                    (else ((car procs) env)
                          (execute-sequence (cdr procs) env))))
            (let ((procs (map analyze exps)))
              (if (null? procs)
                  (error "Empty sequence -- ANALYZE"))
              (lambda (env) (execute-sequence procs env))))

     Eva Lu Ator explains to Alyssa that the version in the text does
     more of the work of evaluating a sequence at analysis time.
     Alyssa's sequence-execution procedure, rather than having the
     calls to the individual execution procedures built in, loops
     through the procedures in order to call them: In effect, although
     the individual expressions in the sequence have been analyzed, the
     sequence itself has not been.

     Compare the two versions of `analyze-sequence'.  For example,
     consider the common case (typical of procedure bodies) where the
     sequence has just one expression.  What work will the execution
     procedure produced by Alyssa's program do?  What about the
     execution procedure produced by the program in the text above?
     How do the two versions compare for a sequence with two
     expressions?

     *Exercise 4.24:* Design and carry out some experiments to compare
     the speed of the original metacircular evaluator with the version
     in this section.  Use your results to estimate the fraction of time
     that is spent in analysis versus execution for various procedures.

   ---------- Footnotes ----------

   (1) This technique is an integral part of the compilation process,
which we shall discuss in *Note Chapter 5::.  Jonathan Rees wrote a
Scheme interpreter like this in about 1982 for the T project (Rees and
Adams 1982).  Marc Feeley (1986) (see also Feeley and Lapalme 1987)
independently invented this technique in his master's thesis.

   (2) There is, however, an important part of the variable search that
_can_ be done as part of the syntactic analysis.  As we will show in
section *Note 5-5-6::, one can determine the position in the
environment structure where the value of the variable will be found,
thus obviating the need to scan the environment for the entry that
matches the variable.

   (3) See *Note Exercise 4-23:: for some insight into the processing
of sequences.


File: sicp.info,  Node: 4-2,  Next: 4-3,  Prev: 4-1,  Up: Chapter 4

4.2 Variations on a Scheme - Lazy Evaluation
============================================

Now that we have an evaluator expressed as a Lisp program, we can
experiment with alternative choices in language design simply by
modifying the evaluator.  Indeed, new languages are often invented by
first writing an evaluator that embeds the new language within an
existing high-level language.  For example, if we wish to discuss some
aspect of a proposed modification to Lisp with another member of the
Lisp community, we can supply an evaluator that embodies the change.
The recipient can then experiment with the new evaluator and send back
comments as further modifications.  Not only does the high-level
implementation base make it easier to test and debug the evaluator; in
addition, the embedding enables the designer to snarf(1) features from
the underlying language, just as our embedded Lisp evaluator uses
primitives and control structure from the underlying Lisp.  Only later
(if ever) need the designer go to the trouble of building a complete
implementation in a low-level language or in hardware.  In this section
and the next we explore some variations on Scheme that provide
significant additional expressive power.

* Menu:

* 4-2-1::            Normal Order and Applicative Order
* 4-2-2::            An Interpreter with Lazy Evaluation
* 4-2-3::            Streams as Lazy Lists

   ---------- Footnotes ----------

   (1) Snarf: "To grab, especially a large document or file for the
purpose of using it either with or without the owner's permission."
Snarf down: "To snarf, sometimes with the connotation of absorbing,
processing, or understanding."  (These definitions were snarfed from
Steele et al. 1983.  See also Raymond 1993.)


File: sicp.info,  Node: 4-2-1,  Next: 4-2-2,  Prev: 4-2,  Up: 4-2

4.2.1 Normal Order and Applicative Order
----------------------------------------

In section *Note 1-1::, where we began our discussion of models of
evaluation, we noted that Scheme is an "applicative-order" language,
namely, that all the arguments to Scheme procedures are evaluated when
the procedure is applied.  In contrast, "normal-order" languages delay
evaluation of procedure arguments until the actual argument values are
needed.  Delaying evaluation of procedure arguments until the last
possible moment (e.g., until they are required by a primitive
operation) is called evaluation "lazy evaluation".(1)  Consider the
procedure

     (define (try a b)
       (if (= a 0) 1 b))

   Evaluating `(try 0 (/ 1 0))' generates an error in Scheme.  With lazy
evaluation, there would be no error.  Evaluating the expression would
return 1, because the argument `(/ 1 0)' would never be evaluated.

   An example that exploits lazy evaluation is the definition of a
procedure `unless'

     (define (unless condition usual-value exceptional-value)
       (if condition exceptional-value usual-value))

that can be used in expressions such as

     (unless (= b 0)
             (/ a b)
             (begin (display "exception: returning 0")
                    0))

   This won't work in an applicative-order language because both the
usual value and the exceptional value will be evaluated before `unless'
is called (compare *Note Exercise 1-6::).  An advantage of lazy
evaluation is that some procedures, such as `unless', can do useful
computation even if evaluation of some of their arguments would produce
errors or would not terminate.

   If the body of a procedure is entered before an argument has been
evaluated we say that the procedure is "non-strict" in that argument.
If the argument is evaluated before the body of the procedure is
entered we say that the procedure is "strict" in that argument.(2)  In
a purely applicative-order language, all procedures are strict in each
argument.  In a purely normal-order language, all compound procedures
are non-strict in each argument, and primitive procedures may be either
strict or non-strict.  There are also languages (see *Note Exercise
4-31::) that give programmers detailed control over the strictness of
the procedures they define.

   A striking example of a procedure that can usefully be made
non-strict is `cons' (or, in general, almost any constructor for data
structures).  One can do useful computation, combining elements to form
data structures and operating on the resulting data structures, even if
the values of the elements are not known.  It makes perfect sense, for
instance, to compute the length of a list without knowing the values of
the individual elements in the list.  We will exploit this idea in
section *Note 4-2-3:: to implement the streams of *Note Chapter 3:: as
lists formed of non-strict `cons' pairs.

     *Exercise 4.25:* Suppose that (in ordinary applicative-order
     Scheme) we define `unless' as shown above and then define
     `factorial' in terms of `unless' as

          (define (factorial n)
            (unless (= n 1)
                    (* n (factorial (- n 1)))
                    1))

     What happens if we attempt to evaluate `(factorial 5)'?  Will our
     definitions work in a normal-order language?

     *Exercise 4.26:* Ben Bitdiddle and Alyssa P. Hacker disagree over
     the importance of lazy evaluation for implementing things such as
     `unless'.  Ben points out that it's possible to implement `unless'
     in applicative order as a special form.  Alyssa counters that, if
     one did that, `unless' would be merely syntax, not a procedure
     that could be used in conjunction with higher-order procedures.
     Fill in the details on both sides of the argument.  Show how to
     implement `unless' as a derived expression (like `cond' or `let'),
     and give an example of a situation where it might be useful to
     have `unless' available as a procedure, rather than as a special
     form.

   ---------- Footnotes ----------

   (1) The difference between the "lazy" terminology and the
"normal-order" terminology is somewhat fuzzy.  Generally, "lazy" refers
to the mechanisms of particular evaluators, while "normal-order" refers
to the semantics of languages, independent of any particular evaluation
strategy.  But this is not a hard-and-fast distinction, and the two
terminologies are often used interchangeably.

   (2) The "strict" versus "non-strict" terminology means essentially
the same thing as "applicative-order" versus "normal-order," except
that it refers to individual procedures and arguments rather than to
the language as a whole.  At a conference on programming languages you
might hear someone say, "The normal-order language Hassle has certain
strict primitives.  Other procedures take their arguments by lazy
evaluation."


File: sicp.info,  Node: 4-2-2,  Next: 4-2-3,  Prev: 4-2-1,  Up: 4-2

4.2.2 An Interpreter with Lazy Evaluation
-----------------------------------------

In this section we will implement a normal-order language that is the
same as Scheme except that compound procedures are non-strict in each
argument.  Primitive procedures will still be strict.  It is not
difficult to modify the evaluator of section *Note 4-1-1:: so that the
language it interprets behaves this way.  Almost all the required
changes center around procedure application.

   The basic idea is that, when applying a procedure, the interpreter
must determine which arguments are to be evaluated and which are to be
delayed.  The delayed arguments are not evaluated; instead, they are
transformed into objects called "thunks".(1) The thunk must contain the
information required to produce the value of the argument when it is
needed, as if it had been evaluated at the time of the application.
Thus, the thunk must contain the argument expression and the
environment in which the procedure application is being evaluated.

   The process of evaluating the expression in a thunk is called "forcing".(2)
In general, a thunk will be forced only when its value is needed: when
it is passed to a primitive procedure that will use the value of the
thunk; when it is the value of a predicate of a conditional; and when
it is the value of an operator that is about to be applied as a
procedure.  One design choice we have available is whether or not to "memoize"
thunks, as we did with delayed objects in section *Note 3-5-1::.  With
memoization, the first time a thunk is forced, it stores the value that
is computed.  Subsequent forcings simply return the stored value
without repeating the computation.  We'll make our interpreter memoize,
because this is more efficient for many applications.  There are tricky
considerations here, however.(3)

Modifying the evaluator
.......................

The main difference between the lazy evaluator and the one in section
*Note 4-1:: is in the handling of procedure applications in `eval' and
`apply'.

   The `application?' clause of `eval' becomes

     ((application? exp)
      (apply (actual-value (operator exp) env)
             (operands exp)
             env))

   This is almost the same as the `application?' clause of `eval' in
section *Note 4-1-1::.  For lazy evaluation, however, we call `apply'
with the operand expressions, rather than the arguments produced by
evaluating them.  Since we will need the environment to construct
thunks if the arguments are to be delayed, we must pass this as well.
We still evaluate the operator, because `apply' needs the actual
procedure to be applied in order to dispatch on its type (primitive
versus compound) and apply it.

   Whenever we need the actual value of an expression, we use

     (define (actual-value exp env)
       (force-it (eval exp env)))

instead of just `eval', so that if the expression's value is a thunk, it
will be forced.

   Our new version of `apply' is also almost the same as the version in
section *Note 4-1-1::.  The difference is that `eval' has passed in
unevaluated operand expressions: For primitive procedures (which are
strict), we evaluate all the arguments before applying the primitive;
for compound procedures (which are non-strict) we delay all the
arguments before applying the procedure.

     (define (apply procedure arguments env)
       (cond ((primitive-procedure? procedure)
              (apply-primitive-procedure
               procedure
               (list-of-arg-values arguments env)))  ; changed
             ((compound-procedure? procedure)
              (eval-sequence
               (procedure-body procedure)
               (extend-environment
                (procedure-parameters procedure)
                (list-of-delayed-args arguments env) ; changed
                (procedure-environment procedure))))
             (else
              (error
               "Unknown procedure type -- APPLY" procedure))))

   The procedures that process the arguments are just like
`list-of-values' from section *Note 4-1-1::, except that
`list-of-delayed-args' delays the arguments instead of evaluating them,
and `list-of-arg-values' uses `actual-value' instead of `eval':

     (define (list-of-arg-values exps env)
       (if (no-operands? exps)
           '()
           (cons (actual-value (first-operand exps) env)
                 (list-of-arg-values (rest-operands exps)
                                     env))))

     (define (list-of-delayed-args exps env)
       (if (no-operands? exps)
           '()
           (cons (delay-it (first-operand exps) env)
                 (list-of-delayed-args (rest-operands exps)
                                       env))))

   The other place we must change the evaluator is in the handling of
`if', where we must use `actual-value' instead of `eval' to get the
value of the predicate expression before testing whether it is true or
false:

     (define (eval-if exp env)
       (if (true? (actual-value (if-predicate exp) env))
           (eval (if-consequent exp) env)
           (eval (if-alternative exp) env)))

   Finally, we must change the `driver-loop' procedure (section *Note
4-1-4::) to use `actual-value' instead of `eval', so that if a delayed
value is propagated back to the read-eval-print loop, it will be forced
before being printed.  We also change the prompts to indicate that this
is the lazy evaluator:

     (define input-prompt ";;; L-Eval input:")
     (define output-prompt ";;; L-Eval value:")

     (define (driver-loop)
       (prompt-for-input input-prompt)
       (let ((input (read)))
         (let ((output
                (actual-value input the-global-environment)))
           (announce-output output-prompt)
           (user-print output)))
       (driver-loop))

   With these changes made, we can start the evaluator and test it.  The
successful evaluation of the `try' expression discussed in section
*Note 4-2-1:: indicates that the interpreter is performing lazy
evaluation:

     (define the-global-environment (setup-environment))

     (driver-loop)

     ;;; L-Eval input:
     (define (try a b)
       (if (= a 0) 1 b))
     ;;; L-Eval value:
     ok

     ;;; L-Eval input:
     (try 0 (/ 1 0))
     ;;; L-Eval value:
     1

Representing thunks
...................

Our evaluator must arrange to create thunks when procedures are applied
to arguments and to force these thunks later.  A thunk must package an
expression together with the environment, so that the argument can be
produced later.  To force the thunk, we simply extract the expression
and environment from the thunk and evaluate the expression in the
environment.  We use `actual-value' rather than `eval' so that in case
the value of the expression is itself a thunk, we will force that, and
so on, until we reach something that is not a thunk:

     (define (force-it obj)
       (if (thunk? obj)
           (actual-value (thunk-exp obj) (thunk-env obj))
           obj))

   One easy way to package an expression with an environment is to make
a list containing the expression and the environment.  Thus, we create
a thunk as follows:

     (define (delay-it exp env)
       (list 'thunk exp env))

     (define (thunk? obj)
       (tagged-list? obj 'thunk))

     (define (thunk-exp thunk) (cadr thunk))

     (define (thunk-env thunk) (caddr thunk))

   Actually, what we want for our interpreter is not quite this, but
rather thunks that have been memoized.  When a thunk is forced, we will
turn it into an evaluated thunk by replacing the stored expression with
its value and changing the `thunk' tag so that it can be recognized as
already evaluated.(4)

     (define (evaluated-thunk? obj)
       (tagged-list? obj 'evaluated-thunk))

     (define (thunk-value evaluated-thunk) (cadr evaluated-thunk))

     (define (force-it obj)
       (cond ((thunk? obj)
              (let ((result (actual-value
                             (thunk-exp obj)
                             (thunk-env obj))))
                (set-car! obj 'evaluated-thunk)
                (set-car! (cdr obj) result)  ; replace `exp' with its value
                (set-cdr! (cdr obj) '())     ; forget unneeded `env'
                result))
             ((evaluated-thunk? obj)
              (thunk-value obj))
             (else obj)))

   Notice that the same `delay-it' procedure works both with and without
memoization.

     *Exercise 4.27:* Suppose we type in the following definitions to
     the lazy evaluator:

          (define count 0)

          (define (id x)
            (set! count (+ count 1))
            x)

     Give the missing values in the following sequence of interactions,
     and explain your answers.(5)

          (define w (id (id 10)))

          ;;; L-Eval input:
          count
          ;;; L-Eval value:
          <RESPONSE>

          ;;; L-Eval input:
          w
          ;;; L-Eval value:
          <RESPONSE>

          ;;; L-Eval input:
          count
          ;;; L-Eval value:
          <RESPONSE>

     *Exercise 4.28:* `Eval' uses `actual-value' rather than `eval' to
     evaluate the operator before passing it to `apply', in order to
     force the value of the operator.  Give an example that
     demonstrates the need for this forcing.

     *Exercise 4.29:* Exhibit a program that you would expect to run
     much more slowly without memoization than with memoization.  Also,
     consider the following interaction, where the `id' procedure is
     defined as in *Note Exercise 4-27:: and `count' starts at 0:

          (define (square x)
            (* x x))

          ;;; L-Eval input:
          (square (id 10))
          ;;; L-Eval value:
          <RESPONSE>

          ;;; L-Eval input:
          count
          ;;; L-Eval value:
          <RESPONSE>

     Give the responses both when the evaluator memoizes and when it
     does not.

     *Exercise 4.30:* Cy D. Fect, a reformed C programmer, is worried
     that some side effects may never take place, because the lazy
     evaluator doesn't force the expressions in a sequence.  Since the
     value of an expression in a sequence other than the last one is
     not used (the expression is there only for its effect, such as
     assigning to a variable or printing), there can be no subsequent
     use of this value (e.g., as an argument to a primitive procedure)
     that will cause it to be forced.  Cy thus thinks that when
     evaluating sequences, we must force all expressions in the
     sequence except the final one.  He proposes to modify
     `eval-sequence' from section *Note 4-1-1:: to use `actual-value'
     rather than `eval':

          (define (eval-sequence exps env)
            (cond ((last-exp? exps) (eval (first-exp exps) env))
                  (else (actual-value (first-exp exps) env)
                        (eval-sequence (rest-exps exps) env))))

       a. Ben Bitdiddle thinks Cy is wrong.  He shows Cy the `for-each'
          procedure described in *Note Exercise 2-23::, which gives an
          important example of a sequence with side effects:

               (define (for-each proc items)
                 (if (null? items)
                     'done
                     (begin (proc (car items))
                            (for-each proc (cdr items)))))

          He claims that the evaluator in the text (with the original
          `eval-sequence') handles this correctly:

               ;;; L-Eval input:
               (for-each (lambda (x) (newline) (display x))
                         (list 57 321 88))
               57
               321
               88
               ;;; L-Eval value:
               done

          Explain why Ben is right about the behavior of `for-each'.

       b. Cy agrees that Ben is right about the `for-each' example, but
          says that that's not the kind of program he was thinking
          about when he proposed his change to `eval-sequence'.  He
          defines the following two procedures in the lazy evaluator:

               (define (p1 x)
                 (set! x (cons x '(2)))
                 x)

               (define (p2 x)
                 (define (p e)
                   e
                   x)
                 (p (set! x (cons x '(2)))))

          What are the values of `(p1 1)' and `(p2 1)' with the original
          `eval-sequence'?  What would the values be with Cy's proposed
          change to `eval-sequence'?

       c. Cy also points out that changing `eval-sequence' as he
          proposes does not affect the behavior of the example in part
          a.  Explain why this is true.

       d. How do you think sequences ought to be treated in the lazy
          evaluator?  Do you like Cy's approach, the approach in the
          text, or some other approach?


     *Exercise 4.31:* The approach taken in this section is somewhat
     unpleasant, because it makes an incompatible change to Scheme.  It
     might be nicer to implement lazy evaluation as an "upward-compatible
     extension", that is, so that ordinary Scheme programs will work as
     before.  We can do this by extending the syntax of procedure
     declarations to let the user control whether or not arguments are
     to be delayed.  While we're at it, we may as well also give the
     user the choice between delaying with and without memoization.
     For example, the definition

          (define (f a (b lazy) c (d lazy-memo))
            ...)

     would define `f' to be a procedure of four arguments, where the
     first and third arguments are evaluated when the procedure is
     called, the second argument is delayed, and the fourth argument is
     both delayed and memoized.  Thus, ordinary procedure definitions
     will produce the same behavior as ordinary Scheme, while adding
     the `lazy-memo' declaration to each parameter of every compound
     procedure will produce the behavior of the lazy evaluator defined
     in this section. Design and implement the changes required to
     produce such an extension to Scheme.  You will have to implement
     new syntax procedures to handle the new syntax for `define'.  You
     must also arrange for `eval' or `apply' to determine when
     arguments are to be delayed, and to force or delay arguments
     accordingly, and you must arrange for forcing to memoize or not,
     as appropriate.

   ---------- Footnotes ----------

   (1) The word "thunk" was invented by an informal working group that
was discussing the implementation of call-by-name in Algol 60.  They
observed that most of the analysis of ("thinking about") the expression
could be done at compile time; thus, at run time, the expression would
already have been "thunk" about (Ingerman et al. 1960).

   (2) This is analogous to the use of `force' on the delayed objects
that were introduced in *Note Chapter 3:: to represent streams.  The
critical difference between what we are doing here and what we did in
*Note Chapter 3:: is that we are building delaying and forcing into the
evaluator, and thus making this uniform and automatic throughout the
language.

   (3) Lazy evaluation combined with memoization is sometimes referred
to as "call-by-need" argument passing, in contrast to "call-by-name"
argument passing.  (Call-by-name, introduced in Algol 60, is similar to
non-memoized lazy evaluation.)  As language designers, we can build our
evaluator to memoize, not to memoize, or leave this an option for
programmers (*Note Exercise 4-31::).  As you might expect from *Note
Chapter 3::, these choices raise issues that become both subtle and
confusing in the presence of assignments.  (See *Note Exercise 4-27::
and *Note Exercise 4-29::.)  An excellent article by Clinger (1982)
attempts to clarify the multiple dimensions of confusion that arise
here.

   (4) Notice that we also erase the `env' from the thunk once the
expression's value has been computed.  This makes no difference in the
values returned by the interpreter.  It does help save space, however,
because removing the reference from the thunk to the `env' once it is
no longer needed allows this structure to be "garbage-collected" and
its space recycled, as we will discuss in section *Note 5-3::.

   Similarly, we could have allowed unneeded environments in the
memoized delayed objects of section *Note 3-5-1:: to be
garbage-collected, by having `memo-proc' do something like `(set! proc
'())' to discard the procedure `proc' (which includes the environment
in which the `delay' was evaluated) after storing its value.

   (5) This exercise demonstrates that the interaction between lazy
evaluation and side effects can be very confusing.  This is just what
you might expect from the discussion in *Note Chapter 3::.


File: sicp.info,  Node: 4-2-3,  Prev: 4-2-2,  Up: 4-2

4.2.3 Streams as Lazy Lists
---------------------------

In section *Note 3-5-1::, we showed how to implement streams as delayed
lists.  We introduced special forms `delay' and `cons-stream', which
allowed us to construct a "promise" to compute the `cdr' of a stream,
without actually fulfilling that promise until later.  We could use
this general technique of introducing special forms whenever we need
more control over the evaluation process, but this is awkward.  For one
thing, a special form is not a first-class object like a procedure, so
we cannot use it together with higher-order procedures.(1)
Additionally, we were forced to create streams as a new kind of data
object similar but not identical to lists, and this required us to
reimplement many ordinary list operations (`map', `append', and so on)
for use with streams.

   With lazy evaluation, streams and lists can be identical, so there
is no need for special forms or for separate list and stream
operations.  All we need to do is to arrange matters so that `cons' is
non-strict.  One way to accomplish this is to extend the lazy evaluator
to allow for non-strict primitives, and to implement `cons' as one of
these.  An easier way is to recall (section *Note 2-1-3::) that there
is no fundamental need to implement `cons' as a primitive at all.
Instead, we can represent pairs as procedures:(2)

     (define (cons x y)
       (lambda (m) (m x y)))

     (define (car z)
       (z (lambda (p q) p)))

     (define (cdr z)
       (z (lambda (p q) q)))

   In terms of these basic operations, the standard definitions of the
list operations will work with infinite lists (streams) as well as
finite ones, and the stream operations can be implemented as list
operations.  Here are some examples:

     (define (list-ref items n)
       (if (= n 0)
           (car items)
           (list-ref (cdr items) (- n 1))))

     (define (map proc items)
       (if (null? items)
           '()
           (cons (proc (car items))
                 (map proc (cdr items)))))

     (define (scale-list items factor)
       (map (lambda (x) (* x factor))
            items))

     (define (add-lists list1 list2)
       (cond ((null? list1) list2)
             ((null? list2) list1)
             (else (cons (+ (car list1) (car list2))
                         (add-lists (cdr list1) (cdr list2))))))

     (define ones (cons 1 ones))

     (define integers (cons 1 (add-lists ones integers)))

     ;;; L-Eval input:
     (list-ref integers 17)
     ;;; L-Eval value:
     18

   Note that these lazy lists are even lazier than the streams of *Note
Chapter 3::: The `car' of the list, as well as the `cdr', is
delayed.(3)  In fact, even accessing the `car' or `cdr' of a lazy pair
need not force the value of a list element.  The value will be forced
only when it is really needed - e.g., for use as the argument of a
primitive, or to be printed as an answer.

   Lazy pairs also help with the problem that arose with streams in
section *Note 3-5-4::, where we found that formulating stream models of
systems with loops may require us to sprinkle our programs with
explicit `delay' operations, beyond the ones supplied by `cons-stream'.
With lazy evaluation, all arguments to procedures are delayed
uniformly.  For instance, we can implement procedures to integrate
lists and solve differential equations as we originally intended in
section *Note 3-5-4:::

     (define (integral integrand initial-value dt)
       (define int
         (cons initial-value
               (add-lists (scale-list integrand dt)
                         int)))
       int)

     (define (solve f y0 dt)
       (define y (integral dy y0 dt))
       (define dy (map f y))
       y)

     ;;; L-Eval input:
     (list-ref (solve (lambda (x) x) 1 0.001) 1000)
     ;;; L-Eval value:
     2.716924

     *Exercise 4.32:* Give some examples that illustrate the difference
     between the streams of *Note Chapter 3:: and the "lazier" lazy
     lists described in this section.  How can you take advantage of
     this extra laziness?

     *Exercise 4.33:* Ben Bitdiddle tests the lazy list implementation
     given above by evaluating the expression

          (car '(a b c))

     To his surprise, this produces an error.  After some thought, he
     realizes that the "lists" obtained by reading in quoted
     expressions are different from the lists manipulated by the new
     definitions of `cons', `car', and `cdr'.  Modify the evaluator's
     treatment of quoted expressions so that quoted lists typed at the
     driver loop will produce true lazy lists.

     *Exercise 4.34:* Modify the driver loop for the evaluator so that
     lazy pairs and lists will print in some reasonable way.  (What are
     you going to do about infinite lists?)  You may also need to modify
     the representation of lazy pairs so that the evaluator can
     identify them in order to print them.

   ---------- Footnotes ----------

   (1) This is precisely the issue with the `unless' procedure, as in
*Note Exercise 4-26::.

   (2) This is the procedural representation described in *Note
Exercise 2-4::.  Essentially any procedural representation (e.g., a
message-passing implementation) would do as well.  Notice that we can
install these definitions in the lazy evaluator simply by typing them
at the driver loop.  If we had originally included `cons', `car', and
`cdr' as primitives in the global environment, they will be redefined.
(Also see *Note Exercise 4-33:: and *Note Exercise 4-34::.)

   (3) This permits us to create delayed versions of more general kinds
of list structures, not just sequences.  Hughes 1990 discusses some
applications of "lazy trees."


File: sicp.info,  Node: 4-3,  Next: 4-4,  Prev: 4-2,  Up: Chapter 4

4.3 Variations on a Scheme - Nondeterministic Computing
=======================================================

In this section, we extend the Scheme evaluator to support a programming
paradigm called "nondeterministic computing" by building into the
evaluator a facility to support automatic search.  This is a much more
profound change to the language than the introduction of lazy
evaluation in section *Note 4-2::.

   Nondeterministic computing, like stream processing, is useful for
"generate and test" applications.  Consider the task of starting with
two lists of positive integers and finding a pair of integers--one from
the first list and one from the second list--whose sum is prime.  We
saw how to handle this with finite sequence operations in section *Note
2-2-3:: and with infinite streams in section *Note 3-5-3::.  Our
approach was to generate the sequence of all possible pairs and filter
these to select the pairs whose sum is prime.  Whether we actually
generate the entire sequence of pairs first as in *Note Chapter 2::, or
interleave the generating and filtering as in *Note Chapter 3::, is
immaterial to the essential image of how the computation is organized.

   The nondeterministic approach evokes a different image.  Imagine
simply that we choose (in some way) a number from the first list and a
number from the second list and require (using some mechanism) that
their sum be prime.  This is expressed by following procedure:

     (define (prime-sum-pair list1 list2)
       (let ((a (an-element-of list1))
             (b (an-element-of list2)))
         (require (prime? (+ a b)))
         (list a b)))

   It might seem as if this procedure merely restates the problem,
rather than specifying a way to solve it.  Nevertheless, this is a
legitimate nondeterministic program.(1)

   The key idea here is that expressions in a nondeterministic language
can have more than one possible value.  For instance, `an-element-of'
might return any element of the given list.  Our nondeterministic
program evaluator will work by automatically choosing a possible value
and keeping track of the choice.  If a subsequent requirement is not
met, the evaluator will try a different choice, and it will keep trying
new choices until the evaluation succeeds, or until we run out of
choices.  Just as the lazy evaluator freed the programmer from the
details of how values are delayed and forced, the nondeterministic
program evaluator will free the programmer from the details of how
choices are made.

   It is instructive to contrast the different images of time evoked by
nondeterministic evaluation and stream processing.  Stream processing
uses lazy evaluation to decouple the time when the stream of possible
answers is assembled from the time when the actual stream elements are
produced.  The evaluator supports the illusion that all the possible
answers are laid out before us in a timeless sequence.  With
nondeterministic evaluation, an expression represents the exploration
of a set of possible worlds, each determined by a set of choices.  Some
of the possible worlds lead to dead ends, while others have useful
values.  The nondeterministic program evaluator supports the illusion
that time branches, and that our programs have different possible
execution histories.  When we reach a dead end, we can revisit a
previous choice point and proceed along a different branch.

   The nondeterministic program evaluator implemented below is called
the `amb' evaluator because it is based on a new special form called
`amb'.  We can type the above definition of `prime-sum-pair' at the
`amb' evaluator driver loop (along with definitions of `prime?',
`an-element-of', and `require') and run the procedure as follows:

     ;;; Amb-Eval input:
     (prime-sum-pair '(1 3 5 8) '(20 35 110))
     ;;; Starting a new problem
     ;;; Amb-Eval value:
     (3 20)

   The value returned was obtained after the evaluator repeatedly chose
elements from each of the lists, until a successful choice was made.

   Section *Note 4-3-1:: introduces `amb' and explains how it supports
nondeterminism through the evaluator's automatic search mechanism.
Section *Note 4-3-2:: presents examples of nondeterministic programs,
and section *Note 4-3-3:: gives the details of how to implement the
`amb' evaluator by modifying the ordinary Scheme evaluator.

* Menu:

* 4-3-1::            Amb and Search
* 4-3-2::            Examples of Nondeterministic Programs
* 4-3-3::            Implementing the `Amb' Evaluator

   ---------- Footnotes ----------

   (1) We assume that we have previously defined a procedure `prime?'
that tests whether numbers are prime.  Even with `prime?' defined, the
`prime-sum-pair' procedure may look suspiciously like the unhelpful
"pseudo-Lisp" attempt to define the square-root function, which we
described at the beginning of section *Note 1-1-7::.  In fact, a
square-root procedure along those lines can actually be formulated as a
nondeterministic program.  By incorporating a search mechanism into the
evaluator, we are eroding the distinction between purely declarative
descriptions and imperative specifications of how to compute answers.
We'll go even farther in this direction in section *Note 4-4::.


File: sicp.info,  Node: 4-3-1,  Next: 4-3-2,  Prev: 4-3,  Up: 4-3

4.3.1 Amb and Search
--------------------

To extend Scheme to support nondeterminism, we introduce a new special
form called `amb'.(1) The expression

     (amb <E_1> <E_2> ... <E_N>)

returns the value of one of the n expressions <E_I> "ambiguously."  For
example, the expression

     (list (amb 1 2 3) (amb 'a 'b))

can have six possible values:

     `(1 a)' `(1 b)' `(2 a)' `(2 b)' `(3 a)' `(3 b)'

   `Amb' with a single choice produces an ordinary (single) value.

   `Amb' with no choices--the expression `(amb)'--is an expression with
no acceptable values.  Operationally, we can think of `(amb)' as an
expression that when evaluated causes the computation to "fail": The
computation aborts and no value is produced.  Using this idea, we can
express the requirement that a particular predicate expression `p' must
be true as follows:

     (define (require p)
       (if (not p) (amb)))

   With `amb' and `require', we can implement the `an-element-of'
procedure used above:

     (define (an-element-of items)
       (require (not (null? items)))
       (amb (car items) (an-element-of (cdr items))))

   `An-element-of' fails if the list is empty.  Otherwise it ambiguously
returns either the first element of the list or an element chosen from
the rest of the list.

   We can also express infinite ranges of choices.  The following
procedure potentially returns any integer greater than or equal to some
given n:

     (define (an-integer-starting-from n)
       (amb n (an-integer-starting-from (+ n 1))))

   This is like the stream procedure `integers-starting-from' described
in section *Note 3-5-2::, but with an important difference: The stream
procedure returns an object that represents the sequence of all
integers beginning with n, whereas the `amb' procedure returns a single
integer.(2)

   Abstractly, we can imagine that evaluating an `amb' expression
causes time to split into branches, where the computation continues on
each branch with one of the possible values of the expression.  We say
that `amb' represents a "nondeterministic choice point".  If we had a
machine with a sufficient number of processors that could be
dynamically allocated, we could implement the search in a
straightforward way.  Execution would proceed as in a sequential
machine, until an `amb' expression is encountered.  At this point, more
processors would be allocated and initialized to continue all of the
parallel executions implied by the choice.  Each processor would proceed
sequentially as if it were the only choice, until it either terminates
by encountering a failure, or it further subdivides, or it finishes.(3)

   On the other hand, if we have a machine that can execute only one
process (or a few concurrent processes), we must consider the
alternatives sequentially.  One could imagine modifying an evaluator to
pick at random a branch to follow whenever it encounters a choice
point.  Random choice, however, can easily lead to failing values.  We
might try running the evaluator over and over, making random choices
and hoping to find a non-failing value, but it is better to "systematically
search" all possible execution paths.  The `amb' evaluator that we will
develop and work with in this section implements a systematic search as
follows: When the evaluator encounters an application of `amb', it
initially selects the first alternative.  This selection may itself
lead to a further choice.  The evaluator will always initially choose
the first alternative at each choice point.  If a choice results in a
failure, then the evaluator automagically(4) "backtracks" to the most
recent choice point and tries the next alternative.  If it runs out of
alternatives at any choice point, the evaluator will back up to the
previous choice point and resume from there.  This process leads to a
search strategy known as "depth-first search" or backtracking
"chronological backtracking".(5)

Driver loop
...........

The driver loop for the `amb' evaluator has some unusual properties.  It
reads an expression and prints the value of the first non-failing
execution, as in the `prime-sum-pair' example shown above.  If we want
to see the value of the next successful execution, we can ask the
interpreter to backtrack and attempt to generate a second non-failing
execution.  This is signaled by typing the symbol `try-again'.  If any
expression except `try-again' is given, the interpreter will start a
new problem, discarding the unexplored alternatives in the previous
problem.  Here is a sample interaction:

     ;;; Amb-Eval input:
     (prime-sum-pair '(1 3 5 8) '(20 35 110))
     ;;; Starting a new problem
     ;;; Amb-Eval value:
     (3 20)

     ;;; Amb-Eval input:
     try-again
     ;;; Amb-Eval value:
     (3 110)

     ;;; Amb-Eval input:
     try-again
     ;;; Amb-Eval value:
     (8 35)

     ;;; Amb-Eval input:
     try-again
     ;;; There are no more values of
     (prime-sum-pair (quote (1 3 5 8)) (quote (20 35 110)))

     ;;; Amb-Eval input:
     (prime-sum-pair '(19 27 30) '(11 36 58))
     ;;; Starting a new problem
     ;;; Amb-Eval value:
     (30 11)

     *Exercise 4.35:* Write a procedure `an-integer-between' that
     returns an integer between two given bounds.  This can be used to
     implement a procedure that finds Pythagorean triples, i.e.,
     triples of integers (i,j,k) between the given bounds such that i
     <= j and i^2 + j^2 = k^2, as follows:

          (define (a-pythagorean-triple-between low high)
            (let ((i (an-integer-between low high)))
              (let ((j (an-integer-between i high)))
                (let ((k (an-integer-between j high)))
                  (require (= (+ (* i i) (* j j)) (* k k)))
                  (list i j k)))))

     *Exercise 4.36:* *Note Exercise 3-69:: discussed how to generate
     the stream of _all_ Pythagorean triples, with no upper bound on
     the size of the integers to be searched.  Explain why simply
     replacing `an-integer-between' by `an-integer-starting-from' in
     the procedure in *Note Exercise 4-35:: is not an adequate way to
     generate arbitrary Pythagorean triples.  Write a procedure that
     actually will accomplish this.  (That is, write a procedure for
     which repeatedly typing `try-again' would in principle eventually
     generate all Pythagorean triples.)

     *Exercise 4.37:* Ben Bitdiddle claims that the following method
     for generating Pythagorean triples is more efficient than the one
     in *Note Exercise 4-35::.  Is he correct?  (Hint: Consider the
     number of possibilities that must be explored.)

          (define (a-pythagorean-triple-between low high)
            (let ((i (an-integer-between low high))
                  (hsq (* high high)))
              (let ((j (an-integer-between i high)))
                (let ((ksq (+ (* i i) (* j j))))
                  (require (>= hsq ksq))
                  (let ((k (sqrt ksq)))
                    (require (integer? k))
                    (list i j k))))))

   ---------- Footnotes ----------

   (1) The idea of `amb' for nondeterministic programming was first
described in 1961 by John McCarthy (see McCarthy 1967).

   (2) In actuality, the distinction between nondeterministically
returning a single choice and returning all choices depends somewhat on
our point of view.  From the perspective of the code that uses the
value, the nondeterministic choice returns a single value.  From the
perspective of the programmer designing the code, the nondeterministic
choice potentially returns all possible values, and the computation
branches so that each value is investigated separately.

   (3) One might object that this is a hopelessly inefficient
mechanism.  It might require millions of processors to solve some
easily stated problem this way, and most of the time most of those
processors would be idle.  This objection should be taken in the
context of history.  Memory used to be considered just such an
expensive commodity.  In 1964 a megabyte of RAM cost about $400,000.
Now every personal computer has many megabytes of RAM, and most of the
time most of that RAM is unused.  It is hard to underestimate the cost
of mass-produced electronics.

   (4) Automagically: "Automatically, but in a way which, for some
reason (typically because it is too complicated, or too ugly, or
perhaps even too trivial), the speaker doesn't feel like explaining."
(Steele 1983, Raymond 1993)

   (5) [Footnote 4.47] The integration of automatic search strategies
into programming languages has had a long and checkered history.  The
first suggestions that nondeterministic algorithms might be elegantly
encoded in a programming language with search and automatic
backtracking came from Robert Floyd (1967).  Carl Hewitt (1969)
invented a programming language called Planner that explicitly
supported automatic chronological backtracking, providing for a
built-in depth-first search strategy.  Sussman, Winograd, and Charniak
(1971) implemented a subset of this language, called MicroPlanner,
which was used to support work in problem solving and robot planning.
Similar ideas, arising from logic and theorem proving, led to the
genesis in Edinburgh and Marseille of the elegant language Prolog
(which we will discuss in section *Note 4-4::).  After sufficient
frustration with automatic search, McDermott and Sussman (1972)
developed a language called Conniver, which included mechanisms for
placing the search strategy under programmer control.  This proved
unwieldy, however, and Sussman and Stallman (1975) found a more
tractable approach while investigating methods of symbolic analysis for
electrical circuits.  They developed a non-chronological backtracking
scheme that was based on tracing out the logical dependencies
connecting facts, a technique that has come to be known as "dependency-directed
backtracking".  Although their method was complex, it produced
reasonably efficient programs because it did little redundant search.
Doyle (1979) and McAllester (1978, 1980) generalized and clarified the
methods of Stallman and Sussman, developing a new paradigm for
formulating search that is now called "truth maintenance".  Modern
problem-solving systems all use some form of truth-maintenance system
as a substrate.  See Forbus and deKleer 1993 for a discussion of
elegant ways to build truth-maintenance systems and applications using
truth maintenance.  Zabih, McAllester, and Chapman 1987 describes a
nondeterministic extension to Scheme that is based on `amb'; it is
similar to the interpreter described in this section, but more
sophisticated, because it uses dependency-directed backtracking rather
than chronological backtracking.  Winston 1992 gives an introduction to
both kinds of backtracking.


File: sicp.info,  Node: 4-3-2,  Next: 4-3-3,  Prev: 4-3-1,  Up: 4-3

4.3.2 Examples of Nondeterministic Programs
-------------------------------------------

Section *Note 4-3-3:: describes the implementation of the `amb'
evaluator.  First, however, we give some examples of how it can be
used.  The advantage of nondeterministic programming is that we can
suppress the details of how search is carried out, thereby expressing
our programs at a higher level of abstraction.

Logic Puzzles
.............

The following puzzle (taken from Dinesman 1968) is typical of a large
class of simple logic puzzles:

     Baker, Cooper, Fletcher, Miller, and Smith live on different
     floors of an apartment house that contains only five floors.
     Baker does not live on the top floor.  Cooper does not live on the
     bottom floor.  Fletcher does not live on either the top or the
     bottom floor.  Miller lives on a higher floor than does Cooper.
     Smith does not live on a floor adjacent to Fletcher's.  Fletcher
     does not live on a floor adjacent to Cooper's.  Where does
     everyone live?

   We can determine who lives on each floor in a straightforward way by
enumerating all the possibilities and imposing the given
restrictions:(1)

     (define (multiple-dwelling)
       (let ((baker (amb 1 2 3 4 5))
             (cooper (amb 1 2 3 4 5))
             (fletcher (amb 1 2 3 4 5))
             (miller (amb 1 2 3 4 5))
             (smith (amb 1 2 3 4 5)))
         (require
          (distinct? (list baker cooper fletcher miller smith)))
         (require (not (= baker 5)))
         (require (not (= cooper 1)))
         (require (not (= fletcher 5)))
         (require (not (= fletcher 1)))
         (require (> miller cooper))
         (require (not (= (abs (- smith fletcher)) 1)))
         (require (not (= (abs (- fletcher cooper)) 1)))
         (list (list 'baker baker)
               (list 'cooper cooper)
               (list 'fletcher fletcher)
               (list 'miller miller)
               (list 'smith smith))))

   Evaluating the expression `(multiple-dwelling)' produces the result

     ((baker 3) (cooper 2) (fletcher 4) (miller 5) (smith 1))

   Although this simple procedure works, it is very slow.  *Note
Exercise 4-39:: and *Note Exercise 4-40:: discuss some possible
improvements.

     *Exercise 4.38:* Modify the multiple-dwelling procedure to omit
     the requirement that Smith and Fletcher do not live on adjacent
     floors.  How many solutions are there to this modified puzzle?

     *Exercise 4.39:* Does the order of the restrictions in the
     multiple-dwelling procedure affect the answer? Does it affect the
     time to find an answer?  If you think it matters, demonstrate a
     faster program obtained from the given one by reordering the
     restrictions.  If you think it does not matter, argue your case.

     *Exercise 4.40:* In the multiple dwelling problem, how many sets
     of assignments are there of people to floors, both before and
     after the requirement that floor assignments be distinct?  It is
     very inefficient to generate all possible assignments of people to
     floors and then leave it to backtracking to eliminate them.  For
     example, most of the restrictions depend on only one or two of the
     person-floor variables, and can thus be imposed before floors have
     been selected for all the people.  Write and demonstrate a much
     more efficient nondeterministic procedure that solves this problem
     based upon generating only those possibilities that are not already
     ruled out by previous restrictions.  (Hint: This will require a
     nest of `let' expressions.)

     *Exercise 4.41:* Write an ordinary Scheme program to solve the
     multiple dwelling puzzle.

     *Exercise 4.42:* Solve the following "Liars" puzzle (from Phillips
     1934):

     Five schoolgirls sat for an examination.  Their parents--so they
     thought--showed an undue degree of interest in the result.  They
     therefore agreed that, in writing home about the examination, each
     girl should make one true statement and one untrue one.  The
     following are the relevant passages from their letters:

        * Betty: "Kitty was second in the examination.  I was only
          third."

        * Ethel: "You'll be glad to hear that I was on top.  Joan was
          second."

        * Joan: "I was third, and poor old Ethel was bottom."

        * Kitty: "I came out second.  Mary was only fourth."

        * Mary: "I was fourth.  Top place was taken by Betty."


     What in fact was the order in which the five girls were placed?

     *Exercise 4.43:* Use the `amb' evaluator to solve the following
     puzzle:(2)

          Mary Ann Moore's father has a yacht and so has each of his
          four friends: Colonel Downing, Mr. Hall, Sir Barnacle Hood,
          and Dr.  Parker.  Each of the five also has one daughter and
          each has named his yacht after a daughter of one of the
          others.  Sir Barnacle's yacht is the Gabrielle, Mr. Moore
          owns the Lorna; Mr. Hall the Rosalind.  The Melissa, owned by
          Colonel Downing, is named after Sir Barnacle's daughter.
          Gabrielle's father owns the yacht that is named after Dr.
          Parker's daughter.  Who is Lorna's father?

     Try to write the program so that it runs efficiently (see *Note
     Exercise 4-40::).  Also determine how many solutions there are if
     we are not told that Mary Ann's last name is Moore.

     *Exercise 4.44:* *Note Exercise 2-42:: described the "eight-queens
     puzzle" of placing queens on a chessboard so that no two attack
     each other.  Write a nondeterministic program to solve this puzzle.

Parsing natural language
........................

Programs designed to accept natural language as input usually start by
attempting to "parse" the input, that is, to match the input against
some grammatical structure.  For example, we might try to recognize
simple sentences consisting of an article followed by a noun followed
by a verb, such as "The cat eats."  To accomplish such an analysis, we
must be able to identify the parts of speech of individual words.  We
could start with some lists that classify various words:(3)

     (define nouns '(noun student professor cat class))

     (define verbs '(verb studies lectures eats sleeps))

     (define articles '(article the a))

   We also need a "grammar", that is, a set of rules describing how
grammatical elements are composed from simpler elements.  A very simple
grammar might stipulate that a sentence always consists of two
pieces--a noun phrase followed by a verb--and that a noun phrase
consists of an article followed by a noun.  With this grammar, the
sentence "The cat eats" is parsed as follows:

     (sentence (noun-phrase (article the) (noun cat))
               (verb eats))

   We can generate such a parse with a simple program that has separate
procedures for each of the grammatical rules.  To parse a sentence, we
identify its two constituent pieces and return a list of these two
elements, tagged with the symbol `sentence':

     (define (parse-sentence)
       (list 'sentence
              (parse-noun-phrase)
              (parse-word verbs)))

   A noun phrase, similarly, is parsed by finding an article followed
by a noun:

     (define (parse-noun-phrase)
       (list 'noun-phrase
             (parse-word articles)
             (parse-word nouns)))

   At the lowest level, parsing boils down to repeatedly checking that
the next unparsed word is a member of the list of words for the
required part of speech.  To implement this, we maintain a global
variable `*unparsed*', which is the input that has not yet been parsed.
Each time we check a word, we require that `*unparsed*' must be
non-empty and that it should begin with a word from the designated
list.  If so, we remove that word from `*unparsed*' and return the word
together with its part of speech (which is found at the head of the
list):(4)

     (define (parse-word word-list)
       (require (not (null? *unparsed*)))
       (require (memq (car *unparsed*) (cdr word-list)))
       (let ((found-word (car *unparsed*)))
         (set! *unparsed* (cdr *unparsed*))
         (list (car word-list) found-word)))

   To start the parsing, all we need to do is set `*unparsed*' to be the
entire input, try to parse a sentence, and check that nothing is left
over:

     (define *unparsed* '())

     (define (parse input)
       (set! *unparsed* input)
       (let ((sent (parse-sentence)))
         (require (null? *unparsed*))
         sent))

   We can now try the parser and verify that it works for our simple
test sentence:

     ;;; Amb-Eval input:
     (parse '(the cat eats))
     ;;; Starting a new problem
     ;;; Amb-Eval value:
     (sentence (noun-phrase (article the) (noun cat)) (verb eats))

   The `amb' evaluator is useful here because it is convenient to
express the parsing constraints with the aid of `require'.  Automatic
search and backtracking really pay off, however, when we consider more
complex grammars where there are choices for how the units can be
decomposed.

   Let's add to our grammar a list of prepositions:

     (define prepositions '(prep for to in by with))

and define a prepositional phrase (e.g., "for the cat") to be a
preposition followed by a noun phrase:

     (define (parse-prepositional-phrase)
       (list 'prep-phrase
             (parse-word prepositions)
             (parse-noun-phrase)))

   Now we can define a sentence to be a noun phrase followed by a verb
phrase, where a verb phrase can be either a verb or a verb phrase
extended by a prepositional phrase:(5)

     (define (parse-sentence)
       (list 'sentence
              (parse-noun-phrase)
              (parse-verb-phrase)))

     (define (parse-verb-phrase)
       (define (maybe-extend verb-phrase)
         (amb verb-phrase
              (maybe-extend (list 'verb-phrase
                                  verb-phrase
                                  (parse-prepositional-phrase)))))
       (maybe-extend (parse-word verbs)))

   While we're at it, we can also elaborate the definition of noun
phrases to permit such things as "a cat in the class."  What we used to
call a noun phrase, we'll now call a simple noun phrase, and a noun
phrase will now be either a simple noun phrase or a noun phrase
extended by a prepositional phrase:

     (define (parse-simple-noun-phrase)
       (list 'simple-noun-phrase
             (parse-word articles)
             (parse-word nouns)))

     (define (parse-noun-phrase)
       (define (maybe-extend noun-phrase)
         (amb noun-phrase
              (maybe-extend (list 'noun-phrase
                                  noun-phrase
                                  (parse-prepositional-phrase)))))
       (maybe-extend (parse-simple-noun-phrase)))

   Our new grammar lets us parse more complex sentences.  For example

     (parse '(the student with the cat sleeps in the class))

produces

     (sentence
      (noun-phrase
       (simple-noun-phrase (article the) (noun student))
       (prep-phrase (prep with)
                    (simple-noun-phrase
                     (article the) (noun cat))))
      (verb-phrase
       (verb sleeps)
       (prep-phrase (prep in)
                    (simple-noun-phrase
                     (article the) (noun class)))))

   Observe that a given input may have more than one legal parse.  In
the sentence "The professor lectures to the student with the cat," it
may be that the professor is lecturing with the cat, or that the
student has the cat.  Our nondeterministic program finds both
possibilities:

     (parse '(the professor lectures to the student with the cat))

produces

     (sentence
      (simple-noun-phrase (article the) (noun professor))
      (verb-phrase
       (verb-phrase
        (verb lectures)
        (prep-phrase (prep to)
                     (simple-noun-phrase
                      (article the) (noun student))))
       (prep-phrase (prep with)
                    (simple-noun-phrase
                     (article the) (noun cat)))))

   Asking the evaluator to try again yields

     (sentence
      (simple-noun-phrase (article the) (noun professor))
      (verb-phrase
       (verb lectures)
       (prep-phrase (prep to)
                    (noun-phrase
                     (simple-noun-phrase
                      (article the) (noun student))
                     (prep-phrase (prep with)
                                  (simple-noun-phrase
                                   (article the) (noun cat)))))))

     *Exercise 4.45:* With the grammar given above, the following
     sentence can be parsed in five different ways: "The professor
     lectures to the student in the class with the cat."  Give the five
     parses and explain the differences in shades of meaning among them.

     *Exercise 4.46:* The evaluators in sections *Note 4-1:: and *Note
     4-2:: do not determine what order operands are evaluated in.  We
     will see that the `amb' evaluator evaluates them from left to
     right.  Explain why our parsing program wouldn't work if the
     operands were evaluated in some other order.

     *Exercise 4.47:* Louis Reasoner suggests that, since a verb phrase
     is either a verb or a verb phrase followed by a prepositional
     phrase, it would be much more straightforward to define the
     procedure `parse-verb-phrase' as follows (and similarly for noun
     phrases):

          (define (parse-verb-phrase)
            (amb (parse-word verbs)
                 (list 'verb-phrase
                       (parse-verb-phrase)
                       (parse-prepositional-phrase))))

     Does this work?  Does the program's behavior change if we
     interchange the order of expressions in the `amb'

     *Exercise 4.48:* Extend the grammar given above to handle more
     complex sentences.  For example, you could extend noun phrases and
     verb phrases to include adjectives and adverbs, or you could
     handle compound sentences.(6)

     *Exercise 4.49:* Alyssa P. Hacker is more interested in generating
     interesting sentences than in parsing them.  She reasons that by
     simply changing the procedure `parse-word' so that it ignores the
     "input sentence" and instead always succeeds and generates an
     appropriate word, we can use the programs we had built for parsing
     to do generation instead.  Implement Alyssa's idea, and show the
     first half-dozen or so sentences generated.(7)

   ---------- Footnotes ----------

   (1) Our program uses the following procedure to determine if the
elements of a list are distinct:

     (define (distinct? items)
       (cond ((null? items) true)
             ((null? (cdr items)) true)
             ((member (car items) (cdr items)) false)
             (else (distinct? (cdr items)))))

   `Member' is like `memq' except that it uses `equal?' instead of
`eq?' to test for equality.

   (2) This is taken from a booklet called "Problematical Recreations,"
published in the 1960s by Litton Industries, where it is attributed to
the `Kansas State Engineer'.

   (3) Here we use the convention that the first element of each list
designates the part of speech for the rest of the words in the list.

   (4) Notice that `parse-word' uses `set!' to modify the unparsed
input list.  For this to work, our `amb' evaluator must undo the
effects of `set!' operations when it backtracks.

   (5) Observe that this definition is recursive--a verb may be
followed by any number of prepositional phrases.

   (6) This kind of grammar can become arbitrarily complex, but it is
only a toy as far as real language understanding is concerned.  Real
natural-language understanding by computer requires an elaborate
mixture of syntactic analysis and interpretation of meaning.  On the
other hand, even toy parsers can be useful in supporting flexible
command languages for programs such as information-retrieval systems.
Winston 1992 discusses computational approaches to real language
understanding and also the applications of simple grammars to command
languages.

   (7) Although Alyssa's idea works just fine (and is surprisingly
simple), the sentences that it generates are a bit boring--they don't
sample the possible sentences of this language in a very interesting
way.  In fact, the grammar is highly recursive in many places, and
Alyssa's technique "falls into" one of these recursions and gets stuck.
See *Note Exercise 4-50:: for a way to deal with this.


File: sicp.info,  Node: 4-3-3,  Prev: 4-3-2,  Up: 4-3

4.3.3 Implementing the `Amb' Evaluator
--------------------------------------

The evaluation of an ordinary Scheme expression may return a value, may
never terminate, or may signal an error.  In nondeterministic Scheme
the evaluation of an expression may in addition result in the discovery
of a dead end, in which case evaluation must backtrack to a previous
choice point.  The interpretation of nondeterministic Scheme is
complicated by this extra case.

   We will construct the `amb' evaluator for nondeterministic Scheme by
modifying the analyzing evaluator of section *Note 4-1-7::.(1)  As in
the analyzing evaluator, evaluation of an expression is accomplished by
calling an execution procedure produced by analysis of that expression.
The difference between the interpretation of ordinary Scheme and the
interpretation of nondeterministic Scheme will be entirely in the
execution procedures.

Execution procedures and continuations
......................................

Recall that the execution procedures for the ordinary evaluator take one
argument: the environment of execution.  In contrast, the execution
procedures in the `amb' evaluator take three arguments: the
environment, and two procedures called "continuation procedures".  The
evaluation of an expression will finish by calling one of these two
continuations: If the evaluation results in a value, the "success
continuation" is called with that value; if the evaluation results in
the discovery of a dead end, the "failure continuation" is called.
Constructing and calling appropriate continuations is the mechanism by
which the nondeterministic evaluator implements backtracking.

   It is the job of the success continuation to receive a value and
proceed with the computation.  Along with that value, the success
continuation is passed another failure continuation, which is to be
called subsequently if the use of that value leads to a dead end.

   It is the job of the failure continuation to try another branch of
the nondeterministic process.  The essence of the nondeterministic
language is in the fact that expressions may represent choices among
alternatives.  The evaluation of such an expression must proceed with
one of the indicated alternative choices, even though it is not known
in advance which choices will lead to acceptable results.  To deal with
this, the evaluator picks one of the alternatives and passes this value
to the success continuation.  Together with this value, the evaluator
constructs and passes along a failure continuation that can be called
later to choose a different alternative.

   A failure is triggered during evaluation (that is, a failure
continuation is called) when a user program explicitly rejects the
current line of attack (for example, a call to `require' may result in
execution of `(amb)', an expression that always fails--see section
*Note 4-3-1::).  The failure continuation in hand at that point will
cause the most recent choice point to choose another alternative.  If
there are no more alternatives to be considered at that choice point, a
failure at an earlier choice point is triggered, and so on.  Failure
continuations are also invoked by the driver loop in response to a
`try-again' request, to find another value of the expression.

   In addition, if a side-effect operation (such as assignment to a
variable) occurs on a branch of the process resulting from a choice, it
may be necessary, when the process finds a dead end, to undo the side
effect before making a new choice.  This is accomplished by having the
side-effect operation produce a failure continuation that undoes the
side effect and propagates the failure.

   In summary, failure continuations are constructed by

   * `amb' expressions--to provide a mechanism to make alternative
     choices if the current choice made by the `amb' expression leads
     to a dead end;

   * the top-level driver--to provide a mechanism to report failure
     when the choices are exhausted;

   * assignments--to intercept failures and undo assignments during
     backtracking.


   Failures are initiated only when a dead end is encountered.  This
occurs

   * if the user program executes `(amb)';

   * if the user types `try-again' at the top-level driver.


   Failure continuations are also called during processing of a failure:

   * When the failure continuation created by an assignment finishes
     undoing a side effect, it calls the failure continuation it
     intercepted, in order to propagate the failure back to the choice
     point that led to this assignment or to the top level.

   * When the failure continuation for an `amb' runs out of choices, it
     calls the failure continuation that was originally given to the
     `amb', in order to propagate the failure back to the previous
     choice point or to the top level.


Structure of the evaluator
..........................

The syntax- and data-representation procedures for the `amb' evaluator,
and also the basic `analyze' procedure, are identical to those in the
evaluator of section *Note 4-1-7::, except for the fact that we need
additional syntax procedures to recognize the `amb' special form:(2)

     (define (amb? exp) (tagged-list? exp 'amb))

     (define (amb-choices exp) (cdr exp))

   We must also add to the dispatch in `analyze' a clause that will
recognize this special form and generate an appropriate execution
procedure:

     ((amb? exp) (analyze-amb exp))

   The top-level procedure `ambeval' (similar to the version of `eval'
given in section *Note 4-1-7::) analyzes the given expression and
applies the resulting execution procedure to the given environment,
together with two given continuations:

     (define (ambeval exp env succeed fail)
       ((analyze exp) env succeed fail))

   A success continuation is a procedure of two arguments: the value
just obtained and another failure continuation to be used if that value
leads to a subsequent failure. A failure continuation is a procedure of
no arguments.  So the general form of an execution procedure is

     (lambda (env succeed fail)
       ;; `succeed' is `(lambda (value fail) ...)'
       ;; `fail' is `(lambda () ...)'
       ...)

   For example, executing

     (ambeval <EXP>
              the-global-environment
              (lambda (value fail) value)
              (lambda () 'failed))

will attempt to evaluate the given expression and will return either the
expression's value (if the evaluation succeeds) or the symbol `failed'
(if the evaluation fails).  The call to `ambeval' in the driver loop
shown below uses much more complicated continuation procedures, which
continue the loop and support the `try-again' request.

   Most of the complexity of the `amb' evaluator results from the
mechanics of passing the continuations around as the execution
procedures call each other.  In going through the following code, you
should compare each of the execution procedures with the corresponding
procedure for the ordinary evaluator given in section *Note 4-1-7::.

Simple expressions
..................

The execution procedures for the simplest kinds of expressions are
essentially the same as those for the ordinary evaluator, except for
the need to manage the continuations.  The execution procedures simply
succeed with the value of the expression, passing along the failure
continuation that was passed to them.

     (define (analyze-self-evaluating exp)
       (lambda (env succeed fail)
         (succeed exp fail)))

     (define (analyze-quoted exp)
       (let ((qval (text-of-quotation exp)))
         (lambda (env succeed fail)
           (succeed qval fail))))

     (define (analyze-variable exp)
       (lambda (env succeed fail)
         (succeed (lookup-variable-value exp env)
                  fail)))

     (define (analyze-lambda exp)
       (let ((vars (lambda-parameters exp))
             (bproc (analyze-sequence (lambda-body exp))))
         (lambda (env succeed fail)
           (succeed (make-procedure vars bproc env)
                    fail))))

   Notice that looking up a variable always "succeeds."  If
`lookup-variable-value' fails to find the variable, it signals an error,
as usual.  Such a "failure" indicates a program bug--a reference to an
unbound variable; it is not an indication that we should try another
nondeterministic choice instead of the one that is currently being
tried.

Conditionals and sequences
..........................

Conditionals are also handled in a similar way as in the ordinary
evaluator.  The execution procedure generated by `analyze-if' invokes
the predicate execution procedure `pproc' with a success continuation
that checks whether the predicate value is true and goes on to execute
either the consequent or the alternative.  If the execution of `pproc'
fails, the original failure continuation for the `if' expression is
called.

     (define (analyze-if exp)
       (let ((pproc (analyze (if-predicate exp)))
             (cproc (analyze (if-consequent exp)))
             (aproc (analyze (if-alternative exp))))
         (lambda (env succeed fail)
           (pproc env
                  ;; success continuation for evaluating the predicate
                  ;; to obtain `pred-value'
                  (lambda (pred-value fail2)
                    (if (true? pred-value)
                        (cproc env succeed fail2)
                        (aproc env succeed fail2)))
                  ;; failure continuation for evaluating the predicate
                  fail))))

   Sequences are also handled in the same way as in the previous
evaluator, except for the machinations in the subprocedure
`sequentially' that are required for passing the continuations.
Namely, to sequentially execute `a' and then `b', we call `a' with a
success continuation that calls `b'.

     (define (analyze-sequence exps)
       (define (sequentially a b)
         (lambda (env succeed fail)
           (a env
              ;; success continuation for calling `a'
              (lambda (a-value fail2)
                (b env succeed fail2))
              ;; failure continuation for calling `a'
              fail)))
       (define (loop first-proc rest-procs)
         (if (null? rest-procs)
             first-proc
             (loop (sequentially first-proc (car rest-procs))
                   (cdr rest-procs))))
       (let ((procs (map analyze exps)))
         (if (null? procs)
             (error "Empty sequence -- ANALYZE"))
         (loop (car procs) (cdr procs))))

Definitions and assignments
...........................

Definitions are another case where we must go to some trouble to manage
the continuations, because it is necessary to evaluate the
definition-value expression before actually defining the new variable.
To accomplish this, the definition-value execution procedure `vproc' is
called with the environment, a success continuation, and the failure
continuation.  If the execution of `vproc' succeeds, obtaining a value
`val' for the defined variable, the variable is defined and the success
is propagated:

     (define (analyze-definition exp)
       (let ((var (definition-variable exp))
             (vproc (analyze (definition-value exp))))
         (lambda (env succeed fail)
           (vproc env
                  (lambda (val fail2)
                    (define-variable! var val env)
                    (succeed 'ok fail2))
                  fail))))

   Assignments are more interesting.  This is the first place where we
really use the continuations, rather than just passing them around.
The execution procedure for assignments starts out like the one for
definitions.  It first attempts to obtain the new value to be assigned
to the variable. If this evaluation of `vproc' fails, the assignment
fails.

   If `vproc' succeeds, however, and we go on to make the assignment,
we must consider the possibility that this branch of the computation
might later fail, which will require us to backtrack out of the
assignment.  Thus, we must arrange to undo the assignment as part of
the backtracking process.(3)

   This is accomplished by giving `vproc' a success continuation
(marked with the comment "*1*" below) that saves the old value of the
variable before assigning the new value to the variable and proceeding
from the assignment.  The failure continuation that is passed along
with the value of the assignment (marked with the comment "*2*" below)
restores the old value of the variable before continuing the failure.
That is, a successful assignment provides a failure continuation that
will intercept a subsequent failure; whatever failure would otherwise
have called `fail2' calls this procedure instead, to undo the
assignment before actually calling `fail2'.

     (define (analyze-assignment exp)
       (let ((var (assignment-variable exp))
             (vproc (analyze (assignment-value exp))))
         (lambda (env succeed fail)
           (vproc env
                  (lambda (val fail2)        ; *1*
                    (let ((old-value
                           (lookup-variable-value var env)))
                      (set-variable-value! var val env)
                      (succeed 'ok
                               (lambda ()    ; *2*
                                 (set-variable-value! var
                                                      old-value
                                                      env)
                                 (fail2)))))
                  fail))))

Procedure applications
......................

The execution procedure for applications contains no new ideas except
for the technical complexity of managing the continuations.  This
complexity arises in `analyze-application', due to the need to keep
track of the success and failure continuations as we evaluate the
operands.  We use a procedure `get-args' to evaluate the list of
operands, rather than a simple `map' as in the ordinary evaluator.

     (define (analyze-application exp)
       (let ((fproc (analyze (operator exp)))
             (aprocs (map analyze (operands exp))))
         (lambda (env succeed fail)
           (fproc env
                  (lambda (proc fail2)
                    (get-args aprocs
                              env
                              (lambda (args fail3)
                                (execute-application
                                 proc args succeed fail3))
                              fail2))
                  fail))))

   In `get-args', notice how `cdr'ing down the list of `aproc'
execution procedures and `cons'ing up the resulting list of `args' is
accomplished by calling each `aproc' in the list with a success
continuation that recursively calls `get-args'.  Each of these recursive
calls to `get-args' has a success continuation whose value is the
`cons' of the newly obtained argument onto the list of accumulated
arguments:

     (define (get-args aprocs env succeed fail)
       (if (null? aprocs)
           (succeed '() fail)
           ((car aprocs) env
                         ;; success continuation for this `aproc'
                         (lambda (arg fail2)
                           (get-args (cdr aprocs)
                                     env
                                     ;; success continuation for recursive
                                     ;; call to `get-args'
                                     (lambda (args fail3)
                                       (succeed (cons arg args)
                                                fail3))
                                     fail2))
                         fail)))

   The actual procedure application, which is performed by
`execute-application', is accomplished in the same way as for the
ordinary evaluator, except for the need to manage the continuations.

     (define (execute-application proc args succeed fail)
       (cond ((primitive-procedure? proc)
              (succeed (apply-primitive-procedure proc args)
                       fail))
             ((compound-procedure? proc)
              ((procedure-body proc)
               (extend-environment (procedure-parameters proc)
                                   args
                                   (procedure-environment proc))
               succeed
               fail))
             (else
              (error
               "Unknown procedure type -- EXECUTE-APPLICATION"
               proc))))

Evaluating `amb' expressions
............................

The `amb' special form is the key element in the nondeterministic
language.  Here we see the essence of the interpretation process and
the reason for keeping track of the continuations.  The execution
procedure for `amb' defines a loop `try-next' that cycles through the
execution procedures for all the possible values of the `amb'
expression.  Each execution procedure is called with a failure
continuation that will try the next one.  When there are no more
alternatives to try, the entire `amb' expression fails.

     (define (analyze-amb exp)
       (let ((cprocs (map analyze (amb-choices exp))))
         (lambda (env succeed fail)
           (define (try-next choices)
             (if (null? choices)
                 (fail)
                 ((car choices) env
                                succeed
                                (lambda ()
                                  (try-next (cdr choices))))))
           (try-next cprocs))))

Driver loop
...........

The driver loop for the `amb' evaluator is complex, due to the mechanism
that permits the user to try again in evaluating an expression.  The
driver uses a procedure called `internal-loop', which takes as argument
a procedure `try-again'.  The intent is that calling `try-again' should
go on to the next untried alternative in the nondeterministic
evaluation.  `Internal-loop' either calls `try-again' in response to
the user typing `try-again' at the driver loop, or else starts a new
evaluation by calling `ambeval'.

   The failure continuation for this call to `ambeval' informs the user
that there are no more values and re-invokes the driver loop.

   The success continuation for the call to `ambeval' is more subtle.
We print the obtained value and then invoke the internal loop again
with a `try-again' procedure that will be able to try the next
alternative.  This `next-alternative' procedure is the second argument
that was passed to the success continuation.  Ordinarily, we think of
this second argument as a failure continuation to be used if the
current evaluation branch later fails.  In this case, however, we have
completed a successful evaluation, so we can invoke the "failure"
alternative branch in order to search for additional successful
evaluations.

     (define input-prompt ";;; Amb-Eval input:")
     (define output-prompt ";;; Amb-Eval value:")

     (define (driver-loop)
       (define (internal-loop try-again)
         (prompt-for-input input-prompt)
         (let ((input (read)))
           (if (eq? input 'try-again)
               (try-again)
               (begin
                 (newline)
                 (display ";;; Starting a new problem ")
                 (ambeval input
                          the-global-environment
                          ;; `ambeval' success
                          (lambda (val next-alternative)
                            (announce-output output-prompt)
                            (user-print val)
                            (internal-loop next-alternative))
                          ;; `ambeval' failure
                          (lambda ()
                            (announce-output
                             ";;; There are no more values of")
                            (user-print input)
                            (driver-loop)))))))
       (internal-loop
        (lambda ()
          (newline)
          (display ";;; There is no current problem")
          (driver-loop))))

   The initial call to `internal-loop' uses a `try-again' procedure that
complains that there is no current problem and restarts the driver
loop.  This is the behavior that will happen if the user types
`try-again' when there is no evaluation in progress.

     *Exercise 4.50:* Implement a new special form `ramb' that is like
     `amb' except that it searches alternatives in a random order,
     rather than from left to right.  Show how this can help with
     Alyssa's problem in *Note Exercise 4-49::.

     *Exercise 4.51:* Implement a new kind of assignment called
     `permanent-set!' that is not undone upon failure.  For example, we
     can choose two distinct elements from a list and count the number
     of trials required to make a successful choice as follows:

          (define count 0)

          (let ((x (an-element-of '(a b c)))
                (y (an-element-of '(a b c))))
            (permanent-set! count (+ count 1))
            (require (not (eq? x y)))
            (list x y count))
          ;;; Starting a new problem
          ;;; Amb-Eval value:
          (a b 2)

          ;;; Amb-Eval input:
          try-again
          ;;; Amb-Eval value:
          (a c 3)

     What values would have been displayed if we had used `set!' here
     rather than `permanent-set!' ?

     *Exercise 4.52:* Implement a new construct called `if-fail' that
     permits the user to catch the failure of an expression.  `If-fail'
     takes two expressions.  It evaluates the first expression as usual
     and returns as usual if the evaluation succeeds.  If the evaluation
     fails, however, the value of the second expression is returned, as
     in the following example:

          ;;; Amb-Eval input:
          (if-fail (let ((x (an-element-of '(1 3 5))))
                     (require (even? x))
                     x)
                   'all-odd)
          ;;; Starting a new problem
          ;;; Amb-Eval value:
          all-odd

          ;;; Amb-Eval input:
          (if-fail (let ((x (an-element-of '(1 3 5 8))))
                     (require (even? x))
                     x)
                   'all-odd)
          ;;; Starting a new problem
          ;;; Amb-Eval value:
          8

     *Exercise 4.53:* With `permanent-set!' as described in *Note
     Exercise 4-51:: and `if-fail' as in *Note Exercise 4-52::, what
     will be the result of evaluating

          (let ((pairs '()))
            (if-fail (let ((p (prime-sum-pair '(1 3 5 8) '(20 35 110))))
                       (permanent-set! pairs (cons p pairs))
                       (amb))
                     pairs))

     *Exercise 4.54:* If we had not realized that `require' could be
     implemented as an ordinary procedure that uses `amb', to be
     defined by the user as part of a nondeterministic program, we
     would have had to implement it as a special form.  This would
     require syntax procedures

          (define (require? exp) (tagged-list? exp 'require))

          (define (require-predicate exp) (cadr exp))

     and a new clause in the dispatch in `analyze'

          ((require? exp) (analyze-require exp))

     as well the procedure `analyze-require' that handles `require'
     expressions.  Complete the following definition of
     `analyze-require'.

          (define (analyze-require exp)
            (let ((pproc (analyze (require-predicate exp))))
              (lambda (env succeed fail)
                (pproc env
                       (lambda (pred-value fail2)
                         (if <??>
                             <??>
                             (succeed 'ok fail2)))
                       fail))))

   ---------- Footnotes ----------

   (1) We chose to implement the lazy evaluator in section *Note 4-2::
as a modification of the ordinary metacircular evaluator of section
*Note 4-1-1::.  In contrast, we will base the `amb' evaluator on the
analyzing evaluator of section *Note 4-1-7::, because the execution
procedures in that evaluator provide a convenient framework for
implementing backtracking.

   (2) We assume that the evaluator supports `let' (see *Note Exercise
4-22::), which we have used in our nondeterministic programs.

   (3) We didn't worry about undoing definitions, since we can assume
that internal definitions are scanned out (section *Note 4-1-6::).


File: sicp.info,  Node: 4-4,  Prev: 4-3,  Up: Chapter 4

4.4 Logic Programming
=====================

In *Note Chapter 1:: we stressed that computer science deals with
imperative (how to) knowledge, whereas mathematics deals with
declarative (what is) knowledge.  Indeed, programming languages require
that the programmer express knowledge in a form that indicates the
step-by-step methods for solving particular problems.  On the other
hand, high-level languages provide, as part of the language
implementation, a substantial amount of methodological knowledge that
frees the user from concern with numerous details of how a specified
computation will progress.

   Most programming languages, including Lisp, are organized around
computing the values of mathematical functions.  Expression-oriented
languages (such as Lisp, Fortran, and Algol) capitalize on the "pun"
that an expression that describes the value of a function may also be
interpreted as a means of computing that value.  Because of this, most
programming languages are strongly biased toward unidirectional
computations (computations with well-defined inputs and outputs).
There are, however, radically different programming languages that
relax this bias.  We saw one such example in section *Note 3-3-5::,
where the objects of computation were arithmetic constraints.  In a
constraint system the direction and the order of computation are not so
well specified; in carrying out a computation the system must therefore
provide more detailed "how to" knowledge than would be the case with an
ordinary arithmetic computation.  This does not mean, however, that the
user is released altogether from the responsibility of providing
imperative knowledge.  There are many constraint networks that
implement the same set of constraints, and the user must choose from
the set of mathematically equivalent networks a suitable network to
specify a particular computation.

   The nondeterministic program evaluator of section *Note 4-3:: also
moves away from the view that programming is about constructing
algorithms for computing unidirectional functions.  In a
nondeterministic language, expressions can have more than one value,
and, as a result, the computation is dealing with relations rather than
with single-valued functions.  Logic programming extends this idea by
combining a relational vision of programming with a powerful kind of
symbolic pattern matching called "unification".(1)

   This approach, when it works, can be a very powerful way to write
programs.  Part of the power comes from the fact that a single "what
is" fact can be used to solve a number of different problems that would
have different "how to" components.  As an example, consider the
`append' operation, which takes two lists as arguments and combines
their elements to form a single list.  In a procedural language such as
Lisp, we could define `append' in terms of the basic list constructor
`cons', as we did in section *Note 2-2-1:::

     (define (append x y)
       (if (null? x)
           y
           (cons (car x) (append (cdr x) y))))

   This procedure can be regarded as a translation into Lisp of the
following two rules, the first of which covers the case where the first
list is empty and the second of which handles the case of a nonempty
list, which is a `cons' of two parts:

   * For any list `y', the empty list and `y' `append' to form `y'.

   * For any `u', `v', `y', and `z', `(cons u v)' and `y' `append' to
     form `(cons u z)' if `v' and `y' `append' to form `z'.(2)


   Using the `append' procedure, we can answer questions such as

     Find the `append' of `(a b)' and `(c d)'.

   But the same two rules are also sufficient for answering the
following sorts of questions, which the procedure can't answer:

     Find a list `y' that `append's with `(a b)' to produce `(a b c d)'.

     Find all `x' and `y' that `append' to form `(a b c d)'.

   In a logic programming language, the programmer writes an `append'
"procedure" by stating the two rules about `append' given above.  "How
to" knowledge is provided automatically by the interpreter to allow this
single pair of rules to be used to answer all three types of questions
about `append'.(3)

   Contemporary logic programming languages (including the one we
implement here) have substantial deficiencies, in that their general
"how to" methods can lead them into spurious infinite loops or other
undesirable behavior.  Logic programming is an active field of research
in computer science.(4)

   Earlier in this chapter we explored the technology of implementing
interpreters and described the elements that are essential to an
interpreter for a Lisp-like language (indeed, to an interpreter for any
conventional language).  Now we will apply these ideas to discuss an
interpreter for a logic programming language.  We call this language
the "query language", because it is very useful for retrieving
information from data bases by formulating "queries", or questions,
expressed in the language.  Even though the query language is very
different from Lisp, we will find it convenient to describe the
language in terms of the same general framework we have been using all
along: as a collection of primitive elements, together with means of
combination that enable us to combine simple elements to create more
complex elements and means of abstraction that enable us to regard
complex elements as single conceptual units.  An interpreter for a
logic programming language is considerably more complex than an
interpreter for a language like Lisp.  Nevertheless, we will see that
our query-language interpreter contains many of the same elements found
in the interpreter of section *Note 4-1::.  In particular, there will
be an "eval" part that classifies expressions according to type and an
"apply" part that implements the language's abstraction mechanism
(procedures in the case of Lisp, and "rules" in the case of logic
programming).  Also, a central role is played in the implementation by
a frame data structure, which determines the correspondence between
symbols and their associated values.  One additional interesting aspect
of our query-language implementation is that we make substantial use of
streams, which were introduced in *Note Chapter 3::.

* Menu:

* 4-4-1::            Deductive Information Retrieval
* 4-4-2::            How the Query System Works
* 4-4-3::            Is Logic Programming Mathematical Logic?
* 4-4-4::            Implementing the Query System

   ---------- Footnotes ----------

   (1) Logic programming has grown out of a long history of research in
automatic theorem proving.  Early theorem-proving programs could
accomplish very little, because they exhaustively searched the space of
possible proofs.  The major breakthrough that made such a search
plausible was the discovery in the early 1960s of the "unification
algorithm" and the principle "resolution principle" (Robinson 1965).
Resolution was used, for example, by Green and Raphael (1968) (see also
Green 1969) as the basis for a deductive question-answering system.
During most of this period, researchers concentrated on algorithms that
are guaranteed to find a proof if one exists.  Such algorithms were
difficult to control and to direct toward a proof.  Hewitt (1969)
recognized the possibility of merging the control structure of a
programming language with the operations of a logic-manipulation system,
leading to the work in automatic search mentioned in section *Note
4-3-1:: (footnote *Note Footnote 4-47::).  At the same time that this
was being done, Colmerauer, in Marseille, was developing rule-based
systems for manipulating natural language (see Colmerauer et al. 1973).
He invented a programming language called Prolog for representing
those rules.  Kowalski (1973; 1979), in Edinburgh, recognized that
execution of a Prolog program could be interpreted as proving theorems
(using a proof technique called linear Horn-clause resolution).  The
merging of the last two strands led to the logic-programming movement.
Thus, in assigning credit for the development of logic programming, the
French can point to Prolog's genesis at the University of Marseille,
while the British can highlight the work at the University of
Edinburgh.  According to people at MIT, logic programming was developed
by these groups in an attempt to figure out what Hewitt was talking
about in his brilliant but impenetrable Ph.D. thesis.  For a history of
logic programming, see Robinson 1983.

   (2) To see the correspondence between the rules and the procedure,
let `x' in the procedure (where `x' is nonempty) correspond to `(cons u
v)' in the rule.  Then `z' in the rule corresponds to the `append' of
`(cdr x)' and `y'.

   (3) This certainly does not relieve the user of the entire problem
of how to compute the answer.  There are many different mathematically
equivalent sets of rules for formulating the `append' relation, only
some of which can be turned into effective devices for computing in any
direction.  In addition, sometimes "what is" information gives no clue
"how to" compute an answer.  For example, consider the problem of
computing the y such that y^2 = x.

   (4) Interest in logic programming peaked during the early 80s when
the Japanese government began an ambitious project aimed at building
superfast computers optimized to run logic programming languages.  The
speed of such computers was to be measured in LIPS (Logical Inferences
Per Second) rather than the usual FLOPS (FLoating-point Operations Per
Second).  Although the project succeeded in developing hardware and
software as originally planned, the international computer industry
moved in a different direction.  See Feigenbaum and Shrobe 1993 for an
overview evaluation of the Japanese project.  The logic programming
community has also moved on to consider relational programming based on
techniques other than simple pattern matching, such as the ability to
deal with numerical constraints such as the ones illustrated in the
constraint-propagation system of section *Note 3-3-5::.


File: sicp.info,  Node: 4-4-1,  Next: 4-4-2,  Prev: 4-4,  Up: 4-4

4.4.1 Deductive Information Retrieval
-------------------------------------

Logic programming excels in providing interfaces to data bases for
information retrieval.  The query language we shall implement in this
chapter is designed to be used in this way.

   In order to illustrate what the query system does, we will show how
it can be used to manage the data base of personnel records for
Microshaft, a thriving high-technology company in the Boston area.  The
language provides pattern-directed access to personnel information and
can also take advantage of general rules in order to make logical
deductions.

A sample data base
..................

The personnel data base for Microshaft contains "assertions" about
company personnel.  Here is the information about Ben Bitdiddle, the
resident computer wizard:

     (address (Bitdiddle Ben) (Slumerville (Ridge Road) 10))
     (job (Bitdiddle Ben) (computer wizard))
     (salary (Bitdiddle Ben) 60000)

   Each assertion is a list (in this case a triple) whose elements can
themselves be lists.

   As resident wizard, Ben is in charge of the company's computer
division, and he supervises two programmers and one technician.  Here
is the information about them:

     (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78))
     (job (Hacker Alyssa P) (computer programmer))
     (salary (Hacker Alyssa P) 40000)
     (supervisor (Hacker Alyssa P) (Bitdiddle Ben))

     (address (Fect Cy D) (Cambridge (Ames Street) 3))
     (job (Fect Cy D) (computer programmer))
     (salary (Fect Cy D) 35000)
     (supervisor (Fect Cy D) (Bitdiddle Ben))

     (address (Tweakit Lem E) (Boston (Bay State Road) 22))
     (job (Tweakit Lem E) (computer technician))
     (salary (Tweakit Lem E) 25000)
     (supervisor (Tweakit Lem E) (Bitdiddle Ben))

   There is also a programmer trainee, who is supervised by Alyssa:

     (address (Reasoner Louis) (Slumerville (Pine Tree Road) 80))
     (job (Reasoner Louis) (computer programmer trainee))
     (salary (Reasoner Louis) 30000)
     (supervisor (Reasoner Louis) (Hacker Alyssa P))

   All of these people are in the computer division, as indicated by
the word `computer' as the first item in their job descriptions.

   Ben is a high-level employee.  His supervisor is the company's big
wheel himself:

     (supervisor (Bitdiddle Ben) (Warbucks Oliver))

     (address (Warbucks Oliver) (Swellesley (Top Heap Road)))
     (job (Warbucks Oliver) (administration big wheel))
     (salary (Warbucks Oliver) 150000)

   Besides the computer division supervised by Ben, the company has an
accounting division, consisting of a chief accountant and his assistant:

     (address (Scrooge Eben) (Weston (Shady Lane) 10))
     (job (Scrooge Eben) (accounting chief accountant))
     (salary (Scrooge Eben) 75000)
     (supervisor (Scrooge Eben) (Warbucks Oliver))

     (address (Cratchet Robert) (Allston (N Harvard Street) 16))
     (job (Cratchet Robert) (accounting scrivener))
     (salary (Cratchet Robert) 18000)
     (supervisor (Cratchet Robert) (Scrooge Eben))

   There is also a secretary for the big wheel:

     (address (Aull DeWitt) (Slumerville (Onion Square) 5))
     (job (Aull DeWitt) (administration secretary))
     (salary (Aull DeWitt) 25000)
     (supervisor (Aull DeWitt) (Warbucks Oliver))

   The data base also contains assertions about which kinds of jobs can
be done by people holding other kinds of jobs.  For instance, a
computer wizard can do the jobs of both a computer programmer and a
computer technician:

     (can-do-job (computer wizard) (computer programmer))
     (can-do-job (computer wizard) (computer technician))

   A computer programmer could fill in for a trainee:

     (can-do-job (computer programmer)
                 (computer programmer trainee))

   Also, as is well known,

     (can-do-job (administration secretary)
                 (administration big wheel))

Simple queries
..............

The query language allows users to retrieve information from the data
base by posing queries in response to the system's prompt.  For
example, to find all computer programmers one can say

     ;;; Query input:
     (job ?x (computer programmer))

   The system will respond with the following items:

     ;;; Query results:
     (job (Hacker Alyssa P) (computer programmer))
     (job (Fect Cy D) (computer programmer))

   The input query specifies that we are looking for entries in the
data base that match a certain "pattern".  In this example, the pattern
specifies entries consisting of three items, of which the first is the
literal symbol `job', the second can be anything, and the third is the
literal list `(computer programmer)'.  The "anything" that can be the
second item in the matching list is specified by a "pattern variable",
`?x'.  The general form of a pattern variable is a symbol, taken to be
the name of the variable, preceded by a question mark.  We will see
below why it is useful to specify names for pattern variables rather
than just putting `?' into patterns to represent "anything."  The
system responds to a simple query by showing all entries in the data
base that match the specified pattern.

   A pattern can have more than one variable.  For example, the query

     (address ?x ?y)

will list all the employees' addresses.

   A pattern can have no variables, in which case the query simply
determines whether that pattern is an entry in the data base.  If so,
there will be one match; if not, there will be no matches.

   The same pattern variable can appear more than once in a query,
specifying that the same "anything" must appear in each position.  This
is why variables have names.  For example,

     (supervisor ?x ?x)

finds all people who supervise themselves (though there are no such
assertions in our sample data base).

   The query

     (job ?x (computer ?type))

matches all job entries whose third item is a two-element list whose
first item is `computer':

     (job (Bitdiddle Ben) (computer wizard))
     (job (Hacker Alyssa P) (computer programmer))
     (job (Fect Cy D) (computer programmer))
     (job (Tweakit Lem E) (computer technician))

   This same pattern does _not_ match

     (job (Reasoner Louis) (computer programmer trainee))

because the third item in the entry is a list of three elements, and the
pattern's third item specifies that there should be two elements.  If
we wanted to change the pattern so that the third item could be any
list beginning with `computer', we could specify(1)

     (job ?x (computer . ?type))

   For example,

     (computer . ?type)

matches the data

     (computer programmer trainee)

with `?type' as the list `(programmer trainee)'.  It also matches the
data

     (computer programmer)

with `?type' as the list `(programmer)', and matches the data

     (computer)

with `?type' as the empty list `()'.

   We can describe the query language's processing of simple queries as
follows:

   * The system finds all assignments to variables in the query pattern
     that "satisfy" the pattern--that is, all sets of values for the
     variables such that if the pattern variables are "instantiated
     with" (replaced by) the values, the result is in the data base.

   * The system responds to the query by listing all instantiations of
     the query pattern with the variable assignments that satisfy it.


   Note that if the pattern has no variables, the query reduces to a
determination of whether that pattern is in the data base.  If so, the
empty assignment, which assigns no values to variables, satisfies that
pattern for that data base.

     *Exercise 4.55:* Give simple queries that retrieve the following
     information from the data base:

       1. all people supervised by Ben Bitdiddle;

       2. the names and jobs of all people in the accounting division;

       3. the names and addresses of all people who live in Slumerville.


Compound queries
................

Simple queries form the primitive operations of the query language.  In
order to form compound operations, the query language provides means of
combination.  One thing that makes the query language a logic
programming language is that the means of combination mirror the means
of combination used in forming logical expressions: `and', `or', and
`not'.  (Here `and', `or', and `not' are not the Lisp primitives, but
rather operations built into the query language.)

   We can use `and' as follows to find the addresses of all the computer
programmers:

     (and (job ?person (computer programmer))
          (address ?person ?where))

   The resulting output is

     (and (job (Hacker Alyssa P) (computer programmer))
          (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78)))

     (and (job (Fect Cy D) (computer programmer))
          (address (Fect Cy D) (Cambridge (Ames Street) 3)))

   In general,

     (and <QUERY_1> <QUERY_2> ... <QUERY_N>)

is satisfied by all sets of values for the pattern variables that
simultaneously satisfy <QUERY_1> ... <QUERY_N>.

   As for simple queries, the system processes a compound query by
finding all assignments to the pattern variables that satisfy the
query, then displaying instantiations of the query with those values.

   Another means of constructing compound queries is through `or'.  For
example,

     (or (supervisor ?x (Bitdiddle Ben))
         (supervisor ?x (Hacker Alyssa P)))

will find all employees supervised by Ben Bitdiddle or Alyssa P.
Hacker:

     (or (supervisor (Hacker Alyssa P) (Bitdiddle Ben))
         (supervisor (Hacker Alyssa P) (Hacker Alyssa P)))

     (or (supervisor (Fect Cy D) (Bitdiddle Ben))
         (supervisor (Fect Cy D) (Hacker Alyssa P)))

     (or (supervisor (Tweakit Lem E) (Bitdiddle Ben))
         (supervisor (Tweakit Lem E) (Hacker Alyssa P)))

     (or (supervisor (Reasoner Louis) (Bitdiddle Ben))
         (supervisor (Reasoner Louis) (Hacker Alyssa P)))

   In general,

     (or <QUERY_1> <QUERY_2> ... <QUERY_N>)

is satisfied by all sets of values for the pattern variables that
satisfy at least one of <QUERY_1> ... <QUERY_N>.

   Compound queries can also be formed with `not'. For example,

     (and (supervisor ?x (Bitdiddle Ben))
          (not (job ?x (computer programmer))))

finds all people supervised by Ben Bitdiddle who are not computer
programmers.  In general,

     (not <QUERY_1>)

is satisfied by all assignments to the pattern variables that do not
satisfy <QUERY_1>.(2)

   The final combining form is called `lisp-value'.  When `lisp-value'
is the first element of a pattern, it specifies that the next element
is a Lisp predicate to be applied to the rest of the (instantiated)
elements as arguments.  In general,

     (lisp-value <PREDICATE> <ARG_1> ... <ARG_N>)

will be satisfied by assignments to the pattern variables for which the
<PREDICATE> applied to the instantiated <ARG_1> ...  <ARG_N> is true.
For example, to find all people whose salary is greater than $30,000 we
could write(3)

     (and (salary ?person ?amount)
          (lisp-value > ?amount 30000))

     *Exercise 4.56:* Formulate compound queries that retrieve the
     following information:

       a. the names of all people who are supervised by Ben Bitdiddle,
          together with their addresses;

       b. all people whose salary is less than Ben Bitdiddle's,
          together with their salary and Ben Bitdiddle's salary;

       c. all people who are supervised by someone who is not in the
          computer division, together with the supervisor's name and
          job.


Rules
.....

In addition to primitive queries and compound queries, the query
language provides means for abstracting queries.  These are given by "rules".
The rule

     (rule (lives-near ?person-1 ?person-2)
           (and (address ?person-1 (?town . ?rest-1))
                (address ?person-2 (?town . ?rest-2))
                (not (same ?person-1 ?person-2))))

specifies that two people live near each other if they live in the same
town.  The final `not' clause prevents the rule from saying that all
people live near themselves.  The `same' relation is defined by a very
simple rule:(4)

     (rule (same ?x ?x))

   The following rule declares that a person is a "wheel" in an
organization if he supervises someone who is in turn a supervisor:

     (rule (wheel ?person)
           (and (supervisor ?middle-manager ?person)
                (supervisor ?x ?middle-manager)))

   The general form of a rule is

     (rule <CONCLUSION> <BODY>)

where <CONCLUSION> is a pattern and <BODY> is any query.(5) We can
think of a rule as representing a large (even infinite) set of
assertions, namely all instantiations of the rule conclusion with
variable assignments that satisfy the rule body.  When we described
simple queries (patterns), we said that an assignment to variables
satisfies a pattern if the instantiated pattern is in the data base.
But the pattern needn't be explicitly in the data base as an assertion.
It can be an implicit assertion implied by a rule.  For example, the
query

     (lives-near ?x (Bitdiddle Ben))

results in

     (lives-near (Reasoner Louis) (Bitdiddle Ben))
     (lives-near (Aull DeWitt) (Bitdiddle Ben))

   To find all computer programmers who live near Ben Bitdiddle, we can
ask

     (and (job ?x (computer programmer))
          (lives-near ?x (Bitdiddle Ben)))

   As in the case of compound procedures, rules can be used as parts of
other rules (as we saw with the `lives-near' rule above) or even be
defined recursively.  For instance, the rule

     (rule (outranked-by ?staff-person ?boss)
           (or (supervisor ?staff-person ?boss)
               (and (supervisor ?staff-person ?middle-manager)
                    (outranked-by ?middle-manager ?boss))))

says that a staff person is outranked by a boss in the organization if
the boss is the person's supervisor or (recursively) if the person's
supervisor is outranked by the boss.

     *Exercise 4.57:* Define a rule that says that person 1 can replace
     person 2 if either person 1 does the same job as person 2 or
     someone who does person 1's job can also do person 2's job, and if
     person 1 and person 2 are not the same person. Using your rule,
     give queries that find the following:

       a. all people who can replace Cy D. Fect;

       b. all people who can replace someone who is being paid more
          than they are, together with the two salaries.


     *Exercise 4.58:* Define a rule that says that a person is a "big
     shot" in a division if the person works in the division but does
     not have a supervisor who works in the division.

     *Exercise 4.59:* Ben Bitdiddle has missed one meeting too many.
     Fearing that his habit of forgetting meetings could cost him his
     job, Ben decides to do something about it.  He adds all the weekly
     meetings of the firm to the Microshaft data base by asserting the
     following:

          (meeting accounting (Monday 9am))
          (meeting administration (Monday 10am))
          (meeting computer (Wednesday 3pm))
          (meeting administration (Friday 1pm))

     Each of the above assertions is for a meeting of an entire
     division.  Ben also adds an entry for the company-wide meeting
     that spans all the divisions.  All of the company's employees
     attend this meeting.

          (meeting whole-company (Wednesday 4pm))

       a. On Friday morning, Ben wants to query the data base for all
          the meetings that occur that day.  What query should he use?

       b. Alyssa P. Hacker is unimpressed.  She thinks it would be much
          more useful to be able to ask for her meetings by specifying
          her name.  So she designs a rule that says that a person's
          meetings include all `whole-company' meetings plus all
          meetings of that person's division.  Fill in the body of
          Alyssa's rule.

               (rule (meeting-time ?person ?day-and-time)
                     <RULE-BODY>)

       c. Alyssa arrives at work on Wednesday morning and wonders what
          meetings she has to attend that day.  Having defined the
          above rule, what query should she make to find this out?


     *Exercise 4.60:* By giving the query

          (lives-near ?person (Hacker Alyssa P))

     Alyssa P. Hacker is able to find people who live near her, with
     whom she can ride to work.  On the other hand, when she tries to
     find all pairs of people who live near each other by querying

          (lives-near ?person-1 ?person-2)

     she notices that each pair of people who live near each other is
     listed twice; for example,

          (lives-near (Hacker Alyssa P) (Fect Cy D))
          (lives-near (Fect Cy D) (Hacker Alyssa P))

     Why does this happen?  Is there a way to find a list of people who
     live near each other, in which each pair appears only once?
     Explain.

Logic as programs
.................

We can regard a rule as a kind of logical implication: _If_ an
assignment of values to pattern variables satisfies the body, _then_ it
satisfies the conclusion.  Consequently, we can regard the query
language as having the ability to perform "logical deductions" based
upon the rules.  As an example, consider the `append' operation
described at the beginning of section *Note 4-4::.  As we said,
`append' can be characterized by the following two rules:

   * For any list `y', the empty list and `y' `append' to form `y'.

   * For any `u', `v', `y', and `z', `(cons u v)' and `y' `append' to
     form `(cons u z)' if `v' and `y' `append' to form `z'.


   To express this in our query language, we define two rules for a
relation

     (append-to-form x y z)

which we can interpret to mean "`x' and `y' `append' to form `z'":

     (rule (append-to-form () ?y ?y))

     (rule (append-to-form (?u . ?v) ?y (?u . ?z))
           (append-to-form ?v ?y ?z))

   The first rule has no body, which means that the conclusion holds
for any value of `?y'.  Note how the second rule makes use of
dotted-tail notation to name the `car' and `cdr' of a list.

   Given these two rules, we can formulate queries that compute the
`append' of two lists:

     ;;; Query input:
     (append-to-form (a b) (c d) ?z)
     ;;; Query results:
     (append-to-form (a b) (c d) (a b c d))

   What is more striking, we can use the same rules to ask the question
"Which list, when `append'ed to `(a b)', yields `(a b c d)'?"  This is
done as follows:

     ;;; Query input:
     (append-to-form (a b) ?y (a b c d))
     ;;; Query results:
     (append-to-form (a b) (c d) (a b c d))

   We can also ask for all pairs of lists that `append' to form `(a b c
d)':

     ;;; Query input:
     (append-to-form ?x ?y (a b c d))
     ;;; Query results:
     (append-to-form () (a b c d) (a b c d))
     (append-to-form (a) (b c d) (a b c d))
     (append-to-form (a b) (c d) (a b c d))
     (append-to-form (a b c) (d) (a b c d))
     (append-to-form (a b c d) () (a b c d))

   The query system may seem to exhibit quite a bit of intelligence in
using the rules to deduce the answers to the queries above.  Actually,
as we will see in the next section, the system is following a
well-determined algorithm in unraveling the rules.  Unfortunately,
although the system works impressively in the `append' case, the
general methods may break down in more complex cases, as we will see in
section *Note 4-4-3::.

     *Exercise 4.61:* The following rules implement a `next-to'
     relation that finds adjacent elements of a list:

          (rule (?x next-to ?y in (?x ?y . ?u)))

          (rule (?x next-to ?y in (?v . ?z))
                (?x next-to ?y in ?z))

     What will the response be to the following queries?

          (?x next-to ?y in (1 (2 3) 4))

          (?x next-to 1 in (2 1 3 1))

     *Exercise 4.62:* Define rules to implement the `last-pair'
     operation of *Note Exercise 2-17::, which returns a list
     containing the last element of a nonempty list.  Check your rules
     on queries such as `(last-pair (3) ?x)', `(last-pair (1 2 3) ?x)',
     and `(last-pair (2 ?x) (3))'.  Do your rules work correctly on
     queries such as `(last-pair ?x (3))' ?

     *Exercise 4.63:* The following data base (see Genesis 4) traces
     the genealogy of the descendants of Ada back to Adam, by way of
     Cain:

          (son Adam Cain)
          (son Cain Enoch)
          (son Enoch Irad)
          (son Irad Mehujael)
          (son Mehujael Methushael)
          (son Methushael Lamech)
          (wife Lamech Ada)
          (son Ada Jabal)
          (son Ada Jubal)

     Formulate rules such as "If S is the son of f, and f is the son of
     G, then S is the grandson of G" and "If W is the wife of M, and S
     is the son of W, then S is the son of M" (which was supposedly
     more true in biblical times than today) that will enable the query
     system to find the grandson of Cain; the sons of Lamech; the
     grandsons of Methushael.  (See *Note Exercise 4-69:: for some
     rules to deduce more complicated relationships.)

   ---------- Footnotes ----------

   (1) This uses the dotted-tail notation introduced in *Note Exercise
2-20::.

   (2) Actually, this description of `not' is valid only for simple
cases.  The real behavior of `not' is more complex.  We will examine
`not''s peculiarities in sections *Note 4-4-2:: and *Note 4-4-3::.

   (3) `Lisp-value' should be used only to perform an operation not
provided in the query language.  In particular, it should not be used
to test equality (since that is what the matching in the query language
is designed to do) or inequality (since that can be done with the
`same' rule shown below).

   (4) Notice that we do not need `same' in order to make two things be
the same: We just use the same pattern variable for each--in effect, we
have one thing instead of two things in the first place.  For example,
see `?town' in the `lives-near' rule and `?middle-manager' in the
`wheel' rule below.  `Same' is useful when we want to force two things
to be different, such as `?person-1' and `?person-2' in the
`lives-near' rule.  Although using the same pattern variable in two
parts of a query forces the same value to appear in both places, using
different pattern variables does not force different values to appear.
(The values assigned to different pattern variables may be the same or
different.)

   (5) We will also allow rules without bodies, as in `same', and we
will interpret such a rule to mean that the rule conclusion is
satisfied by any values of the variables.


File: sicp.info,  Node: 4-4-2,  Next: 4-4-3,  Prev: 4-4-1,  Up: 4-4

4.4.2 How the Query System Works
--------------------------------

In section *Note 4-4-4:: we will present an implementation of the query
interpreter as a collection of procedures.  In this section we give an
overview that explains the general structure of the system independent
of low-level implementation details.  After describing the
implementation of the interpreter, we will be in a position to
understand some of its limitations and some of the subtle ways in which
the query language's logical operations differ from the operations of
mathematical logic.

   It should be apparent that the query evaluator must perform some
kind of search in order to match queries against facts and rules in the
data base.  One way to do this would be to implement the query system
as a nondeterministic program, using the `amb' evaluator of section
*Note 4-3:: (see *Note Exercise 4-78::).  Another possibility is to
manage the search with the aid of streams.  Our implementation follows
this second approach.

   The query system is organized around two central operations called "pattern
matching" and "unification".  We first describe pattern matching and
explain how this operation, together with the organization of
information in terms of streams of frames, enables us to implement both
simple and compound queries.  We next discuss unification, a
generalization of pattern matching needed to implement rules.  Finally,
we show how the entire query interpreter fits together through a
procedure that classifies expressions in a manner analogous to the way
`eval' classifies expressions for the interpreter described in section
*Note 4-1::.

Pattern matching
................

A "pattern matcher" is a program that tests whether some datum fits a
specified pattern.  For example, the data list `((a b) c (a b))' matches
the pattern `(?x c ?x)' with the pattern variable `?x' bound to `(a
b)'.  The same data list matches the pattern `(?x ?y ?z)' with `?x' and
`?z' both bound to `(a b)' and `?y' bound to `c'.  It also matches the
pattern `((?x ?y) c (?x ?y))' with `?x' bound to `a' and `?y' bound to
`b'.  However, it does not match the pattern `(?x a ?y)', since that
pattern specifies a list whose second element is the symbol `a'.

   The pattern matcher used by the query system takes as inputs a
pattern, a datum, and a "frame" that specifies bindings for various
pattern variables.  It checks whether the datum matches the pattern in
a way that is consistent with the bindings already in the frame.  If
so, it returns the given frame augmented by any bindings that may have
been determined by the match.  Otherwise, it indicates that the match
has failed.

   For example, using the pattern `(?x ?y ?x)' to match `(a b a)' given
an empty frame will return a frame specifying that `?x' is bound to `a'
and `?y' is bound to `b'.  Trying the match with the same pattern, the
same datum, and a frame specifying that `?y' is bound to `a' will fail.
Trying the match with the same pattern, the same datum, and a frame in
which `?y' is bound to `b' and `?x' is unbound will return the given
frame augmented by a binding of `?x' to `a'.

   The pattern matcher is all the mechanism that is needed to process
simple queries that don't involve rules.  For instance, to process the
query

     (job ?x (computer programmer))

we scan through all assertions in the data base and select those that
match the pattern with respect to an initially empty frame.  For each
match we find, we use the frame returned by the match to instantiate
the pattern with a value for `?x'.

Streams of frames
.................

The testing of patterns against frames is organized through the use of
streams.  Given a single frame, the matching process runs through the
data-base entries one by one.  For each data-base entry, the matcher
generates either a special symbol indicating that the match has failed
or an extension to the frame.  The results for all the data-base
entries are collected into a stream, which is passed through a filter
to weed out the failures.  The result is a stream of all the frames
that extend the given frame via a match to some assertion in the data
base.(1)

   In our system, a query takes an input stream of frames and performs
the above matching operation for every frame in the stream, as
indicated in *Note Figure 4-4::.  That is, for each frame in the input
stream, the query generates a new stream consisting of all extensions
to that frame by matches to assertions in the data base.  All these
streams are then combined to form one huge stream, which contains all
possible extensions of every frame in the input stream.  This stream is
the output of the query.

     *Figure 4.4:* A query processes a stream of frames.

                                            output stream
            input stream   +-------------+  of frames,
            of frames      |    query    |  filtered and extended
          ---------------->|             +------------------------->
                           | (job ?x ?y) |
                           +-------------+
                                  ^
                                  |
                         stream of assertions
                            from data base

   To answer a simple query, we use the query with an input stream
consisting of a single empty frame.  The resulting output stream
contains all extensions to the empty frame (that is, all answers to our
query).  This stream of frames is then used to generate a stream of
copies of the original query pattern with the variables instantiated by
the values in each frame, and this is the stream that is finally
printed.

Compound queries
................

The real elegance of the stream-of-frames implementation is evident
when we deal with compound queries.  The processing of compound queries
makes use of the ability of our matcher to demand that a match be
consistent with a specified frame.  For example, to handle the `and' of
two queries, such as

     (and (can-do-job ?x (computer programmer trainee))
          (job ?person ?x))

(informally, "Find all people who can do the job of a computer
programmer trainee"), we first find all entries that match the pattern

     (can-do-job ?x (computer programmer trainee))

   This produces a stream of frames, each of which contains a binding
for `?x'.  Then for each frame in the stream we find all entries that
match

     (job ?person ?x)

in a way that is consistent with the given binding for `?x'.  Each such
match will produce a frame containing bindings for `?x' and `?person'.
The `and' of two queries can be viewed as a series combination of the
two component queries, as shown in *Note Figure 4-5::.  The frames that
pass through the first query filter are filtered and further extended
by the second query.

     *Figure 4.5:* The `and' combination of two queries is produced by
     operating on the stream of frames in series.

                          +----------------------+
                          |       (and A B)      |
            input stream  |                      |  output stream
            of frames     |   +---+       +---+  |  of frames
          ------------------->| A +------>| B +-------------------->
                          |   +---+       +---+  |
                          |     ^           ^    |
                          |     |           |    |
                          |     +-----*-----+    |
                          +-----------|----------+
                                      |
                                  data base

   *Note Figure 4-6:: shows the analogous method for computing the `or'
of two queries as a parallel combination of the two component queries.
The input stream of frames is extended separately by each query.  The
two resulting streams are then merged to produce the final output
stream.

     *Figure 4.6:* The `or' combination of two queries is produced by
     operating on the stream of frames in parallel and merging the
     results.

                     +---------------------------+
                     |          (or A B)         |
                     |    +---+                  |
          input      | +->| A |------------+     |  output
          stream of  | |  +---+            V     |  stream of
          frames     | |    ^          +-------+ |  frames
          -------------*    |          | merge +--------------->
                     | |    |          +-------+ |
                     | |    |              ^     |
                     | |    |   +---+      |     |
                     | +------->| B +------+     |
                     |      |   +---+            |
                     |      |     ^              |
                     |      |     |              |
                     |      +--*--+              |
                     +---------|-----------------+
                               |
                           data base

   Even from this high-level description, it is apparent that the
processing of compound queries can be slow.  For example, since a query
may produce more than one output frame for each input frame, and each
query in an `and' gets its input frames from the previous query, an
`and' query could, in the worst case, have to perform a number of
matches that is exponential in the number of queries (see *Note
Exercise 4-76::).(2) Though systems for handling only simple queries
are quite practical, dealing with complex queries is extremely
difficult.(3)

   From the stream-of-frames viewpoint, the `not' of some query acts as
a filter that removes all frames for which the query can be satisfied.
For instance, given the pattern

     (not (job ?x (computer programmer)))

we attempt, for each frame in the input stream, to produce extension
frames that satisfy `(job ?x (computer programmer))'.  We remove from
the input stream all frames for which such extensions exist.  The
result is a stream consisting of only those frames in which the binding
for `?x' does not satisfy `(job ?x (computer programmer))'.  For
example, in processing the query

     (and (supervisor ?x ?y)
          (not (job ?x (computer programmer))))

the first clause will generate frames with bindings for `?x' and `?y'.
The `not' clause will then filter these by removing all frames in which
the binding for `?x' satisfies the restriction that `?x' is a computer
programmer.(4)

   The `lisp-value' special form is implemented as a similar filter on
frame streams.  We use each frame in the stream to instantiate any
variables in the pattern, then apply the Lisp predicate.  We remove
from the input stream all frames for which the predicate fails.

Unification
...........

In order to handle rules in the query language, we must be able to find
the rules whose conclusions match a given query pattern.  Rule
conclusions are like assertions except that they can contain variables,
so we will need a generalization of pattern matching--called "unification"--in
which both the "pattern" and the "datum" may contain variables.

   A unifier takes two patterns, each containing constants and
variables, and determines whether it is possible to assign values to
the variables that will make the two patterns equal.  If so, it returns
a frame containing these bindings.  For example, unifying `(?x a ?y)'
and `(?y ?z a)' will specify a frame in which `?x', `?y', and `?z' must
all be bound to `a'.  On the other hand, unifying `(?x ?y a)' and `(?x
b ?y)' will fail, because there is no value for `?y' that can make the
two patterns equal.  (For the second elements of the patterns to be
equal, `?y' would have to be `b'; however, for the third elements to be
equal, `?y' would have to be `a'.)  The unifier used in the query
system, like the pattern matcher, takes a frame as input and performs
unifications that are consistent with this frame.

   The unification algorithm is the most technically difficult part of
the query system.  With complex patterns, performing unification may
seem to require deduction.  To unify `(?x ?x)' and `((a ?y c) (a b
?z))', for example, the algorithm must infer that `?x' should be `(a b
c)', `?y' should be `b', and `?z' should be `c'.  We may think of this
process as solving a set of equations among the pattern components.  In
general, these are simultaneous equations, which may require substantial
manipulation to solve.(5) For example, unifying `(?x ?x)' and `((a ?y
c) (a b ?z))' may be thought of as specifying the simultaneous equations

     ?x  =  (a ?y c)
     ?x  =  (a b ?z)

   These equations imply that

     (a ?y c)  =  (a b ?z)

which in turn implies that

     a  =  a, ?y  =  b, c  =  ?z,

and hence that

     ?x  =  (a b c)

   In a successful pattern match, all pattern variables become bound,
and the values to which they are bound contain only constants.  This is
also true of all the examples of unification we have seen so far.  In
general, however, a successful unification may not completely determine
the variable values; some variables may remain unbound and others may
be bound to values that contain variables.

   Consider the unification of `(?x a)' and `((b ?y) ?z)'.  We can
deduce that `?x = (b ?y)' and `a = ?z', but we cannot further solve for
`?x' or `?y'.  The unification doesn't fail, since it is certainly
possible to make the two patterns equal by assigning values to `?x' and
`?y'.  Since this match in no way restricts the values `?y' can take
on, no binding for `?y' is put into the result frame.  The match does,
however, restrict the value of `?x'.  Whatever value `?y' has, `?x'
must be `(b ?y)'.  A binding of `?x' to the pattern `(b ?y)' is thus
put into the frame.  If a value for `?y' is later determined and added
to the frame (by a pattern match or unification that is required to be
consistent with this frame), the previously bound `?x' will refer to
this value.(6)

Applying rules
..............

Unification is the key to the component of the query system that makes
inferences from rules. To see how this is accomplished, consider
processing a query that involves applying a rule, such as

     (lives-near ?x (Hacker Alyssa P))

   To process this query, we first use the ordinary pattern-match
procedure described above to see if there are any assertions in the
data base that match this pattern.  (There will not be any in this
case, since our data base includes no direct assertions about who lives
near whom.)  The next step is to attempt to unify the query pattern
with the conclusion of each rule.  We find that the pattern unifies
with the conclusion of the rule

     (rule (lives-near ?person-1 ?person-2)
           (and (address ?person-1 (?town . ?rest-1))
                (address ?person-2 (?town . ?rest-2))
                (not (same ?person-1 ?person-2))))

resulting in a frame specifying that `?person-2' is bound to `(Hacker
Alyssa P)' and that `?x' should be bound to (have the same value as)
`?person-1'.  Now, relative to this frame, we evaluate the compound
query given by the body of the rule.  Successful matches will extend
this frame by providing a binding for `?person-1', and consequently a
value for `?x', which we can use to instantiate the original query
pattern.

   In general, the query evaluator uses the following method to apply a
rule when trying to establish a query pattern in a frame that specifies
bindings for some of the pattern variables:

   * Unify the query with the conclusion of the rule to form, if
     successful, an extension of the original frame.

   * Relative to the extended frame, evaluate the query formed by the
     body of the rule.


   Notice how similar this is to the method for applying a procedure in
the `eval'/`apply' evaluator for Lisp:

   * Bind the procedure's parameters to its arguments to form a frame
     that extends the original procedure environment.

   * Relative to the extended environment, evaluate the expression
     formed by the body of the procedure.


   The similarity between the two evaluators should come as no
surprise.  Just as procedure definitions are the means of abstraction
in Lisp, rule definitions are the means of abstraction in the query
language.  In each case, we unwind the abstraction by creating
appropriate bindings and evaluating the rule or procedure body relative
to these.

Simple queries
..............

We saw earlier in this section how to evaluate simple queries in the
absence of rules.  Now that we have seen how to apply rules, we can
describe how to evaluate simple queries by using both rules and
assertions.

   Given the query pattern and a stream of frames, we produce, for each
frame in the input stream, two streams:

   * a stream of extended frames obtained by matching the pattern
     against all assertions in the data base (using the pattern
     matcher), and

   * a stream of extended frames obtained by applying all possible
     rules (using the unifier).(7)


   Appending these two streams produces a stream that consists of all
the ways that the given pattern can be satisfied consistent with the
original frame.  These streams (one for each frame in the input stream)
are now all combined to form one large stream, which therefore consists
of all the ways that any of the frames in the original input stream can
be extended to produce a match with the given pattern.

The query evaluator and the driver loop
.......................................

Despite the complexity of the underlying matching operations, the
system is organized much like an evaluator for any language.  The
procedure that coordinates the matching operations is called `qeval',
and it plays a role analogous to that of the `eval' procedure for Lisp.
`Qeval' takes as inputs a query and a stream of frames.  Its output is
a stream of frames, corresponding to successful matches to the query
pattern, that extend some frame in the input stream, as indicated in
*Note Figure 4-4::.  Like `eval', `qeval' classifies the different
types of expressions (queries) and dispatches to an appropriate
procedure for each.  There is a procedure for each special form (`and',
`or', `not', and `lisp-value') and one for simple queries.

   The driver loop, which is analogous to the `driver-loop' procedure
for the other evaluators in this chapter, reads queries from the
terminal.  For each query, it calls `qeval' with the query and a stream
that consists of a single empty frame.  This will produce the stream of
all possible matches (all possible extensions to the empty frame).  For
each frame in the resulting stream, it instantiates the original query
using the values of the variables found in the frame.  This stream of
instantiated queries is then printed.(8)

   The driver also checks for the special command `assert!', which
signals that the input is not a query but rather an assertion or rule
to be added to the data base.  For instance,

     (assert! (job (Bitdiddle Ben) (computer wizard)))

     (assert! (rule (wheel ?person)
                    (and (supervisor ?middle-manager ?person)
                         (supervisor ?x ?middle-manager))))

   ---------- Footnotes ----------

   (1) Because matching is generally very expensive, we would like to
avoid applying the full matcher to every element of the data base.
This is usually arranged by breaking up the process into a fast, coarse
match and the final match.  The coarse match filters the data base to
produce a small set of candidates for the final match.  With care, we
can arrange our data base so that some of the work of coarse matching
can be done when the data base is constructed rather then when we want
to select the candidates.  This is called "indexing" the data base.
There is a vast technology built around data-base-indexing schemes.
Our implementation, described in section *Note 4-4-4::, contains a
simple-minded form of such an optimization.

   (2) But this kind of exponential explosion is not common in `and'
queries because the added conditions tend to reduce rather than expand
the number of frames produced.

   (3) There is a large literature on data-base-management systems that
is concerned with how to handle complex queries efficiently.

   (4) There is a subtle difference between this filter implementation
of `not' and the usual meaning of `not' in mathematical logic.  See
section *Note 4-4-3::.

   (5) In one-sided pattern matching, all the equations that contain
pattern variables are explicit and already solved for the unknown (the
pattern variable).

   (6) Another way to think of unification is that it generates the
most general pattern that is a specialization of the two input
patterns.  That is, the unification of `(?x a)' and `((b ?y) ?z)' is
`((b ?y) a)', and the unification of `(?x a ?y)' and `(?y ?z a)',
discussed above, is `(a a a)'.  For our implementation, it is more
convenient to think of the result of unification as a frame rather than
a pattern.

   (7) Since unification is a generalization of matching, we could
simplify the system by using the unifier to produce both streams.
Treating the easy case with the simple matcher, however, illustrates
how matching (as opposed to full-blown unification) can be useful in
its own right.

   (8) The reason we use streams (rather than lists) of frames is that
the recursive application of rules can generate infinite numbers of
values that satisfy a query.  The delayed evaluation embodied in
streams is crucial here: The system will print responses one by one as
they are generated, regardless of whether there are a finite or
infinite number of responses.


File: sicp.info,  Node: 4-4-3,  Next: 4-4-4,  Prev: 4-4-2,  Up: 4-4

4.4.3 Is Logic Programming Mathematical Logic?
----------------------------------------------

The means of combination used in the query language may at first seem
identical to the operations `and', `or', and `not' of mathematical
logic, and the application of query-language rules is in fact
accomplished through a legitimate method of inference.(1) This
identification of the query language with mathematical logic is not
really valid, though, because the query language provides a structure
"control structure" that interprets the logical statements
procedurally.  We can often take advantage of this control structure.
For example, to find all of the supervisors of programmers we could
formulate a query in either of two logically equivalent forms:

     (and (job ?x (computer programmer))
          (supervisor ?x ?y))

or

     (and (supervisor ?x ?y)
          (job ?x (computer programmer)))

   If a company has many more supervisors than programmers (the usual
case), it is better to use the first form rather than the second
because the data base must be scanned for each intermediate result
(frame) produced by the first clause of the `and'.

   The aim of logic programming is to provide the programmer with
techniques for decomposing a computational problem into two separate
problems: "what" is to be computed, and "how" this should be computed.
This is accomplished by selecting a subset of the statements of
mathematical logic that is powerful enough to be able to describe
anything one might want to compute, yet weak enough to have a
controllable procedural interpretation.  The intention here is that, on
the one hand, a program specified in a logic programming language
should be an effective program that can be carried out by a computer.
Control ("how" to compute) is effected by using the order of evaluation
of the language.  We should be able to arrange the order of clauses and
the order of subgoals within each clause so that the computation is
done in an order deemed to be effective and efficient.  At the same
time, we should be able to view the result of the computation ("what"
to compute) as a simple consequence of the laws of logic.

   Our query language can be regarded as just such a procedurally
interpretable subset of mathematical logic.  An assertion represents a
simple fact (an atomic proposition).  A rule represents the implication
that the rule conclusion holds for those cases where the rule body
holds.  A rule has a natural procedural interpretation: To establish
the conclusion of the rule, establish the body of the rule.  Rules,
therefore, specify computations.  However, because rules can also be
regarded as statements of mathematical logic, we can justify any
"inference" accomplished by a logic program by asserting that the same
result could be obtained by working entirely within mathematical
logic.(2)

Infinite loops
..............

A consequence of the procedural interpretation of logic programs is
that it is possible to construct hopelessly inefficient programs for
solving certain problems.  An extreme case of inefficiency occurs when
the system falls into infinite loops in making deductions.  As a simple
example, suppose we are setting up a data base of famous marriages,
including

     (assert! (married Minnie Mickey))

   If we now ask

     (married Mickey ?who)

we will get no response, because the system doesn't know that if A is
married to B, then B is married to A.  So we assert the rule

     (assert! (rule (married ?x ?y)
                    (married ?y ?x)))

and again query

     (married Mickey ?who)

   Unfortunately, this will drive the system into an infinite loop, as
follows:

   * The system finds that the `married' rule is applicable; that is,
     the rule conclusion `(married ?x ?y)' successfully unifies with
     the query pattern `(married Mickey ?who)' to produce a frame in
     which `?x' is bound to `Mickey' and `?y' is bound to `?who'.  So
     the interpreter proceeds to evaluate the rule body `(married ?y
     ?x)' in this frame--in effect, to process the query `(married ?who
     Mickey)'.

   * One answer appears directly as an assertion in the data base:
     `(married Minnie Mickey)'.

   * The `married' rule is also applicable, so the interpreter again
     evaluates the rule body, which this time is equivalent to
     `(married Mickey ?who)'.


   The system is now in an infinite loop.  Indeed, whether the system
will find the simple answer `(married Minnie Mickey)' before it goes
into the loop depends on implementation details concerning the order in
which the system checks the items in the data base.  This is a very
simple example of the kinds of loops that can occur.  Collections of
interrelated rules can lead to loops that are much harder to
anticipate, and the appearance of a loop can depend on the order of
clauses in an `and' (see *Note Exercise 4-64::) or on low-level details
concerning the order in which the system processes queries.(3)

Problems with `not'
...................

Another quirk in the query system concerns `not'.  Given the data base
of section *Note 4-4-1::, consider the following two queries:

     (and (supervisor ?x ?y)
          (not (job ?x (computer programmer))))

     (and (not (job ?x (computer programmer)))
          (supervisor ?x ?y))

   These two queries do not produce the same result.  The first query
begins by finding all entries in the data base that match `(supervisor
?x ?y)', and then filters the resulting frames by removing the ones in
which the value of `?x' satisfies `(job ?x (computer programmer))'.
The second query begins by filtering the incoming frames to remove
those that can satisfy `(job ?x (computer programmer))'.  Since the
only incoming frame is empty, it checks the data base to see if there
are any patterns that satisfy `(job ?x (computer programmer))'.  Since
there generally are entries of this form, the `not' clause filters out
the empty frame and returns an empty stream of frames.  Consequently,
the entire compound query returns an empty stream.

   The trouble is that our implementation of `not' really is meant to
serve as a filter on values for the variables.  If a `not' clause is
processed with a frame in which some of the variables remain unbound
(as does `?x' in the example above), the system will produce unexpected
results. Similar problems occur with the use of `lisp-value'--the Lisp
predicate can't work if some of its arguments are unbound.  See *Note
Exercise 4-77::.

   There is also a much more serious way in which the `not' of the query
language differs from the `not' of mathematical logic.  In logic, we
interpret the statement "not P" to mean that P is not true.  In the
query system, however, "not P" means that P is not deducible from the
knowledge in the data base.  For example, given the personnel data base
of section *Note 4-4-1::, the system would happily deduce all sorts of
`not' statements, such as that Ben Bitdiddle is not a baseball fan,
that it is not raining outside, and that 2 + 2 is not 4.(4) In other
words, the `not' of logic programming languages reflects the so-called "closed
world assumption" that all relevant information has been included in
the data base.(5)

     *Exercise 4.64:* Louis Reasoner mistakenly deletes the
     `outranked-by' rule (section *Note 4-4-1::) from the data base.
     When he realizes this, he quickly reinstalls it.  Unfortunately,
     he makes a slight change in the rule, and types it in as

          (rule (outranked-by ?staff-person ?boss)
                (or (supervisor ?staff-person ?boss)
                    (and (outranked-by ?middle-manager ?boss)
                         (supervisor ?staff-person ?middle-manager))))

     Just after Louis types this information into the system, DeWitt
     Aull comes by to find out who outranks Ben Bitdiddle. He issues
     the query

          (outranked-by (Bitdiddle Ben) ?who)

     After answering, the system goes into an infinite loop.  Explain
     why.

     *Exercise 4.65:* Cy D. Fect, looking forward to the day when he
     will rise in the organization, gives a query to find all the
     wheels (using the `wheel' rule of section *Note 4-4-1::):

          (wheel ?who)

     To his surprise, the system responds

          ;;; Query results:
          (wheel (Warbucks Oliver))
          (wheel (Bitdiddle Ben))
          (wheel (Warbucks Oliver))
          (wheel (Warbucks Oliver))
          (wheel (Warbucks Oliver))

     Why is Oliver Warbucks listed four times?

     *Exercise 4.66:* Ben has been generalizing the query system to
     provide statistics about the company.  For example, to find the
     total salaries of all the computer programmers one will be able to
     say

          (sum ?amount
               (and (job ?x (computer programmer))
                    (salary ?x ?amount)))

     In general, Ben's new system allows expressions of the form

          (accumulation-function <VARIABLE>
                                 <QUERY PATTERN>)

     where `accumulation-function' can be things like `sum', `average',
     or `maximum'.  Ben reasons that it should be a cinch to implement
     this.  He will simply feed the query pattern to `qeval'.  This
     will produce a stream of frames.  He will then pass this stream
     through a mapping function that extracts the value of the
     designated variable from each frame in the stream and feed the
     resulting stream of values to the accumulation function.  Just as
     Ben completes the implementation and is about to try it out, Cy
     walks by, still puzzling over the `wheel' query result in exercise
     *Note Exercise 4-65::.  When Cy shows Ben the system's response,
     Ben groans, "Oh, no, my simple accumulation scheme won't work!"

     What has Ben just realized?  Outline a method he can use to
     salvage the situation.

     *Exercise 4.67:* Devise a way to install a loop detector in the
     query system so as to avoid the kinds of simple loops illustrated
     in the text and in *Note Exercise 4-64::.  The general idea is that
     the system should maintain some sort of history of its current
     chain of deductions and should not begin processing a query that
     it is already working on.  Describe what kind of information
     (patterns and frames) is included in this history, and how the
     check should be made.  (After you study the details of the
     query-system implementation in section *Note 4-4-4::, you may want
     to modify the system to include your loop detector.)

     *Exercise 4.68:* Define rules to implement the `reverse' operation
     of *Note Exercise 2-18::, which returns a list containing the same
     elements as a given list in reverse order.  (Hint: Use
     `append-to-form'.)  Can your rules answer both `(reverse (1 2 3)
     ?x)' and `(reverse ?x (1 2 3))' ?

     *Exercise 4.69:* Beginning with the data base and the rules you
     formulated in *Note Exercise 4-63::, devise a rule for adding
     "greats" to a grandson relationship. This should enable the system
     to deduce that Irad is the great-grandson of Adam, or that Jabal
     and Jubal are the great-great-great-great-great-grandsons of Adam.
     (Hint: Represent the fact about Irad, for example, as `((great
     grandson) Adam Irad)'.  Write rules that determine if a list ends
     in the word `grandson'.  Use this to express a rule that allows
     one to derive the relationship `((great .  ?rel) ?x ?y)', where
     `?rel' is a list ending in `grandson'.)  Check your rules on
     queries such as `((great grandson) ?g ?ggs)' and `(?relationship
     Adam Irad)'.

   ---------- Footnotes ----------

   (1) That a particular method of inference is legitimate is not a
trivial assertion.  One must prove that if one starts with true
premises, only true conclusions can be derived.  The method of
inference represented by rule applications is "modus ponens", the
familiar method of inference that says that if A is true and _A implies
B_ is true, then we may conclude that B is true.

   (2) We must qualify this statement by agreeing that, in speaking of
the "inference" accomplished by a logic program, we assume that the
computation terminates.  Unfortunately, even this qualified statement
is false for our implementation of the query language (and also false
for programs in Prolog and most other current logic programming
languages) because of our use of `not' and `lisp-value'.  As we will
describe below, the `not' implemented in the query language is not
always consistent with the `not' of mathematical logic, and
`lisp-value' introduces additional complications.  We could implement a
language consistent with mathematical logic by simply removing `not'
and `lisp-value' from the language and agreeing to write programs using
only simple queries, `and', and `or'.  However, this would greatly
restrict the expressive power of the language.  One of the major
concerns of research in logic programming is to find ways to achieve
more consistency with mathematical logic without unduly sacrificing
expressive power.

   (3) This is not a problem of the logic but one of the procedural
interpretation of the logic provided by our interpreter.  We could
write an interpreter that would not fall into a loop here.  For
example, we could enumerate all the proofs derivable from our
assertions and our rules in a breadth-first rather than a depth-first
order.  However, such a system makes it more difficult to take
advantage of the order of deductions in our programs.  One attempt to
build sophisticated control into such a program is described in deKleer
et al. 1977.  Another technique, which does not lead to such serious
control problems, is to put in special knowledge, such as detectors for
particular kinds of loops (*Note Exercise 4-67::).  However, there can
be no general scheme for reliably preventing a system from going down
infinite paths in performing deductions.  Imagine a diabolical rule of
the form "To show P(x) is true, show that P(f(x)) is true," for some
suitably chosen function f.

   (4) Consider the query `(not (baseball-fan (Bitdiddle Ben)))'.  The
system finds that `(baseball-fan (Bitdiddle Ben))' is not in the data
base, so the empty frame does not satisfy the pattern and is not
filtered out of the initial stream of frames.  The result of the query
is thus the empty frame, which is used to instantiate the input query
to produce `(not (baseball-fan (Bitdiddle Ben)))'.

   (5) A discussion and justification of this treatment of `not' can be
found in the article by Clark (1978).


File: sicp.info,  Node: 4-4-4,  Prev: 4-4-3,  Up: 4-4

4.4.4 Implementing the Query System
-----------------------------------

Section *Note 4-4-2:: described how the query system works. Now we fill
in the details by presenting a complete implementation of the system.

* Menu:

* 4-4-4-1::          The Driver Loop and Instantiation
* 4-4-4-2::          The Evaluator
* 4-4-4-3::          Finding Assertions by Pattern Matching
* 4-4-4-4::          Rules and Unification
* 4-4-4-5::          Maintaining the Data Base
* 4-4-4-6::          Stream Operations
* 4-4-4-7::          Query Syntax Procedures
* 4-4-4-8::          Frames and Bindings


File: sicp.info,  Node: 4-4-4-1,  Next: 4-4-4-2,  Prev: 4-4-4,  Up: 4-4-4

4.4.4.1 The Driver Loop and Instantiation
.........................................

The driver loop for the query system repeatedly reads input
expressions.  If the expression is a rule or assertion to be added to
the data base, then the information is added.  Otherwise the expression
is assumed to be a query.  The driver passes this query to the
evaluator `qeval' together with an initial frame stream consisting of a
single empty frame.  The result of the evaluation is a stream of frames
generated by satisfying the query with variable values found in the
data base.  These frames are used to form a new stream consisting of
copies of the original query in which the variables are instantiated
with values supplied by the stream of frames, and this final stream is
printed at the terminal:

     (define input-prompt ";;; Query input:")
     (define output-prompt ";;; Query results:")

     (define (query-driver-loop)
       (prompt-for-input input-prompt)
       (let ((q (query-syntax-process (read))))
         (cond ((assertion-to-be-added? q)
                (add-rule-or-assertion! (add-assertion-body q))
                (newline)
                (display "Assertion added to data base.")
                (query-driver-loop))
               (else
                (newline)
                (display output-prompt)
                (display-stream
                 (stream-map
                  (lambda (frame)
                    (instantiate q
                                 frame
                                 (lambda (v f)
                                   (contract-question-mark v))))
                  (qeval q (singleton-stream '()))))
                (query-driver-loop)))))

   Here, as in the other evaluators in this chapter, we use an abstract
syntax for the expressions of the query language.  The implementation
of the expression syntax, including the predicate
`assertion-to-be-added?' and the selector `add-assertion-body', is
given in section *Note 4-4-4-7::.  `Add-rule-or-assertion!' is defined
in section *Note 4-4-4-5::.

   Before doing any processing on an input expression, the driver loop
transforms it syntactically into a form that makes the processing more
efficient.  This involves changing the representation of pattern
variables.  When the query is instantiated, any variables that remain
unbound are transformed back to the input representation before being
printed.  These transformations are performed by the two procedures
`query-syntax-process' and `contract-question-mark' (section *Note
4-4-4-7::).

   To instantiate an expression, we copy it, replacing any variables in
the expression by their values in a given frame.  The values are
themselves instantiated, since they could contain variables (for
example, if `?x' in `exp' is bound to `?y' as the result of unification
and `?y' is in turn bound to 5).  The action to take if a variable
cannot be instantiated is given by a procedural argument to
`instantiate'.

     (define (instantiate exp frame unbound-var-handler)
       (define (copy exp)
         (cond ((var? exp)
                (let ((binding (binding-in-frame exp frame)))
                  (if binding
                      (copy (binding-value binding))
                      (unbound-var-handler exp frame))))
               ((pair? exp)
                (cons (copy (car exp)) (copy (cdr exp))))
               (else exp)))
       (copy exp))

   The procedures that manipulate bindings are defined in section *Note
4-4-4-8::.


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4.4.4.2 The Evaluator
.....................

The `qeval' procedure, called by the `query-driver-loop', is the basic
evaluator of the query system.  It takes as inputs a query and a stream
of frames, and it returns a stream of extended frames.  It identifies
special forms by a data-directed dispatch using `get' and `put', just
as we did in implementing generic operations in *Note Chapter 2::.  Any
query that is not identified as a special form is assumed to be a
simple query, to be processed by `simple-query'.

     (define (qeval query frame-stream)
       (let ((qproc (get (type query) 'qeval)))
         (if qproc
             (qproc (contents query) frame-stream)
             (simple-query query frame-stream))))

   `Type' and `contents', defined in section *Note 4-4-4-7::, implement
the abstract syntax of the special forms.

Simple queries
..............

The `simple-query' procedure handles simple queries.  It takes as
arguments a simple query (a pattern) together with a stream of frames,
and it returns the stream formed by extending each frame by all
data-base matches of the query.

     (define (simple-query query-pattern frame-stream)
       (stream-flatmap
        (lambda (frame)
          (stream-append-delayed
           (find-assertions query-pattern frame)
           (delay (apply-rules query-pattern frame))))
        frame-stream))

   For each frame in the input stream, we use `find-assertions' (section
*Note 4-4-4-3::) to match the pattern against all assertions in the
data base, producing a stream of extended frames, and we use
`apply-rules' (section *Note 4-4-4-4::) to apply all possible rules,
producing another stream of extended frames.  These two streams are
combined (using `stream-append-delayed', section *Note 4-4-4-6::) to
make a stream of all the ways that the given pattern can be satisfied
consistent with the original frame (see *Note Exercise 4-71::).  The
streams for the individual input frames are combined using
`stream-flatmap' (section *Note 4-4-4-6::) to form one large stream of
all the ways that any of the frames in the original input stream can be
extended to produce a match with the given pattern.

Compound queries
................

`And' queries are handled as illustrated in *Note Figure 4-5:: by the
`conjoin' procedure.  `Conjoin' takes as inputs the conjuncts and the
frame stream and returns the stream of extended frames.  First,
`conjoin' processes the stream of frames to find the stream of all
possible frame extensions that satisfy the first query in the
conjunction.  Then, using this as the new frame stream, it recursively
applies `conjoin' to the rest of the queries.

     (define (conjoin conjuncts frame-stream)
       (if (empty-conjunction? conjuncts)
           frame-stream
           (conjoin (rest-conjuncts conjuncts)
                    (qeval (first-conjunct conjuncts)
                           frame-stream))))

   The expression

     (put 'and 'qeval conjoin)

sets up `qeval' to dispatch to `conjoin' when an `and' form is
encountered.

   `Or' queries are handled similarly, as shown in *Note Figure 4-6::.
The output streams for the various disjuncts of the `or' are computed
separately and merged using the `interleave-delayed' procedure from
section *Note 4-4-4-6::.  (See *Note Exercise 4-71:: and *Note Exercise
4-72::.)

     (define (disjoin disjuncts frame-stream)
       (if (empty-disjunction? disjuncts)
           the-empty-stream
           (interleave-delayed
            (qeval (first-disjunct disjuncts) frame-stream)
            (delay (disjoin (rest-disjuncts disjuncts)
                            frame-stream)))))

     (put 'or 'qeval disjoin)

   The predicates and selectors for the syntax of conjuncts and
disjuncts are given in section *Note 4-4-4-7::.

Filters
.......

`Not' is handled by the method outlined in section *Note 4-4-2::.  We
attempt to extend each frame in the input stream to satisfy the query
being negated, and we include a given frame in the output stream only
if it cannot be extended.

     (define (negate operands frame-stream)
       (stream-flatmap
        (lambda (frame)
          (if (stream-null? (qeval (negated-query operands)
                                   (singleton-stream frame)))
              (singleton-stream frame)
              the-empty-stream))
        frame-stream))

     (put 'not 'qeval negate)

   `Lisp-value' is a filter similar to `not'.  Each frame in the stream
is used to instantiate the variables in the pattern, the indicated
predicate is applied, and the frames for which the predicate returns
false are filtered out of the input stream.  An error results if there
are unbound pattern variables.

     (define (lisp-value call frame-stream)
       (stream-flatmap
        (lambda (frame)
          (if (execute
               (instantiate
                call
                frame
                (lambda (v f)
                  (error "Unknown pat var -- LISP-VALUE" v))))
              (singleton-stream frame)
              the-empty-stream))
        frame-stream))

     (put 'lisp-value 'qeval lisp-value)

   `Execute', which applies the predicate to the arguments, must `eval'
the predicate expression to get the procedure to apply.  However, it
must not evaluate the arguments, since they are already the actual
arguments, not expressions whose evaluation (in Lisp) will produce the
arguments.  Note that `execute' is implemented using `eval' and `apply'
from the underlying Lisp system.

     (define (execute exp)
       (apply (eval (predicate exp) user-initial-environment)
              (args exp)))

   The `always-true' special form provides for a query that is always
satisfied.  It ignores its contents (normally empty) and simply passes
through all the frames in the input stream.  `Always-true' is used by
the `rule-body' selector (section *Note 4-4-4-7::) to provide bodies
for rules that were defined without bodies (that is, rules whose
conclusions are always satisfied).

     (define (always-true ignore frame-stream) frame-stream)

     (put 'always-true 'qeval always-true)

   The selectors that define the syntax of `not' and `lisp-value' are
given in section *Note 4-4-4-7::.


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4.4.4.3 Finding Assertions by Pattern Matching
..............................................

`Find-assertions', called by `simple-query' (section *Note 4-4-4-2::),
takes as input a pattern and a frame.  It returns a stream of frames,
each extending the given one by a data-base match of the given pattern.
It uses `fetch-assertions' (section *Note 4-4-4-5::) to get a stream
of all the assertions in the data base that should be checked for a
match against the pattern and the frame.  The reason for
`fetch-assertions' here is that we can often apply simple tests that
will eliminate many of the entries in the data base from the pool of
candidates for a successful match.  The system would still work if we
eliminated `fetch-assertions' and simply checked a stream of all
assertions in the data base, but the computation would be less efficient
because we would need to make many more calls to the matcher.

     (define (find-assertions pattern frame)
       (stream-flatmap (lambda (datum)
                         (check-an-assertion datum pattern frame))
                       (fetch-assertions pattern frame)))

   `Check-an-assertion' takes as arguments a pattern, a data object
(assertion), and a frame and returns either a one-element stream
containing the extended frame or `the-empty-stream' if the match fails.

     (define (check-an-assertion assertion query-pat query-frame)
       (let ((match-result
              (pattern-match query-pat assertion query-frame)))
         (if (eq? match-result 'failed)
             the-empty-stream
             (singleton-stream match-result))))

   The basic pattern matcher returns either the symbol `failed' or an
extension of the given frame.  The basic idea of the matcher is to
check the pattern against the data, element by element, accumulating
bindings for the pattern variables.  If the pattern and the data object
are the same, the match succeeds and we return the frame of bindings
accumulated so far.  Otherwise, if the pattern is a variable we extend
the current frame by binding the variable to the data, so long as this
is consistent with the bindings already in the frame.  If the pattern
and the data are both pairs, we (recursively) match the `car' of the
pattern against the `car' of the data to produce a frame; in this frame
we then match the `cdr' of the pattern against the `cdr' of the data.
If none of these cases are applicable, the match fails and we return
the symbol `failed'.

     (define (pattern-match pat dat frame)
       (cond ((eq? frame 'failed) 'failed)
             ((equal? pat dat) frame)
             ((var? pat) (extend-if-consistent pat dat frame))
             ((and (pair? pat) (pair? dat))
              (pattern-match (cdr pat)
                             (cdr dat)
                             (pattern-match (car pat)
                                            (car dat)
                                            frame)))
             (else 'failed)))

   Here is the procedure that extends a frame by adding a new binding,
if this is consistent with the bindings already in the frame:

     (define (extend-if-consistent var dat frame)
       (let ((binding (binding-in-frame var frame)))
         (if binding
             (pattern-match (binding-value binding) dat frame)
             (extend var dat frame))))

   If there is no binding for the variable in the frame, we simply add
the binding of the variable to the data.  Otherwise we match, in the
frame, the data against the value of the variable in the frame.  If the
stored value contains only constants, as it must if it was stored
during pattern matching by `extend-if-consistent', then the match
simply tests whether the stored and new values are the same.  If so, it
returns the unmodified frame; if not, it returns a failure indication.
The stored value may, however, contain pattern variables if it was
stored during unification (see section *Note 4-4-4-4::).  The recursive
match of the stored pattern against the new data will add or check
bindings for the variables in this pattern.  For example, suppose we
have a frame in which `?x' is bound to `(f ?y)' and `?y' is unbound,
and we wish to augment this frame by a binding of `?x' to `(f b)'.  We
look up `?x' and find that it is bound to `(f ?y)'.  This leads us to
match `(f ?y)' against the proposed new value `(f b)' in the same
frame.  Eventually this match extends the frame by adding a binding of
`?y' to `b'.  `?X' remains bound to `(f ?y)'.  We never modify a stored
binding and we never store more than one binding for a given variable.

   The procedures used by `extend-if-consistent' to manipulate bindings
are defined in section *Note 4-4-4-8::.

Patterns with dotted tails
..........................

If a pattern contains a dot followed by a pattern variable, the pattern
variable matches the rest of the data list (rather than the next
element of the data list), just as one would expect with the
dotted-tail notation described in *Note Exercise 2-20::.  Although the
pattern matcher we have just implemented doesn't look for dots, it does
behave as we want.  This is because the Lisp `read' primitive, which is
used by `query-driver-loop' to read the query and represent it as a
list structure, treats dots in a special way.

   When `read' sees a dot, instead of making the next item be the next
element of a list (the `car' of a `cons' whose `cdr' will be the rest
of the list) it makes the next item be the `cdr' of the list structure.
For example, the list structure produced by `read' for the pattern
`(computer ?type)' could be constructed by evaluating the expression
`(cons 'computer (cons '?type '()))', and that for `(computer . ?type)'
could be constructed by evaluating the expression `(cons 'computer
'?type)'.

   Thus, as `pattern-match' recursively compares `car's and `cdr's of a
data list and a pattern that had a dot, it eventually matches the
variable after the dot (which is a `cdr' of the pattern) against a
sublist of the data list, binding the variable to that list.  For
example, matching the pattern `(computer . ?type)' against `(computer
programmer trainee)' will match `?type' against the list `(programmer
trainee)'.


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4.4.4.4 Rules and Unification
.............................

`Apply-rules' is the rule analog of `find-assertions' (section *Note
4-4-4-3::).  It takes as input a pattern and a frame, and it forms a
stream of extension frames by applying rules from the data base.
`Stream-flatmap' maps `apply-a-rule' down the stream of possibly
applicable rules (selected by `fetch-rules', section *Note 4-4-4-5::)
and combines the resulting streams of frames.

     (define (apply-rules pattern frame)
       (stream-flatmap (lambda (rule)
                         (apply-a-rule rule pattern frame))
                       (fetch-rules pattern frame)))

   `Apply-a-rule' applies rules using the method outlined in section
*Note 4-4-2::.  It first augments its argument frame by unifying the
rule conclusion with the pattern in the given frame.  If this succeeds,
it evaluates the rule body in this new frame.

   Before any of this happens, however, the program renames all the
variables in the rule with unique new names.  The reason for this is to
prevent the variables for different rule applications from becoming
confused with each other.  For instance, if two rules both use a
variable named `?x', then each one may add a binding for `?x' to the
frame when it is applied.  These two `?x''s have nothing to do with
each other, and we should not be fooled into thinking that the two
bindings must be consistent.  Rather than rename variables, we could
devise a more clever environment structure; however, the renaming
approach we have chosen here is the most straightforward, even if not
the most efficient.  (See *Note Exercise 4-79::.)  Here is the
`apply-a-rule' procedure:

     (define (apply-a-rule rule query-pattern query-frame)
       (let ((clean-rule (rename-variables-in rule)))
         (let ((unify-result
                (unify-match query-pattern
                             (conclusion clean-rule)
                             query-frame)))
           (if (eq? unify-result 'failed)
               the-empty-stream
               (qeval (rule-body clean-rule)
                      (singleton-stream unify-result))))))

   The selectors `rule-body' and `conclusion' that extract parts of a
rule are defined in section *Note 4-4-4-7::.

   We generate unique variable names by associating a unique identifier
(such as a number) with each rule application and combining this
identifier with the original variable names.  For example, if the
rule-application identifier is 7, we might change each `?x' in the rule
to `?x-7' and each `?y' in the rule to `?y-7'.  (`Make-new-variable' and
`new-rule-application-id' are included with the syntax procedures in
section *Note 4-4-4-7::.)

     (define (rename-variables-in rule)
       (let ((rule-application-id (new-rule-application-id)))
         (define (tree-walk exp)
           (cond ((var? exp)
                  (make-new-variable exp rule-application-id))
                 ((pair? exp)
                  (cons (tree-walk (car exp))
                        (tree-walk (cdr exp))))
                 (else exp)))
         (tree-walk rule)))

   The unification algorithm is implemented as a procedure that takes
as inputs two patterns and a frame and returns either the extended
frame or the symbol `failed'.  The unifier is like the pattern matcher
except that it is symmetrical--variables are allowed on both sides of
the match.  `Unify-match' is basically the same as `pattern-match',
except that there is extra code (marked "`***'" below) to handle the
case where the object on the right side of the match is a variable.

     (define (unify-match p1 p2 frame)
       (cond ((eq? frame 'failed) 'failed)
             ((equal? p1 p2) frame)
             ((var? p1) (extend-if-possible p1 p2 frame))
             ((var? p2) (extend-if-possible p2 p1 frame))  ; ***
             ((and (pair? p1) (pair? p2))
              (unify-match (cdr p1)
                           (cdr p2)
                           (unify-match (car p1)
                                        (car p2)
                                        frame)))
             (else 'failed)))

   In unification, as in one-sided pattern matching, we want to accept
a proposed extension of the frame only if it is consistent with
existing bindings.  The procedure `extend-if-possible' used in
unification is the same as the `extend-if-consistent' used in pattern
matching except for two special checks, marked "`***'" in the program
below.  In the first case, if the variable we are trying to match is
not bound, but the value we are trying to match it with is itself a
(different) variable, it is necessary to check to see if the value is
bound, and if so, to match its value.  If both parties to the match are
unbound, we may bind either to the other.

   The second check deals with attempts to bind a variable to a pattern
that includes that variable.  Such a situation can occur whenever a
variable is repeated in both patterns.  Consider, for example, unifying
the two patterns `(?x ?x)' and `(?y <EXPRESSION INVOLVING `?Y'>)' in a
frame where both `?x' and `?y' are unbound.  First `?x' is matched
against `?y', making a binding of `?x' to `?y'.  Next, the same `?x' is
matched against the given expression involving `?y'.  Since `?x' is
already bound to `?y', this results in matching `?y' against the
expression.  If we think of the unifier as finding a set of values for
the pattern variables that make the patterns the same, then these
patterns imply instructions to find a `?y' such that `?y' is equal to
the expression involving `?y'.  There is no general method for solving
such equations, so we reject such bindings; these cases are recognized
by the predicate `depends-on?'.(1)  On the other hand, we do not want
to reject attempts to bind a variable to itself.  For example, consider
unifying `(?x ?x)' and `(?y ?y)'.  The second attempt to bind `?x' to
`?y' matches `?y' (the stored value of `?x') against `?y' (the new
value of `?x').  This is taken care of by the `equal?'  clause of
`unify-match'.

     (define (extend-if-possible var val frame)
       (let ((binding (binding-in-frame var frame)))
         (cond (binding
                (unify-match
                 (binding-value binding) val frame))
               ((var? val)                      ; ***
                (let ((binding (binding-in-frame val frame)))
                  (if binding
                      (unify-match
                       var (binding-value binding) frame)
                      (extend var val frame))))
               ((depends-on? val var frame)     ; ***
                'failed)
               (else (extend var val frame)))))

   `Depends-on?' is a predicate that tests whether an expression
proposed to be the value of a pattern variable depends on the variable.
This must be done relative to the current frame because the expression
may contain occurrences of a variable that already has a value that
depends on our test variable.  The structure of `depends-on?' is a
simple recursive tree walk in which we substitute for the values of
variables whenever necessary.

     (define (depends-on? exp var frame)
       (define (tree-walk e)
         (cond ((var? e)
                (if (equal? var e)
                    true
                    (let ((b (binding-in-frame e frame)))
                      (if b
                          (tree-walk (binding-value b))
                          false))))
               ((pair? e)
                (or (tree-walk (car e))
                    (tree-walk (cdr e))))
               (else false)))
       (tree-walk exp))

   ---------- Footnotes ----------

   (1) In general, unifying `?y' with an expression involving `?y'
would require our being able to find a fixed point of the equation `?y'
= <EXPRESSION INVOLVING `?Y'>.  It is sometimes possible to
syntactically form an expression that appears to be the solution.  For
example, `?y' = `(f ?y)' seems to have the fixed point `(f (f (f ...
)))', which we can produce by beginning with the expression `(f ?y)'
and repeatedly substituting `(f ?y)' for `?y'.  Unfortunately, not
every such equation has a meaningful fixed point.  The issues that
arise here are similar to the issues of manipulating infinite series in
mathematics.  For example, we know that 2 is the solution to the
equation y = 1 + y/2.  Beginning with the expression 1 + y/2 and
repeatedly substituting 1 + y/2 for y gives

     2 = y = 1 + y/2 = 1 + (1 + y/2)/2 = 1 + 1/2 + y/4 = ...

which leads to

     2 = 1 + 1/2 + 1/4 + 1/8 + ...

However, if we try the same manipulation beginning with the observation
that - 1 is the solution to the equation y = 1 + 2y, we obtain

     -1 = y = 1 + 2y = 1 + 2(1 + 2y) = 1 + 2 + 4y = ...

which leads to

     -1 = 1 + 2 + 4 + 8 + ...

Although the formal manipulations used in deriving these two equations
are identical, the first result is a valid assertion about infinite
series but the second is not.  Similarly, for our unification results,
reasoning with an arbitrary syntactically constructed expression may
lead to errors.


File: sicp.info,  Node: 4-4-4-5,  Next: 4-4-4-6,  Prev: 4-4-4-4,  Up: 4-4-4

4.4.4.5 Maintaining the Data Base
.................................

One important problem in designing logic programming languages is that
of arranging things so that as few irrelevant data-base entries as
possible will be examined in checking a given pattern.  In our system,
in addition to storing all assertions in one big stream, we store all
assertions whose `car's are constant symbols in separate streams, in a
table indexed by the symbol.  To fetch an assertion that may match a
pattern, we first check to see if the `car' of the pattern is a
constant symbol.  If so, we return (to be tested using the matcher) all
the stored assertions that have the same `car'.  If the pattern's `car'
is not a constant symbol, we return all the stored assertions.
Cleverer methods could also take advantage of information in the frame,
or try also to optimize the case where the `car' of the pattern is not
a constant symbol.  We avoid building our criteria for indexing (using
the `car', handling only the case of constant symbols) into the program;
instead we call on predicates and selectors that embody our criteria.

     (define THE-ASSERTIONS the-empty-stream)

     (define (fetch-assertions pattern frame)
       (if (use-index? pattern)
           (get-indexed-assertions pattern)
           (get-all-assertions)))

     (define (get-all-assertions) THE-ASSERTIONS)

     (define (get-indexed-assertions pattern)
       (get-stream (index-key-of pattern) 'assertion-stream))

   `Get-stream' looks up a stream in the table and returns an empty
stream if nothing is stored there.

     (define (get-stream key1 key2)
       (let ((s (get key1 key2)))
         (if s s the-empty-stream)))

   Rules are stored similarly, using the `car' of the rule conclusion.
Rule conclusions are arbitrary patterns, however, so they differ from
assertions in that they can contain variables.  A pattern whose `car'
is a constant symbol can match rules whose conclusions start with a
variable as well as rules whose conclusions have the same `car'.  Thus,
when fetching rules that might match a pattern whose `car' is a
constant symbol we fetch all rules whose conclusions start with a
variable as well as those whose conclusions have the same `car' as the
pattern.  For this purpose we store all rules whose conclusions start
with a variable in a separate stream in our table, indexed by the
symbol `?'.

     (define THE-RULES the-empty-stream)

     (define (fetch-rules pattern frame)
       (if (use-index? pattern)
           (get-indexed-rules pattern)
           (get-all-rules)))

     (define (get-all-rules) THE-RULES)

     (define (get-indexed-rules pattern)
       (stream-append
        (get-stream (index-key-of pattern) 'rule-stream)
        (get-stream '? 'rule-stream)))

   `Add-rule-or-assertion!' is used by `query-driver-loop' to add
assertions and rules to the data base.  Each item is stored in the
index, if appropriate, and in a stream of all assertions or rules in
the data base.

     (define (add-rule-or-assertion! assertion)
       (if (rule? assertion)
           (add-rule! assertion)
           (add-assertion! assertion)))

     (define (add-assertion! assertion)
       (store-assertion-in-index assertion)
       (let ((old-assertions THE-ASSERTIONS))
         (set! THE-ASSERTIONS
               (cons-stream assertion old-assertions))
         'ok))

     (define (add-rule! rule)
       (store-rule-in-index rule)
       (let ((old-rules THE-RULES))
         (set! THE-RULES (cons-stream rule old-rules))
         'ok))

   To actually store an assertion or a rule, we check to see if it can
be indexed.  If so, we store it in the appropriate stream.

     (define (store-assertion-in-index assertion)
       (if (indexable? assertion)
           (let ((key (index-key-of assertion)))
             (let ((current-assertion-stream
                    (get-stream key 'assertion-stream)))
               (put key
                    'assertion-stream
                    (cons-stream assertion
                                 current-assertion-stream))))))

     (define (store-rule-in-index rule)
       (let ((pattern (conclusion rule)))
         (if (indexable? pattern)
             (let ((key (index-key-of pattern)))
               (let ((current-rule-stream
                      (get-stream key 'rule-stream)))
                 (put key
                      'rule-stream
                      (cons-stream rule
                                   current-rule-stream)))))))

   The following procedures define how the data-base index is used.  A
pattern (an assertion or a rule conclusion) will be stored in the table
if it starts with a variable or a constant symbol.

     (define (indexable? pat)
       (or (constant-symbol? (car pat))
           (var? (car pat))))

   The key under which a pattern is stored in the table is either `?'
(if it starts with a variable) or the constant symbol with which it
starts.

     (define (index-key-of pat)
       (let ((key (car pat)))
         (if (var? key) '? key)))

   The index will be used to retrieve items that might match a pattern
if the pattern starts with a constant symbol.

     (define (use-index? pat)
       (constant-symbol? (car pat)))

     *Exercise 4.70:* What is the purpose of the `let' bindings in the
     procedures `add-assertion!' and `add-rule!' ?  What would be wrong
     with the following implementation of `add-assertion!' ?  Hint:
     Recall the definition of the infinite stream of ones in section
     *Note 3-5-2::: `(define ones (cons-stream 1 ones))'.

          (define (add-assertion! assertion)
            (store-assertion-in-index assertion)
            (set! THE-ASSERTIONS
                  (cons-stream assertion THE-ASSERTIONS))
            'ok)


File: sicp.info,  Node: 4-4-4-6,  Next: 4-4-4-7,  Prev: 4-4-4-5,  Up: 4-4-4

4.4.4.6 Stream Operations
.........................

The query system uses a few stream operations that were not presented in
*Note Chapter 3::.

   `Stream-append-delayed' and `interleave-delayed' are just like
`stream-append' and `interleave' (section *Note 3-5-3::), except that
they take a delayed argument (like the `integral' procedure in section
*Note 3-5-4::).  This postpones looping in some cases (see *Note
Exercise 4-71::).

     (define (stream-append-delayed s1 delayed-s2)
       (if (stream-null? s1)
           (force delayed-s2)
           (cons-stream
            (stream-car s1)
            (stream-append-delayed (stream-cdr s1) delayed-s2))))

     (define (interleave-delayed s1 delayed-s2)
       (if (stream-null? s1)
           (force delayed-s2)
           (cons-stream
            (stream-car s1)
            (interleave-delayed (force delayed-s2)
                                (delay (stream-cdr s1))))))

   `Stream-flatmap', which is used throughout the query evaluator to
map a procedure over a stream of frames and combine the resulting
streams of frames, is the stream analog of the `flatmap' procedure
introduced for ordinary lists in section *Note 2-2-3::.  Unlike
ordinary `flatmap', however, we accumulate the streams with an
interleaving process, rather than simply appending them (see *Note
Exercise 4-72:: and *Note Exercise 4-73::).

     (define (stream-flatmap proc s)
       (flatten-stream (stream-map proc s)))

     (define (flatten-stream stream)
       (if (stream-null? stream)
           the-empty-stream
           (interleave-delayed
            (stream-car stream)
            (delay (flatten-stream (stream-cdr stream))))))

   The evaluator also uses the following simple procedure to generate a
stream consisting of a single element:

     (define (singleton-stream x)
       (cons-stream x the-empty-stream))


File: sicp.info,  Node: 4-4-4-7,  Next: 4-4-4-8,  Prev: 4-4-4-6,  Up: 4-4-4

4.4.4.7 Query Syntax Procedures
...............................

`Type' and `contents', used by `qeval' (section *Note 4-4-4-2::),
specify that a special form is identified by the symbol in its `car'.
They are the same as the `type-tag' and `contents' procedures in
section *Note 2-4-2::, except for the error message.

     (define (type exp)
       (if (pair? exp)
           (car exp)
           (error "Unknown expression TYPE" exp)))

     (define (contents exp)
       (if (pair? exp)
           (cdr exp)
           (error "Unknown expression CONTENTS" exp)))

   The following procedures, used by `query-driver-loop' (in section
*Note 4-4-4-1::), specify that rules and assertions are added to the
data base by expressions of the form `(assert! <RULE-OR-ASSERTION>)':

     (define (assertion-to-be-added? exp)
       (eq? (type exp) 'assert!))

     (define (add-assertion-body exp)
       (car (contents exp)))

   Here are the syntax definitions for the `and', `or', `not', and
`lisp-value' special forms (section *Note 4-4-4-2::):

     (define (empty-conjunction? exps) (null? exps))
     (define (first-conjunct exps) (car exps))
     (define (rest-conjuncts exps) (cdr exps))

     (define (empty-disjunction? exps) (null? exps))
     (define (first-disjunct exps) (car exps))
     (define (rest-disjuncts exps) (cdr exps))

     (define (negated-query exps) (car exps))

     (define (predicate exps) (car exps))
     (define (args exps) (cdr exps))

   The following three procedures define the syntax of rules:

     (define (rule? statement)
       (tagged-list? statement 'rule))

     (define (conclusion rule) (cadr rule))

     (define (rule-body rule)
       (if (null? (cddr rule))
           '(always-true)
           (caddr rule)))

   `Query-driver-loop' (section *Note 4-4-4-1::) calls
`query-syntax-process' to transform pattern variables in the expression,
which have the form `?symbol', into the internal format `(? symbol)'.
That is to say, a pattern such as `(job ?x ?y)' is actually represented
internally by the system as `(job (? x) (? y))'.  This increases the
efficiency of query processing, since it means that the system can
check to see if an expression is a pattern variable by checking whether
the `car' of the expression is the symbol `?', rather than having to
extract characters from the symbol.  The syntax transformation is
accomplished by the following procedure:(1)

     (define (query-syntax-process exp)
       (map-over-symbols expand-question-mark exp))

     (define (map-over-symbols proc exp)
       (cond ((pair? exp)
              (cons (map-over-symbols proc (car exp))
                    (map-over-symbols proc (cdr exp))))
             ((symbol? exp) (proc exp))
             (else exp)))

     (define (expand-question-mark symbol)
       (let ((chars (symbol->string symbol)))
         (if (string=? (substring chars 0 1) "?")
             (list '?
                   (string->symbol
                    (substring chars 1 (string-length chars))))
             symbol)))

   Once the variables are transformed in this way, the variables in a
pattern are lists starting with `?', and the constant symbols (which
need to be recognized for data-base indexing, section *Note 4-4-4-5::)
are just the symbols.

     (define (var? exp)
       (tagged-list? exp '?))

     (define (constant-symbol? exp) (symbol? exp))

   Unique variables are constructed during rule application (in section
*Note 4-4-4-4::) by means of the following procedures.  The unique
identifier for a rule application is a number, which is incremented
each time a rule is applied.

     (define rule-counter 0)

     (define (new-rule-application-id)
       (set! rule-counter (+ 1 rule-counter))
       rule-counter)

     (define (make-new-variable var rule-application-id)
       (cons '? (cons rule-application-id (cdr var))))

   When `query-driver-loop' instantiates the query to print the answer,
it converts any unbound pattern variables back to the right form for
printing, using

     (define (contract-question-mark variable)
       (string->symbol
        (string-append "?"
          (if (number? (cadr variable))
              (string-append (symbol->string (caddr variable))
                             "-"
                             (number->string (cadr variable)))
              (symbol->string (cadr variable))))))

   ---------- Footnotes ----------

   (1) Most Lisp systems give the user the ability to modify the
ordinary `read' procedure to perform such transformations by defining "reader
macro characters".  Quoted expressions are already handled in this way:
The reader automatically translates `'expression' into `(quote
expression)' before the evaluator sees it.  We could arrange for
`?expression' to be transformed into `(? expression)' in the same way;
however, for the sake of clarity we have included the transformation
procedure here explicitly.

   `Expand-question-mark' and `contract-question-mark' use several
procedures with `string' in their names.  These are Scheme primitives.


File: sicp.info,  Node: 4-4-4-8,  Prev: 4-4-4-7,  Up: 4-4-4

4.4.4.8 Frames and Bindings
...........................

Frames are represented as lists of bindings, which are variable-value
pairs:

     (define (make-binding variable value)
       (cons variable value))

     (define (binding-variable binding)
       (car binding))

     (define (binding-value binding)
       (cdr binding))

     (define (binding-in-frame variable frame)
       (assoc variable frame))

     (define (extend variable value frame)
       (cons (make-binding variable value) frame))

     *Exercise 4.71:* Louis Reasoner wonders why the `simple-query' and
     `disjoin' procedures (section *Note 4-4-4-2::) are implemented
     using explicit `delay' operations, rather than being defined as
     follows:

          (define (simple-query query-pattern frame-stream)
            (stream-flatmap
             (lambda (frame)
               (stream-append (find-assertions query-pattern frame)
                              (apply-rules query-pattern frame)))
             frame-stream))

          (define (disjoin disjuncts frame-stream)
            (if (empty-disjunction? disjuncts)
                the-empty-stream
                (interleave
                 (qeval (first-disjunct disjuncts) frame-stream)
                 (disjoin (rest-disjuncts disjuncts) frame-stream))))

     Can you give examples of queries where these simpler definitions
     would lead to undesirable behavior?

     *Exercise 4.72:* Why do `disjoin' and `stream-flatmap' interleave
     the streams rather than simply append them?  Give examples that
     illustrate why interleaving works better.  (Hint: Why did we use
     `interleave' in section *Note 3-5-3::?)

     *Exercise 4.73:* Why does `flatten-stream' use `delay' explicitly?
     What would be wrong with defining it as follows:

          (define (flatten-stream stream)
            (if (stream-null? stream)
                the-empty-stream
                (interleave
                 (stream-car stream)
                 (flatten-stream (stream-cdr stream)))))

     *Exercise 4.74:* Alyssa P. Hacker proposes to use a simpler
     version of `stream-flatmap' in `negate', `lisp-value', and
     `find-assertions'.  She observes that the procedure that is mapped
     over the frame stream in these cases always produces either the
     empty stream or a singleton stream, so no interleaving is needed
     when combining these streams.

       a. Fill in the missing expressions in Alyssa's program.

               (define (simple-stream-flatmap proc s)
                 (simple-flatten (stream-map proc s)))

               (define (simple-flatten stream)
                 (stream-map <??>
                             (stream-filter <??> stream)))

       b. Does the query system's behavior change if we change it in
          this way?


     *Exercise 4.75:* Implement for the query language a new special
     form called `unique'.  `Unique' should succeed if there is
     precisely one item in the data base satisfying a specified query.
     For example,

          (unique (job ?x (computer wizard)))

     should print the one-item stream

          (unique (job (Bitdiddle Ben) (computer wizard)))

     since Ben is the only computer wizard, and

          (unique (job ?x (computer programmer)))

     should print the empty stream, since there is more than one
     computer programmer.  Moreover,

          (and (job ?x ?j) (unique (job ?anyone ?j)))

     should list all the jobs that are filled by only one person, and
     the people who fill them.

     There are two parts to implementing `unique'.  The first is to
     write a procedure that handles this special form, and the second
     is to make `qeval' dispatch to that procedure.  The second part is
     trivial, since `qeval' does its dispatching in a data-directed
     way.  If your procedure is called `uniquely-asserted', all you
     need to do is

          (put 'unique 'qeval uniquely-asserted)

     and `qeval' will dispatch to this procedure for every query whose
     `type' (`car') is the symbol `unique'.

     The real problem is to write the procedure `uniquely-asserted'.
     This should take as input the `contents' (`cdr') of the `unique'
     query, together with a stream of frames.  For each frame in the
     stream, it should use `qeval' to find the stream of all extensions
     to the frame that satisfy the given query.  Any stream that does
     not have exactly one item in it should be eliminated.  The
     remaining streams should be passed back to be accumulated into one
     big stream that is the result of the `unique' query.  This is
     similar to the implementation of the `not' special form.

     Test your implementation by forming a query that lists all people
     who supervise precisely one person.

     *Exercise 4.76:* Our implementation of `and' as a series
     combination of queries (*Note Figure 4-5::) is elegant, but it is
     inefficient because in processing the second query of the `and' we
     must scan the data base for each frame produced by the first
     query.  If the data base has n elements, and a typical query
     produces a number of output frames proportional to n (say n/k),
     then scanning the data base for each frame produced by the first
     query will require n^2/k calls to the pattern matcher.  Another
     approach would be to process the two clauses of the `and'
     separately, then look for all pairs of output frames that are
     compatible.  If each query produces n/k output frames, then this
     means that we must perform n^2/k^2 compatibility checks--a factor
     of k fewer than the number of matches required in our current
     method.

     Devise an implementation of `and' that uses this strategy.  You
     must implement a procedure that takes two frames as inputs, checks
     whether the bindings in the frames are compatible, and, if so,
     produces a frame that merges the two sets of bindings.  This
     operation is similar to unification.

     *Exercise 4.77:* In section *Note 4-4-3:: we saw that `not' and
     `lisp-value' can cause the query language to give "wrong" answers
     if these filtering operations are applied to frames in which
     variables are unbound.  Devise a way to fix this shortcoming.  One
     idea is to perform the filtering in a "delayed" manner by
     appending to the frame a "promise" to filter that is fulfilled
     only when enough variables have been bound to make the operation
     possible.  We could wait to perform filtering until all other
     operations have been performed.  However, for efficiency's sake, we
     would like to perform filtering as soon as possible so as to cut
     down on the number of intermediate frames generated.

     *Exercise 4.78:* Redesign the query language as a nondeterministic
     program to be implemented using the evaluator of section *Note
     4-3::, rather than as a stream process.  In this approach, each
     query will produce a single answer (rather than the stream of all
     answers) and the user can type `try-again' to see more answers.
     You should find that much of the mechanism we built in this
     section is subsumed by nondeterministic search and backtracking.
     You will probably also find, however, that your new query language
     has subtle differences in behavior from the one implemented here.
     Can you find examples that illustrate this difference?

     *Exercise 4.79:* When we implemented the Lisp evaluator in section
     *Note 4-1::, we saw how to use local environments to avoid name
     conflicts between the parameters of procedures.  For example, in
     evaluating

          (define (square x)
            (* x x))

          (define (sum-of-squares x y)
            (+ (square x) (square y)))

          (sum-of-squares 3 4)

     there is no confusion between the `x' in `square' and the `x' in
     `sum-of-squares', because we evaluate the body of each procedure
     in an environment that is specially constructed to contain
     bindings for the local variables.  In the query system, we used a
     different strategy to avoid name conflicts in applying rules.
     Each time we apply a rule we rename the variables with new names
     that are guaranteed to be unique.  The analogous strategy for the
     Lisp evaluator would be to do away with local environments and
     simply rename the variables in the body of a procedure each time
     we apply the procedure.

     Implement for the query language a rule-application method that
     uses environments rather than renaming.  See if you can build on
     your environment structure to create constructs in the query
     language for dealing with large systems, such as the rule analog
     of block-structured procedures.  Can you relate any of this to the
     problem of making deductions in a context (e.g., "If I supposed
     that P were true, then I would be able to deduce A and B.") as a
     method of problem solving?  (This problem is open-ended.  A good
     answer is probably worth a Ph.D.)


File: sicp.info,  Node: Chapter 5,  Next: References,  Prev: Chapter 4,  Up: Top

5 Computing with Register Machines
**********************************

     My aim is to show that the heavenly machine is not a kind of
     divine, live being, but a kind of clockwork (and he who believes
     that a clock has soul attributes the maker's glory to the work),
     insofar as nearly all the manifold motions are caused by a most
     simple and material force, just as all motions of the clock are
     caused by a single weight.

     Johannes Kepler (letter to Herwart von Hohenburg, 1605)

   We began this book by studying processes and by describing processes
in terms of procedures written in Lisp.  To explain the meanings of
these procedures, we used a succession of models of evaluation: the
substitution model of *Note Chapter 1::, the environment model of *Note
Chapter 3::, and the metacircular evaluator of *Note Chapter 4::.  Our
examination of the metacircular evaluator, in particular, dispelled
much of the mystery of how Lisp-like languages are interpreted.  But
even the metacircular evaluator leaves important questions unanswered,
because it fails to elucidate the mechanisms of control in a Lisp
system.  For instance, the evaluator does not explain how the
evaluation of a subexpression manages to return a value to the
expression that uses this value, nor does the evaluator explain how
some recursive procedures generate iterative processes (that is, are
evaluated using constant space) whereas other recursive procedures
generate recursive processes.  These questions remain unanswered
because the metacircular evaluator is itself a Lisp program and hence
inherits the control structure of the underlying Lisp system.  In order
to provide a more complete description of the control structure of the
Lisp evaluator, we must work at a more primitive level than Lisp itself.

   In this chapter we will describe processes in terms of the
step-by-step operation of a traditional computer.  Such a computer, or machine
"register machine", sequentially executes "instructions" that
manipulate the contents of a fixed set of storage elements called "registers".
A typical register-machine instruction applies a primitive operation
to the contents of some registers and assigns the result to another
register.  Our descriptions of processes executed by register machines
will look very much like "machine-language" programs for traditional
computers.  However, instead of focusing on the machine language of any
particular computer, we will examine several Lisp procedures and design
a specific register machine to execute each procedure.  Thus, we will
approach our task from the perspective of a hardware architect rather
than that of a machine-language computer programmer.  In designing
register machines, we will develop mechanisms for implementing
important programming constructs such as recursion.  We will also
present a language for describing designs for register machines.  In
section *Note 5-2:: we will implement a Lisp program that uses these
descriptions to simulate the machines we design.

   Most of the primitive operations of our register machines are very
simple.  For example, an operation might add the numbers fetched from
two registers, producing a result to be stored into a third register.
Such an operation can be performed by easily described hardware.  In
order to deal with list structure, however, we will also use the memory
operations `car', `cdr', and `cons', which require an elaborate
storage-allocation mechanism.  In section *Note 5-3:: we study their
implementation in terms of more elementary operations.

   In section *Note 5-4::, after we have accumulated experience
formulating simple procedures as register machines, we will design a
machine that carries out the algorithm described by the metacircular
evaluator of section *Note 4-1::.  This will fill in the gap in our
understanding of how Scheme expressions are interpreted, by providing
an explicit model for the mechanisms of control in the evaluator.  In
section *Note 5-5:: we will study a simple compiler that translates
Scheme programs into sequences of instructions that can be executed
directly with the registers and operations of the evaluator register
machine.

* Menu:

* 5-1::              Designing Register Machines
* 5-2::              A Register-Machine Simulator
* 5-3::              Storage Allocation and Garbage Collection
* 5-4::              The Explicit-Control Evaluator
* 5-5::              Compilation


File: sicp.info,  Node: 5-1,  Next: 5-2,  Prev: Chapter 5,  Up: Chapter 5

5.1 Designing Register Machines
===============================

To design a register machine, we must design its "data paths"
(registers and operations) and the "controller" that sequences these
operations.  To illustrate the design of a simple register machine, let
us examine Euclid's Algorithm, which is used to compute the greatest
common divisor (GCD) of two integers.  As we saw in section *Note
1-2-5::, Euclid's Algorithm can be carried out by an iterative process,
as specified by the following procedure:

     (define (gcd a b)
       (if (= b 0)
           a
           (gcd b (remainder a b))))

   A machine to carry out this algorithm must keep track of two
numbers, a and b, so let us assume that these numbers are stored in two
registers with those names.  The basic operations required are testing
whether the contents of register `b' is zero and computing the
remainder of the contents of register `a' divided by the contents of
register `b'.  The remainder operation is a complex process, but assume
for the moment that we have a primitive device that computes
remainders.  On each cycle of the GCD algorithm, the contents of
register `a' must be replaced by the contents of register `b', and the
contents of `b' must be replaced by the remainder of the old contents
of `a' divided by the old contents of `b'.  It would be convenient if
these replacements could be done simultaneously, but in our model of
register machines we will assume that only one register can be assigned
a new value at each step.  To accomplish the replacements, our machine
will use a third "temporary" register, which we call `t'.  (First the
remainder will be placed in `t', then the contents of `b' will be
placed in `a', and finally the remainder stored in `t' will be placed
in `b'.)

   We can illustrate the registers and operations required for this
machine by using the data-path diagram shown in *Note Figure 5-1::.  In
this diagram, the registers (`a', `b', and `t') are represented by
rectangles.  Each way to assign a value to a register is indicated by
an arrow with an `X' behind the head, pointing from the source of data
to the register.  We can think of the `X' as a button that, when
pushed, allows the value at the source to "flow" into the designated
register.  The label next to each button is the name we will use to
refer to the button.  The names are arbitrary, and can be chosen to
have mnemonic value (for example, `a<-b' denotes pushing the button
that assigns the contents of register `b' to register `a').  The source
of data for a register can be another register (as in the `a<-b'
assignment), an operation result (as in the `t<-r' assignment), or a
constant (a built-in value that cannot be changed, represented in a
data-path diagram by a triangle containing the constant).

   An operation that computes a value from constants and the contents
of registers is represented in a data-path diagram by a trapezoid
containing a name for the operation.  For example, the box marked `rem'
in *Note Figure 5-1:: represents an operation that computes the
remainder of the contents of the registers `a' and `b' to which it is
attached.  Arrows (without buttons) point from the input registers and
constants to the box, and arrows connect the operation's output value
to registers.  A test is represented by a circle containing a name for
the test.  For example, our GCD machine has an operation that tests
whether the contents of register `b' is zero.  A test also has arrows
from its input registers and constants, but it has no output arrows;
its value is used by the controller rather than by the data paths.
Overall, the data-path diagram shows the registers and operations that
are required for the machine and how they must be connected.  If we
view the arrows as wires and the `X' buttons as switches, the data-path
diagram is very like the wiring diagram for a machine that could be
constructed from electrical components.

     *Figure 5.1:* Data paths for a GCD machine.

                                        ___
          +-----+          +-----+     /   \
          |  a  |<--(X)----|  b  +--->|  =  |
          +--+--+   a<-b   +-+---+     \___/
             |               |  ^        ^
             +------+   +----+  |        |
                    |   |      (X) b<-t  |
                 .--+---+--.    |       / \
                  \  rem  /     |      / O \
                   \_____/      |     +-----+
                      |         |
                     (X) t<-r   |
                      |         |
                      V         |
                   +-----+      |
                   |  t  +------+
                   +-----+

   In order for the data paths to actually compute GCDs, the buttons
must be pushed in the correct sequence.  We will describe this sequence
in terms of a controller diagram, as illustrated in *Note Figure 5-2::.
The elements of the controller diagram indicate how the data-path
components should be operated.  The rectangular boxes in the controller
diagram identify data-path buttons to be pushed, and the arrows
describe the sequencing from one step to the next.  The diamond in the
diagram represents a decision.  One of the two sequencing arrows will
be followed, depending on the value of the data-path test identified in
the diamond.  We can interpret the controller in terms of a physical
analogy: Think of the diagram as a maze in which a marble is rolling.
When the marble rolls into a box, it pushes the data-path button that
is named by the box.  When the marble rolls into a decision node (such
as the test for `b' = 0), it leaves the node on the path determined by
the result of the indicated test.  Taken together, the data paths and
the controller completely describe a machine for computing GCDs.  We
start the controller (the rolling marble) at the place marked `start',
after placing numbers in registers `a' and `b'.  When the controller
reaches `done', we will find the value of the GCD in register `a'.

     *Figure 5.2:* Controller for a GCD machine.

               start
                 |
                 V
                / \ yes
          +--->< = >-----> done
          |     \ /
          |      | no
          |      V
          |  +------+
          |  | t<-r |
          |  +---+--+
          |      |
          |      V
          |  +------+
          |  | a<-b |
          |  +---+--+
          |      |
          |      V
          |  +------+
          +--+ b<-t |
             +------+

     *Exercise 5.1:* Design a register machine to compute factorials
     using the iterative algorithm specified by the following
     procedure.  Draw data-path and controller diagrams for this
     machine.

          (define (factorial n)
            (define (iter product counter)
              (if (> counter n)
                  product
                  (iter (* counter product)
                        (+ counter 1))))
            (iter 1 1))

* Menu:

* 5-1-1::            A Language for Describing Register Machines
* 5-1-2::            Abstraction in Machine Design
* 5-1-3::            Subroutines
* 5-1-4::            Using a Stack to Implement Recursion
* 5-1-5::            Instruction Summary


File: sicp.info,  Node: 5-1-1,  Next: 5-1-2,  Prev: 5-1,  Up: 5-1

5.1.1 A Language for Describing Register Machines
-------------------------------------------------

Data-path and controller diagrams are adequate for representing simple
machines such as GCD, but they are unwieldy for describing large
machines such as a Lisp interpreter.  To make it possible to deal with
complex machines, we will create a language that presents, in textual
form, all the information given by the data-path and controller
diagrams.  We will start with a notation that directly mirrors the
diagrams.

   We define the data paths of a machine by describing the registers
and the operations.  To describe a register, we give it a name and
specify the buttons that control assignment to it.  We give each of
these buttons a name and specify the source of the data that enters the
register under the button's control.  (The source is a register, a
constant, or an operation.)  To describe an operation, we give it a
name and specify its inputs (registers or constants).

   We define the controller of a machine as a sequence of "instructions"
together with "labels" that identify "entry points" in the sequence. An
instruction is one of the following:

   * The name of a data-path button to push to assign a value to a
     register.  (This corresponds to a box in the controller diagram.)

   * A `test' instruction, that performs a specified test.

   * A conditional branch (`branch' instruction) to a location
     indicated by a controller label, based on the result of the
     previous test.  (The test and branch together correspond to a
     diamond in the controller diagram.)  If the test is false, the
     controller should continue with the next instruction in the
     sequence.  Otherwise, the controller should continue with the
     instruction after the label.

   * An unconditional branch (`goto' instruction) naming a controller
     label at which to continue execution.


   The machine starts at the beginning of the controller instruction
sequence and stops when execution reaches the end of the sequence.
Except when a branch changes the flow of control, instructions are
executed in the order in which they are listed.

     *Figure 5.3:* A specification of the GCD machine.

          (data-paths
           (registers
            ((name a)
             (buttons ((name a<-b) (source (register b)))))
            ((name b)
             (buttons ((name b<-t) (source (register t)))))
            ((name t)
             (buttons ((name t<-r) (source (operation rem))))))

           (operations
            ((name rem)
             (inputs (register a) (register b)))
            ((name =)
             (inputs (register b) (constant 0)))))

          (controller
           test-b                           ; label
             (test =)                       ; test
             (branch (label gcd-done))      ; conditional branch
             (t<-r)                         ; button push
             (a<-b)                         ; button push
             (b<-t)                         ; button push
             (goto (label test-b))          ; unconditional branch
           gcd-done)                        ; label


   *Note Figure 5-3:: shows the GCD machine described in this way.  This
example only hints at the generality of these descriptions, since the
GCD machine is a very simple case: Each register has only one button,
and each button and test is used only once in the controller.

   Unfortunately, it is difficult to read such a description.  In order
to understand the controller instructions we must constantly refer back
to the definitions of the button names and the operation names, and to
understand what the buttons do we may have to refer to the definitions
of the operation names.  We will thus transform our notation to combine
the information from the data-path and controller descriptions so that
we see it all together.

   To obtain this form of description, we will replace the arbitrary
button and operation names by the definitions of their behavior.  That
is, instead of saying (in the controller) "Push button `t<-r'" and
separately saying (in the data paths) "Button `t<-r' assigns the value
of the `rem' operation to register `t'" and "The `rem' operation's
inputs are the contents of registers `a' and `b'," we will say (in the
controller) "Push the button that assigns to register `t' the value of
the `rem' operation on the contents of registers `a' and `b'."
Similarly, instead of saying (in the controller) "Perform the `=' test"
and separately saying (in the data paths) "The `=' test operates on the
contents of register `b' and the constant 0," we will say "Perform the
`=' test on the contents of register `b' and the constant 0."  We will
omit the data-path description, leaving only the controller sequence.
Thus, the GCD machine is described as follows:

     (controller
      test-b
        (test (op =) (reg b) (const 0))
        (branch (label gcd-done))
        (assign t (op rem) (reg a) (reg b))
        (assign a (reg b))
        (assign b (reg t))
        (goto (label test-b))
      gcd-done)

   This form of description is easier to read than the kind illustrated
in *Note Figure 5-3::, but it also has disadvantages:

   * It is more verbose for large machines, because complete
     descriptions of the data-path elements are repeated whenever the
     elements are mentioned in the controller instruction sequence.
     (This is not a problem in the GCD example, because each operation
     and button is used only once.)  Moreover, repeating the data-path
     descriptions obscures the actual data-path structure of the
     machine; it is not obvious for a large machine how many registers,
     operations, and buttons there are and how they are interconnected.

   * Because the controller instructions in a machine definition look
     like Lisp expressions, it is easy to forget that they are not
     arbitrary Lisp expressions.  They can notate only legal machine
     operations.  For example, operations can operate directly only on
     constants and the contents of registers, not on the results of
     other operations.


   In spite of these disadvantages, we will use this register-machine
language throughout this chapter, because we will be more concerned
with understanding controllers than with understanding the elements and
connections in data paths.  We should keep in mind, however, that
data-path design is crucial in designing real machines.

     *Exercise 5.2:* Use the register-machine language to describe the
     iterative factorial machine of *Note Exercise 5-1::.

Actions
.......

Let us modify the GCD machine so that we can type in the numbers whose
GCD we want and get the answer printed at our terminal.  We will not
discuss how to make a machine that can read and print, but will assume
(as we do when we use `read' and `display' in Scheme) that they are
available as primitive operations.(1)

   `Read' is like the operations we have been using in that it produces
a value that can be stored in a register.  But `read' does not take
inputs from any registers; its value depends on something that happens
outside the parts of the machine we are designing.  We will allow our
machine's operations to have such behavior, and thus will draw and
notate the use of `read' just as we do any other operation that
computes a value.

   `Print', on the other hand, differs from the operations we have been
using in a fundamental way: It does not produce an output value to be
stored in a register.  Though it has an effect, this effect is not on a
part of the machine we are designing.  We will refer to this kind of
operation as an "action".  We will represent an action in a data-path
diagram just as we represent an operation that computes a value--as a
trapezoid that contains the name of the action.  Arrows point to the
action box from any inputs (registers or constants).  We also associate
a button with the action.  Pushing the button makes the action happen.
To make a controller push an action button we use a new kind of
instruction called `perform'.  Thus, the action of printing the
contents of register `a' is represented in a controller sequence by the
instruction

     (perform (op print) (reg a))

   *Note Figure 5-4:: shows the data paths and controller for the new
GCD machine.  Instead of having the machine stop after printing the
answer, we have made it start over, so that it repeatedly reads a pair
of numbers, computes their GCD, and prints the result.  This structure
is like the driver loops we used in the interpreters of *Note Chapter
4::.

     *Figure 5.4:* A GCD machine that reads inputs and prints results.

                             .--------.
                              \ read /
                               \____/
                                 |
                         +-------*------+
                         |              |
                  a<-rd (X)            (X) b<-rd
                         |              |
                         V              V           ___
                      +-----+        +-----+       /   \
                      |  a  |<--(X)--+  b  +----->|  =  |
                      +-+-+-+  a<-b  +-+---+       \___/
                        | |            |  ^          ^
                     +--+ +----+    +--+  |          |
                     |         |    |    (X) b<-t   / \
                     V         V    V     |        / O \
                .---------.  .---------.  |       /_____\
          --(X)->\ print /    \  rem  /   |
             P    \_____/      \_____/    |
                                  |       |
                                 (X) t<-r |
                                  |       |
                                  V       |
                               +-----+    |
                               |  t  +----+
                               +-----+

           (controller
            gcd-loop
              (assign a (op read))
              (assign b (op read))
            test-b
              (test (op =) (reg b) (const 0))
              (branch (label gcd-done))
              (assign t (op rem) (reg a) (reg b))
              (assign a (reg b))
              (assign b (reg t))
              (goto (label test-b))
            gcd-done
              (perform (op print) (reg a))
              (goto (label gcd-loop)))


   ---------- Footnotes ----------

   (1) This assumption glosses over a great deal of complexity.
Usually a large portion of the implementation of a Lisp system is
dedicated to making reading and printing work.


File: sicp.info,  Node: 5-1-2,  Next: 5-1-3,  Prev: 5-1-1,  Up: 5-1

5.1.2 Abstraction in Machine Design
-----------------------------------

We will often define a machine to include "primitive" operations that
are actually very complex.  For example, in sections *Note 5-4:: and
*Note 5-5:: we will treat Scheme's environment manipulations as
primitive.  Such abstraction is valuable because it allows us to ignore
the details of parts of a machine so that we can concentrate on other
aspects of the design.  The fact that we have swept a lot of complexity
under the rug, however, does not mean that a machine design is
unrealistic.  We can always replace the complex "primitives" by simpler
primitive operations.

   Consider the GCD machine. The machine has an instruction that
computes the remainder of the contents of registers `a' and `b' and
assigns the result to register `t'.  If we want to construct the GCD
machine without using a primitive remainder operation, we must specify
how to compute remainders in terms of simpler operations, such as
subtraction.  Indeed, we can write a Scheme procedure that finds
remainders in this way:

     (define (remainder n d)
       (if (< n d)
           n
           (remainder (- n d) d)))

   We can thus replace the remainder operation in the GCD machine's data
paths with a subtraction operation and a comparison test.  *Note Figure
5-5:: shows the data paths and controller for the elaborated machine.
The instruction

     *Figure 5.5:* Data paths and controller for the elaborated GCD
     machine.

                                              ___
          +-----+         +-----+            /   \
          |  a  |<--(X)---+  b  +-------*-->|  =  |
          +--+--+   a<-b  +-+---+       |    \___/
             |              |  ^        |
            (X) t<-a        |  |        |
             |              | (X) b<-t  |
             V              |  |       _V_
          +-----+           |  |      /   \
          |  t  +-------*---|--*-----|  <  |
          +-----+       |   |         \___/
             ^          V   V
             |        ---------
            (X) t<-d   \  -  /
             |          --+--
             |            |
             +------------+


             start
               |
               V
              / \ yes            +-------+
          +->< = >----> done     | t<-d  |<--+
          |   \ /                +---+---+   |
          |    | no                  |       |
          |    |                     V       |
          |    |   +------+         / \ no   |
          |    +-->| t<-a +------->< < >-----+
          |        +------+         \ /
          |                          | yes
          |      +-------------------+
          |      V
          |  +-------+
          |  | a<-b  |
          |  +---+---+
          |      |
          |      V
          |  +-------+
          +--+ b<-t  |
             +-------+

     (assign t (op rem) (reg a) (reg b))

in the GCD controller definition is replaced by a sequence of
instructions that contains a loop, as shown in *Note Figure 5-6::.

     *Figure 5.6:* Controller instruction sequence for the GCD machine
     in *Note Figure 5-5::.

          (controller
           test-b
             (test (op =) (reg b) (const 0))
             (branch (label gcd-done))
             (assign t (reg a))
           rem-loop
             (test (op <) (reg t) (reg b))
             (branch (label rem-done))
             (assign t (op -) (reg t) (reg b))
             (goto (label rem-loop))
           rem-done
             (assign a (reg b))
             (assign b (reg t))
             (goto (label test-b))
           gcd-done)


     *Exercise 5.3:* Design a machine to compute square roots using
     Newton's method, as described in section *Note 1-1-7:::

          (define (sqrt x)
            (define (good-enough? guess)
              (< (abs (- (square guess) x)) 0.001))
            (define (improve guess)
              (average guess (/ x guess)))
            (define (sqrt-iter guess)
              (if (good-enough? guess)
                  guess
                  (sqrt-iter (improve guess))))
            (sqrt-iter 1.0))

     Begin by assuming that `good-enough?' and `improve' operations are
     available as primitives.  Then show how to expand these in terms
     of arithmetic operations.  Describe each version of the `sqrt'
     machine design by drawing a data-path diagram and writing a
     controller definition in the register-machine language.


File: sicp.info,  Node: 5-1-3,  Next: 5-1-4,  Prev: 5-1-2,  Up: 5-1

5.1.3 Subroutines
-----------------

When designing a machine to perform a computation, we would often
prefer to arrange for components to be shared by different parts of the
computation rather than duplicate the components.  Consider a machine
that includes two GCD computations--one that finds the GCD of the
contents of registers `a' and `b' and one that finds the GCD of the
contents of registers `c' and `d'.  We might start by assuming we have
a primitive `gcd' operation, then expand the two instances of `gcd' in
terms of more primitive operations.  *Note Figure 5-7:: shows just the
GCD portions of the resulting machine's data paths, without showing how
they connect to the rest of the machine.  The figure also shows the
corresponding portions of the machine's controller sequence.

     *Figure 5.7:* Portions of the data paths and controller sequence
     for a machine with two GCD computations.

                                      ___                                 ___
          +-----+        +-----+     /   \    +-----+        +-----+     /   \
          |  a  |<-(X)---+  b  |--->|  =  |   |  c  |<-(X)---+  d  |--->|  =  |
          +--+--+  a<-b  ++----+     \___/    +--+--+  c<-d  ++----+     \___/
             |            |  ^         ^         |            |  ^         ^
             `----.   .---'  |         |         `----.   .---'  |         |
                  V   V     (X) b<-t   |              V   V     (X) d<-t   |
                 -------     |        / \            -------     |        / \
                 \ rem /     |       /_0_\           \ rem /     |       /_0_\
                  --+--      |                        --+--      |
                    |        |                          |        |
                   (X) t<-r  |                         (X) s<-r  |
                    |        |                          |        |
                    V        |                          V        |
                 +-----+     |                       +-----+     |
                 |  t  +-----'                       |  s  +-----'
                 +-----+                             +-----+

          gcd-1
           (test (op =) (reg b) (const 0))
           (branch (label after-gcd-1))
           (assign t (op rem) (reg a) (reg b))
           (assign a (reg b))
           (assign b (reg t))
           (goto (label gcd-1))
          after-gcd-1
             ...
          gcd-2
           (test (op =) (reg d) (const 0))
           (branch (label after-gcd-2))
           (assign s (op rem) (reg c) (reg d))
           (assign c (reg d))
           (assign d (reg s))
           (goto (label gcd-2))
          after-gcd-2


   This machine has two remainder operation boxes and two boxes for
testing equality.  If the duplicated components are complicated, as is
the remainder box, this will not be an economical way to build the
machine.  We can avoid duplicating the data-path components by using
the same components for both GCD computations, provided that doing so
will not affect the rest of the larger machine's computation.  If the
values in registers `a' and `b' are not needed by the time the
controller gets to `gcd-2' (or if these values can be moved to other
registers for safekeeping), we can change the machine so that it uses
registers `a' and `b', rather than registers `c' and `d', in computing
the second GCD as well as the first.  If we do this, we obtain the
controller sequence shown in *Note Figure 5-8::.

   We have removed the duplicate data-path components (so that the data
paths are again as in *Note Figure 5-1::), but the controller now has
two GCD sequences that differ only in their entry-point labels.  It
would be better to replace these two sequences by branches to a single
sequence--a `gcd' "subroutine"--at the end of which we branch back to
the correct place in the main instruction sequence.  We can accomplish
this as follows: Before branching to `gcd', we place a distinguishing
value (such as 0 or 1) into a special register, `continue'.  At the end
of the `gcd' subroutine we return either to `after-gcd-1' or to
`after-gcd-2', depending on the value of the `continue' register.
*Note Figure 5-9:: shows the relevant portion of the resulting
controller sequence, which includes only a single copy of the `gcd'
instructions.

     *Figure 5.8:* Portions of the controller sequence for a machine
     that uses the same data-path components for two different GCD
     computations.

          gcd-1
           (test (op =) (reg b) (const 0))
           (branch (label after-gcd-1))
           (assign t (op rem) (reg a) (reg b))
           (assign a (reg b))
           (assign b (reg t))
           (goto (label gcd-1))
          after-gcd-1
            ...
          gcd-2
           (test (op =) (reg b) (const 0))
           (branch (label after-gcd-2))
           (assign t (op rem) (reg a) (reg b))
           (assign a (reg b))
           (assign b (reg t))
           (goto (label gcd-2))
          after-gcd-2


     *Figure 5.9:* Using a `continue' register to avoid the duplicate
     controller sequence in *Note Figure 5-8::.

          gcd
           (test (op =) (reg b) (const 0))
           (branch (label gcd-done))
           (assign t (op rem) (reg a) (reg b))
           (assign a (reg b))
           (assign b (reg t))
           (goto (label gcd))
          gcd-done
           (test (op =) (reg continue) (const 0))
           (branch (label after-gcd-1))
           (goto (label after-gcd-2))
            ...
          ;; Before branching to `gcd' from the first place where
          ;; it is needed, we place 0 in the `continue' register
           (assign continue (const 0))
           (goto (label gcd))
          after-gcd-1
            ...
          ;; Before the second use of `gcd', we place 1 in the `continue' register
           (assign continue (const 1))
           (goto (label gcd))
          after-gcd-2


     *Figure 5.10:* Assigning labels to the `continue' register
     simplifies and generalizes the strategy shown in *Note Figure
     5-9::.

          gcd
           (test (op =) (reg b) (const 0))
           (branch (label gcd-done))
           (assign t (op rem) (reg a) (reg b))
           (assign a (reg b))
           (assign b (reg t))
           (goto (label gcd))
          gcd-done
           (goto (reg continue))
             ...
          ;; Before calling `gcd', we assign to `continue'
          ;; the label to which `gcd' should return.
           (assign continue (label after-gcd-1))
           (goto (label gcd))
          after-gcd-1
             ...
          ;; Here is the second call to `gcd', with a different continuation.
           (assign continue (label after-gcd-2))
           (goto (label gcd))
          after-gcd-2


   This is a reasonable approach for handling small problems, but it
would be awkward if there were many instances of GCD computations in the
controller sequence.  To decide where to continue executing after the
`gcd' subroutine, we would need tests in the data paths and branch
instructions in the controller for all the places that use `gcd'.  A
more powerful method for implementing subroutines is to have the
`continue' register hold the label of the entry point in the controller
sequence at which execution should continue when the subroutine is
finished.  Implementing this strategy requires a new kind of connection
between the data paths and the controller of a register machine: There
must be a way to assign to a register a label in the controller
sequence in such a way that this value can be fetched from the register
and used to continue execution at the designated entry point.

   To reflect this ability, we will extend the `assign' instruction of
the register-machine language to allow a register to be assigned as
value a label from the controller sequence (as a special kind of
constant).  We will also extend the `goto' instruction to allow
execution to continue at the entry point described by the contents of a
register rather than only at an entry point described by a constant
label.  Using these new constructs we can terminate the `gcd'
subroutine with a branch to the location stored in the `continue'
register.  This leads to the controller sequence shown in *Note Figure
5-10::.

   A machine with more than one subroutine could use multiple
continuation registers (e.g., `gcd-continue', `factorial-continue') or
we could have all subroutines share a single `continue' register.
Sharing is more economical, but we must be careful if we have a
subroutine (`sub1') that calls another subroutine (`sub2').  Unless
`sub1' saves the contents of `continue' in some other register before
setting up `continue' for the call to `sub2', `sub1' will not know
where to go when it is finished.  The mechanism developed in the next
section to handle recursion also provides a better solution to this
problem of nested subroutine calls.


File: sicp.info,  Node: 5-1-4,  Next: 5-1-5,  Prev: 5-1-3,  Up: 5-1

5.1.4 Using a Stack to Implement Recursion
------------------------------------------

With the ideas illustrated so far, we can implement any iterative
process by specifying a register machine that has a register
corresponding to each state variable of the process.  The machine
repeatedly executes a controller loop, changing the contents of the
registers, until some termination condition is satisfied.  At each
point in the controller sequence, the state of the machine
(representing the state of the iterative process) is completely
determined by the contents of the registers (the values of the state
variables).

   Implementing recursive processes, however, requires an additional
mechanism.  Consider the following recursive method for computing
factorials, which we first examined in section *Note 1-2-1:::

     (define (factorial n)
       (if (= n 1)
           1
           (* (factorial (- n 1)) n)))

   As we see from the procedure, computing n! requires computing (n -
1)!.  Our GCD machine, modeled on the procedure

     (define (gcd a b)
       (if (= b 0)
           a
           (gcd b (remainder a b))))

similarly had to compute another GCD.  But there is an important
difference between the `gcd' procedure, which reduces the original
computation to a new GCD computation, and `factorial', which requires
computing another factorial as a subproblem.  In GCD, the answer to the
new GCD computation is the answer to the original problem.  To compute
the next GCD, we simply place the new arguments in the input registers
of the GCD machine and reuse the machine's data paths by executing the
same controller sequence.  When the machine is finished solving the
final GCD problem, it has completed the entire computation.

   In the case of factorial (or any recursive process) the answer to
the new factorial subproblem is not the answer to the original problem.
The value obtained for (n - 1)! must be multiplied by n to get the
final answer.  If we try to imitate the GCD design, and solve the
factorial subproblem by decrementing the `n' register and rerunning the
factorial machine, we will no longer have available the old value of
`n' by which to multiply the result.  We thus need a second factorial
machine to work on the subproblem.  This second factorial computation
itself has a factorial subproblem, which requires a third factorial
machine, and so on.  Since each factorial machine contains another
factorial machine within it, the total machine contains an infinite
nest of similar machines and hence cannot be constructed from a fixed,
finite number of parts.

   Nevertheless, we can implement the factorial process as a register
machine if we can arrange to use the same components for each nested
instance of the machine.  Specifically, the machine that computes n!
should use the same components to work on the subproblem of computing
(n - 1)!, on the subproblem for (n - 2)!, and so on.  This is plausible
because, although the factorial process dictates that an unbounded
number of copies of the same machine are needed to perform a
computation, only one of these copies needs to be active at any given
time.  When the machine encounters a recursive subproblem, it can
suspend work on the main problem, reuse the same physical parts to work
on the subproblem, then continue the suspended computation.

   In the subproblem, the contents of the registers will be different
than they were in the main problem. (In this case the `n' register is
decremented.)  In order to be able to continue the suspended
computation, the machine must save the contents of any registers that
will be needed after the subproblem is solved so that these can be
restored to continue the suspended computation.  In the case of
factorial, we will save the old value of `n', to be restored when we
are finished computing the factorial of the decremented `n' register.(1)

   Since there is no _a priori_ limit on the depth of nested recursive
calls, we may need to save an arbitrary number of register values.
These values must be restored in the reverse of the order in which they
were saved, since in a nest of recursions the last subproblem to be
entered is the first to be finished.  This dictates the use of a "stack",
or "last in, first out" data structure, to save register values.  We
can extend the register-machine language to include a stack by adding
two kinds of instructions: Values are placed on the stack using a
`save' instruction and restored from the stack using a `restore'
instruction.  After a sequence of values has been `save'd on the stack,
a sequence of `restore's will retrieve these values in reverse order.(2)

   With the aid of the stack, we can reuse a single copy of the
factorial machine's data paths for each factorial subproblem.  There is
a similar design issue in reusing the controller sequence that operates
the data paths.  To reexecute the factorial computation, the controller
cannot simply loop back to the beginning, as with an iterative process,
because after solving the (n - 1)! subproblem the machine must still
multiply the result by n.  The controller must suspend its computation
of n!, solve the (n - 1)!  subproblem, then continue its computation of
n!.  This view of the factorial computation suggests the use of the
subroutine mechanism described in section *Note 5-1-3::, which has the
controller use a `continue' register to transfer to the part of the
sequence that solves a subproblem and then continue where it left off
on the main problem.  We can thus make a factorial subroutine that
returns to the entry point stored in the `continue' register.  Around
each subroutine call, we save and restore `continue' just as we do the
`n' register, since each "level" of the factorial computation will use
the same `continue' register.  That is, the factorial subroutine must
put a new value in `continue' when it calls itself for a subproblem,
but it will need the old value in order to return to the place that
called it to solve a subproblem.

   *Note Figure 5-11:: shows the data paths and controller for a
machine that implements the recursive `factorial' procedure.  The
machine has a stack and three registers, called `n', `val', and
`continue'.  To simplify the data-path diagram, we have not named the
register-assignment buttons, only the stack-operation buttons (`sc' and
`sn' to save registers, `rc' and `rn' to restore registers).  To
operate the machine, we put in register `n' the number whose factorial
we wish to compute and start the machine.  When the machine reaches
`fact-done', the computation is finished and the answer will be found
in the `val' register.  In the controller sequence, `n' and `continue'
are saved before each recursive call and restored upon return from the
call.  Returning from a call is accomplished by branching to the
location stored in `continue'.  `Continue' is initialized when the
machine starts so that the last return will go to `fact-done'.  The
`val' register, which holds the result of the factorial computation, is
not saved before the recursive call, because the old contents of `val'
is not useful after the subroutine returns.  Only the new value, which
is the value produced by the subcomputation, is needed.

   Although in principle the factorial computation requires an infinite
machine, the machine in *Note Figure 5-11:: is actually finite except
for the stack, which is potentially unbounded.  Any particular physical
implementation of a stack, however, will be of finite size, and this
will limit the depth of recursive calls that can be handled by the
machine.  This implementation of factorial illustrates the general
strategy for realizing recursive algorithms as ordinary register
machines augmented by stacks.  When a recursive subproblem is
encountered, we save on the stack the registers whose current values
will be required after the subproblem is solved, solve the recursive
subproblem, then restore the saved registers and continue execution on
the main problem.  The `continue' register must always be saved.
Whether there are other registers that need to be saved depends on the
particular machine, since not all recursive computations need the
original values of registers that are modified during solution of the
subproblem (see *Note Exercise 5-4::).

A double recursion
..................

Let us examine a more complex recursive process, the tree-recursive
computation of the Fibonacci numbers, which we introduced in section
*Note 1-2-2:::

     (define (fib n)
       (if (< n 2)
           n
           (+ (fib (- n 1)) (fib (- n 2)))))

   Just as with factorial, we can implement the recursive Fibonacci
computation as a register machine with registers `n', `val', and
`continue'.  The machine is more complex than the one for factorial,
because there are two places in the controller sequence where we need
to perform recursive calls--once to compute Fib(n - 1) and once to
compute Fib(n - 2).  To set up for each of these calls, we save the
registers whose values will be needed later, set the `n' register to
the number whose Fib we need to compute recursively (n - 1 or n - 2),
and assign to `continue' the entry point in the main sequence to which
to return (`afterfib-n-1' or `afterfib-n-2', respectively).  We then go
to `fib-loop'.  When we return from the recursive call, the answer is
in `val'.  *Note Figure 5-12:: shows the controller sequence for this
machine.

     *Figure 5.11:* A recursive factorial machine.

                                       ___
                                      /   \
              +----------*-----------|  =  |
              |          |            \___/
             (X)         |              ^
              |          |              |
              V          |          +---+---+   sn    +-------+
          +-------+      |          |       +---(X)-->|       |
          |  val  |<-(X)-|----------+   n   |         | stack |
          +-----+-+      |          |       |<--(X)---+       |
            ^   |        |          +-------+   rn    +-+-----+
            |   |        |            ^                 |   ^
           (X)  |        |            |                 |   |
            |   |   +----|--------*  (X)                |  (X) sc
            |   |   |    |        |   |             rc (X)  |
            |   |   |    *----.   |   |                 |   |
            |   V   V    |    V   V   |                 V   |
            |  -------   |   -------  |              +------+-+
            |  \  *  /   |   \  -  /  |              |continue+--> controller
            |   --+--    |    --+--   |              +--------+
            |     |      |      |     |               ^      ^
            +-----+      |      +-----+               |      |
                         |                           (X)    (X)
                         |                            |      |
                        / \                   after- / \    / \  fact-
                       /_1_\                  fact  /___\  /___\ done

          (controller
             (assign continue (label fact-done))     ; set up final return address
           fact-loop
             (test (op =) (reg n) (const 1))
             (branch (label base-case))
             ;; Set up for the recursive call by saving `n' and `continue'.
             ;; Set up `continue' so that the computation will continue
             ;; at `after-fact' when the subroutine returns.
             (save continue)
             (save n)
             (assign n (op -) (reg n) (const 1))
             (assign continue (label after-fact))
             (goto (label fact-loop))
           after-fact
             (restore n)
             (restore continue)
             (assign val (op *) (reg n) (reg val))   ; `val' now contains n(n - 1)!
             (goto (reg continue))                   ; return to caller
           base-case
             (assign val (const 1))                  ; base case: 1! = 1
             (goto (reg continue))                   ; return to caller
           fact-done)


     *Figure 5.12:* Controller for a machine to compute Fibonacci
     numbers.

          (controller
             (assign continue (label fib-done))
           fib-loop
             (test (op <) (reg n) (const 2))
             (branch (label immediate-answer))
             ;; set up to compute _Fib_(n - 1)
             (save continue)
             (assign continue (label afterfib-n-1))
             (save n)                           ; save old value of `n'
             (assign n (op -) (reg n) (const 1)); clobber `n' to n - 1
             (goto (label fib-loop))            ; perform recursive call
           afterfib-n-1                         ; upon return, `val' contains _Fib_(n - 1)
             (restore n)
             (restore continue)
             ;; set up to compute _Fib_(n - 2)
             (assign n (op -) (reg n) (const 2))
             (save continue)
             (assign continue (label afterfib-n-2))
             (save val)                         ; save _Fib_(n - 1)
             (goto (label fib-loop))
           afterfib-n-2                         ; upon return, `val' contains _Fib_(n - 2)
             (assign n (reg val))               ; `n' now contains _Fib_(n - 2)
             (restore val)                      ; `val' now contains _Fib_(n - 1)
             (restore continue)
             (assign val                        ;  _Fib_(n - 1) +  _Fib_(n - 2)
                     (op +) (reg val) (reg n))
             (goto (reg continue))              ; return to caller, answer is in `val'
           immediate-answer
             (assign val (reg n))               ; base case:  _Fib_(n) = n
             (goto (reg continue))
           fib-done)


     *Exercise 5.4:* Specify register machines that implement each of
     the following procedures.  For each machine, write a controller
     instruction sequence and draw a diagram showing the data paths.

       a. Recursive exponentiation:

               (define (expt b n)
                 (if (= n 0)
                     1
                     (* b (expt b (- n 1)))))

       b. Iterative exponentiation:

               (define (expt b n)
                 (define (expt-iter counter product)
                   (if (= counter 0)
                       product
                       (expt-iter (- counter 1) (* b product))))
                 (expt-iter n 1))


     *Exercise 5.5:* Hand-simulate the factorial and Fibonacci
     machines, using some nontrivial input (requiring execution of at
     least one recursive call).  Show the contents of the stack at each
     significant point in the execution.

     *Exercise 5.6:* Ben Bitdiddle observes that the Fibonacci
     machine's controller sequence has an extra `save' and an extra
     `restore', which can be removed to make a faster machine.  Where
     are these instructions?

   ---------- Footnotes ----------

   (1) One might argue that we don't need to save the old `n'; after we
decrement it and solve the subproblem, we could simply increment it to
recover the old value.  Although this strategy works for factorial, it
cannot work in general, since the old value of a register cannot always
be computed from the new one.

   (2) In section *Note 5-3:: we will see how to implement a stack in
terms of more primitive operations.


File: sicp.info,  Node: 5-1-5,  Prev: 5-1-4,  Up: 5-1

5.1.5 Instruction Summary
-------------------------

A controller instruction in our register-machine language has one of the
following forms, where each <INPUT_I> is either `(reg <REGISTER-NAME>)'
or `(const <CONSTANT-VALUE>)'.  These instructions were introduced in
section *Note 5-1-1:::

     (assign <REGISTER-NAME> (reg <REGISTER-NAME>))

     (assign <REGISTER-NAME> (const <CONSTANT-VALUE>))

     (assign <REGISTER-NAME> (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)

     (perform (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)

     (test (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)

     (branch (label <LABEL-NAME>))

     (goto (label <LABEL-NAME>))

   The use of registers to hold labels was introduced in section *Note
5-1-3:::

     (assign <REGISTER-NAME> (label <LABEL-NAME>))

     (goto (reg <REGISTER-NAME>))

   Instructions to use the stack were introduced in section *Note
5-1-4:::

     (save <REGISTER-NAME>)

     (restore <REGISTER-NAME>)

   The only kind of <CONSTANT-VALUE> we have seen so far is a number,
but later we will use strings, symbols, and lists.  For example,
`(const "abc")' is the string `"abc"',
`(const abc)' is the symbol `abc',
`(const (a b c))' is the list `(a b c)',
and `(const ())' is the empty list.


File: sicp.info,  Node: 5-2,  Next: 5-3,  Prev: 5-1,  Up: Chapter 5

5.2 A Register-Machine Simulator
================================

In order to gain a good understanding of the design of register
machines, we must test the machines we design to see if they perform as
expected.  One way to test a design is to hand-simulate the operation
of the controller, as in *Note Exercise 5-5::.  But this is extremely
tedious for all but the simplest machines.  In this section we
construct a simulator for machines described in the register-machine
language.  The simulator is a Scheme program with four interface
procedures.  The first uses a description of a register machine to
construct a model of the machine (a data structure whose parts
correspond to the parts of the machine to be simulated), and the other
three allow us to simulate the machine by manipulating the model:

          (make-machine <REGISTER-NAMES> <OPERATIONS> <CONTROLLER>)

     constructs and returns a model of the machine with the given
     registers, operations, and controller.

          (set-register-contents! <MACHINE-MODEL> <REGISTER-NAME> <VALUE>)

     stores a value in a simulated register in the given machine.

          (get-register-contents <MACHINE-MODEL> <REGISTER-NAME>)

     returns the contents of a simulated register in the given machine.

          (start <MACHINE-MODEL>)

     simulates the execution of the given machine, starting from the
     beginning of the controller sequence and stopping when it reaches
     the end of the sequence.

   As an example of how these procedures are used, we can define
`gcd-machine' to be a model of the GCD machine of section *Note 5-1-1::
as follows:

     (define gcd-machine
       (make-machine
        '(a b t)
        (list (list 'rem remainder) (list '= =))
        '(test-b
            (test (op =) (reg b) (const 0))
            (branch (label gcd-done))
            (assign t (op rem) (reg a) (reg b))
            (assign a (reg b))
            (assign b (reg t))
            (goto (label test-b))
          gcd-done)))

   The first argument to `make-machine' is a list of register names.
The next argument is a table (a list of two-element lists) that pairs
each operation name with a Scheme procedure that implements the
operation (that is, produces the same output value given the same input
values).  The last argument specifies the controller as a list of
labels and machine instructions, as in section *Note 5-1::.

   To compute GCDs with this machine, we set the input registers, start
the machine, and examine the result when the simulation terminates:

     (set-register-contents! gcd-machine 'a 206)
     done

     (set-register-contents! gcd-machine 'b 40)
     done

     (start gcd-machine)
     done

     (get-register-contents gcd-machine 'a)
     2

   This computation will run much more slowly than a `gcd' procedure
written in Scheme, because we will simulate low-level machine
instructions, such as `assign', by much more complex operations.

     *Exercise 5.7:* Use the simulator to test the machines you
     designed in *Note Exercise 5-4::.

* Menu:

* 5-2-1::            The Machine Model
* 5-2-2::            The Assembler
* 5-2-3::            Generating Execution Procedures for Instructions
* 5-2-4::            Monitoring Machine Performance


File: sicp.info,  Node: 5-2-1,  Next: 5-2-2,  Prev: 5-2,  Up: 5-2

5.2.1 The Machine Model
-----------------------

The machine model generated by `make-machine' is represented as a
procedure with local state using the message-passing techniques
developed in *Note Chapter 3::.  To build this model, `make-machine'
begins by calling the procedure `make-new-machine' to construct the
parts of the machine model that are common to all register machines.
This basic machine model constructed by `make-new-machine' is
essentially a container for some registers and a stack, together with
an execution mechanism that processes the controller instructions one
by one.

   `Make-machine' then extends this basic model (by sending it
messages) to include the registers, operations, and controller of the
particular machine being defined.  First it allocates a register in the
new machine for each of the supplied register names and installs the
designated operations in the machine.  Then it uses an "assembler"
(described below in section *Note 5-2-2::) to transform the controller
list into instructions for the new machine and installs these as the
machine's instruction sequence.  `Make-machine' returns as its value
the modified machine model.

     (define (make-machine register-names ops controller-text)
       (let ((machine (make-new-machine)))
         (for-each (lambda (register-name)
                     ((machine 'allocate-register) register-name))
                   register-names)
         ((machine 'install-operations) ops)
         ((machine 'install-instruction-sequence)
          (assemble controller-text machine))
         machine))

Registers
.........

We will represent a register as a procedure with local state, as in
*Note Chapter 3::.  The procedure `make-register' creates a register
that holds a value that can be accessed or changed:

     (define (make-register name)
       (let ((contents '*unassigned*))
         (define (dispatch message)
           (cond ((eq? message 'get) contents)
                 ((eq? message 'set)
                  (lambda (value) (set! contents value)))
                 (else
                  (error "Unknown request -- REGISTER" message))))
         dispatch))

   The following procedures are used to access registers:

     (define (get-contents register)
       (register 'get))

     (define (set-contents! register value)
       ((register 'set) value))

The stack
.........

We can also represent a stack as a procedure with local state.  The
procedure `make-stack' creates a stack whose local state consists of a
list of the items on the stack.  A stack accepts requests to `push' an
item onto the stack, to `pop' the top item off the stack and return it,
and to `initialize' the stack to empty.

     (define (make-stack)
       (let ((s '()))
         (define (push x)
           (set! s (cons x s)))
         (define (pop)
           (if (null? s)
               (error "Empty stack -- POP")
               (let ((top (car s)))
                 (set! s (cdr s))
                 top)))
         (define (initialize)
           (set! s '())
           'done)
         (define (dispatch message)
           (cond ((eq? message 'push) push)
                 ((eq? message 'pop) (pop))
                 ((eq? message 'initialize) (initialize))
                 (else (error "Unknown request -- STACK"
                              message))))
         dispatch))

   The following procedures are used to access stacks:

     (define (pop stack)
       (stack 'pop))

     (define (push stack value)
       ((stack 'push) value))

The basic machine
.................

The `make-new-machine' procedure, shown in *Note Figure 5-13::,
constructs an object whose local state consists of a stack, an
initially empty instruction sequence, a list of operations that
initially contains an operation to initialize the stack, and a "register
table" that initially contains two registers, named `flag' and `pc'
(for "program counter").  The internal procedure `allocate-register'
adds new entries to the register table, and the internal procedure
`lookup-register' looks up registers in the table.

   The `flag' register is used to control branching in the simulated
machine.  `Test' instructions set the contents of `flag' to the result
of the test (true or false).  `Branch' instructions decide whether or
not to branch by examining the contents of `flag'.

   The `pc' register determines the sequencing of instructions as the
machine runs.  This sequencing is implemented by the internal procedure
`execute'.  In the simulation model, each machine instruction is a data
structure that includes a procedure of no arguments, called the procedure
"instruction execution procedure", such that calling this procedure
simulates executing the instruction.  As the simulation runs, `pc'
points to the place in the instruction sequence beginning with the next
instruction to be executed.  `Execute' gets that instruction, executes
it by calling the instruction execution procedure, and repeats this
cycle until there are no more instructions to execute (i.e., until `pc'
points to the end of the instruction sequence).

     *Figure 5.13:* The `make-new-machine' procedure, which implements
     the basic machine model.

          (define (make-new-machine)
            (let ((pc (make-register 'pc))
                  (flag (make-register 'flag))
                  (stack (make-stack))
                  (the-instruction-sequence '()))
              (let ((the-ops
                     (list (list 'initialize-stack
                                 (lambda () (stack 'initialize)))))
                    (register-table
                     (list (list 'pc pc) (list 'flag flag))))
                (define (allocate-register name)
                  (if (assoc name register-table)
                      (error "Multiply defined register: " name)
                      (set! register-table
                            (cons (list name (make-register name))
                                  register-table)))
                  'register-allocated)
                (define (lookup-register name)
                  (let ((val (assoc name register-table)))
                    (if val
                        (cadr val)
                        (error "Unknown register:" name))))
                (define (execute)
                  (let ((insts (get-contents pc)))
                    (if (null? insts)
                        'done
                        (begin
                          ((instruction-execution-proc (car insts)))
                          (execute)))))
                (define (dispatch message)
                  (cond ((eq? message 'start)
                         (set-contents! pc the-instruction-sequence)
                         (execute))
                        ((eq? message 'install-instruction-sequence)
                         (lambda (seq) (set! the-instruction-sequence seq)))
                        ((eq? message 'allocate-register) allocate-register)
                        ((eq? message 'get-register) lookup-register)
                        ((eq? message 'install-operations)
                         (lambda (ops) (set! the-ops (append the-ops ops))))
                        ((eq? message 'stack) stack)
                        ((eq? message 'operations) the-ops)
                        (else (error "Unknown request -- MACHINE" message))))
                dispatch)))

   As part of its operation, each instruction execution procedure
modifies `pc' to indicate the next instruction to be executed.
`Branch' and `goto' instructions change `pc' to point to the new
destination.  All other instructions simply advance `pc', making it
point to the next instruction in the sequence.  Observe that each call
to `execute' calls `execute' again, but this does not produce an
infinite loop because running the instruction execution procedure
changes the contents of `pc'.

   `Make-new-machine' returns a `dispatch' procedure that implements
message-passing access to the internal state.  Notice that starting the
machine is accomplished by setting `pc' to the beginning of the
instruction sequence and calling `execute'.

   For convenience, we provide an alternate procedural interface to a
machine's `start' operation, as well as procedures to set and examine
register contents, as specified at the beginning of section *Note 5-2:::

     (define (start machine)
       (machine 'start))

     (define (get-register-contents machine register-name)
       (get-contents (get-register machine register-name)))

     (define (set-register-contents! machine register-name value)
       (set-contents! (get-register machine register-name) value)
       'done)

   These procedures (and many procedures in sections *Note 5-2-2:: and
*Note 5-2-3::) use the following to look up the register with a given
name in a given machine:

     (define (get-register machine reg-name)
       ((machine 'get-register) reg-name))


File: sicp.info,  Node: 5-2-2,  Next: 5-2-3,  Prev: 5-2-1,  Up: 5-2

5.2.2 The Assembler
-------------------

The assembler transforms the sequence of controller expressions for a
machine into a corresponding list of machine instructions, each with
its execution procedure.  Overall, the assembler is much like the
evaluators we studied in *Note Chapter 4::--there is an input language
(in this case, the register-machine language) and we must perform an
appropriate action for each type of expression in the language.

   The technique of producing an execution procedure for each
instruction is just what we used in section *Note 4-1-7:: to speed up
the evaluator by separating analysis from runtime execution.  As we saw
in *Note Chapter 4::, much useful analysis of Scheme expressions could
be performed without knowing the actual values of variables.  Here,
analogously, much useful analysis of register-machine-language
expressions can be performed without knowing the actual contents of
machine registers.  For example, we can replace references to registers
by pointers to the register objects, and we can replace references to
labels by pointers to the place in the instruction sequence that the
label designates.

   Before it can generate the instruction execution procedures, the
assembler must know what all the labels refer to, so it begins by
scanning the controller text to separate the labels from the
instructions.  As it scans the text, it constructs both a list of
instructions and a table that associates each label with a pointer into
that list.  Then the assembler augments the instruction list by
inserting the execution procedure for each instruction.

   The `assemble' procedure is the main entry to the assembler.  It
takes the controller text and the machine model as arguments and
returns the instruction sequence to be stored in the model.  `Assemble'
calls `extract-labels' to build the initial instruction list and label
table from the supplied controller text.  The second argument to
`extract-labels' is a procedure to be called to process these results:
This procedure uses `update-insts!' to generate the instruction
execution procedures and insert them into the instruction list, and
returns the modified list.

     (define (assemble controller-text machine)
       (extract-labels controller-text
         (lambda (insts labels)
           (update-insts! insts labels machine)
           insts)))

   `Extract-labels' takes as arguments a list `text' (the sequence of
controller instruction expressions) and a `receive' procedure.
`Receive' will be called with two values: (1) a list `insts' of
instruction data structures, each containing an instruction from `text';
and (2) a table called `labels', which associates each label from
`text' with the position in the list `insts' that the label designates.

     (define (extract-labels text receive)
       (if (null? text)
           (receive '() '())
           (extract-labels (cdr text)
            (lambda (insts labels)
              (let ((next-inst (car text)))
                (if (symbol? next-inst)
                    (receive insts
                             (cons (make-label-entry next-inst
                                                     insts)
                                   labels))
                    (receive (cons (make-instruction next-inst)
                                   insts)
                             labels)))))))

   `Extract-labels' works by sequentially scanning the elements of the
`text' and accumulating the `insts' and the `labels'.  If an element is
a symbol (and thus a label) an appropriate entry is added to the
`labels' table.  Otherwise the element is accumulated onto the `insts'
list.(1)

   `Update-insts!' modifies the instruction list, which initially
contains only the text of the instructions, to include the
corresponding execution procedures:

     (define (update-insts! insts labels machine)
       (let ((pc (get-register machine 'pc))
             (flag (get-register machine 'flag))
             (stack (machine 'stack))
             (ops (machine 'operations)))
         (for-each
          (lambda (inst)
            (set-instruction-execution-proc!
             inst
             (make-execution-procedure
              (instruction-text inst) labels machine
              pc flag stack ops)))
          insts)))

   The machine instruction data structure simply pairs the instruction
text with the corresponding execution procedure.  The execution
procedure is not yet available when `extract-labels' constructs the
instruction, and is inserted later by `update-insts!'.

     (define (make-instruction text)
       (cons text '()))

     (define (instruction-text inst)
       (car inst))

     (define (instruction-execution-proc inst)
       (cdr inst))

     (define (set-instruction-execution-proc! inst proc)
       (set-cdr! inst proc))

   The instruction text is not used by our simulator, but it is handy
to keep around for debugging (see *Note Exercise 5-16::).

   Elements of the label table are pairs:

     (define (make-label-entry label-name insts)
       (cons label-name insts))

   Entries will be looked up in the table with

     (define (lookup-label labels label-name)
       (let ((val (assoc label-name labels)))
         (if val
             (cdr val)
             (error "Undefined label -- ASSEMBLE" label-name))))

     *Exercise 5.8:* The following register-machine code is ambiguous,
     because the label `here' is defined more than once:

          start
            (goto (label here))
          here
            (assign a (const 3))
            (goto (label there))
          here
            (assign a (const 4))
            (goto (label there))
          there

     With the simulator as written, what will the contents of register
     `a' be when control reaches `there'?  Modify the `extract-labels'
     procedure so that the assembler will signal an error if the same
     label name is used to indicate two different locations.

   ---------- Footnotes ----------

   (1) Using the `receive' procedure here is a way to get
`extract-labels' to effectively return two values--`labels' and
`insts'--without explicitly making a compound data structure to hold
them.  An alternative implementation, which returns an explicit pair of
values, is

     (define (extract-labels text)
       (if (null? text)
           (cons '() '())
           (let ((result (extract-labels (cdr text))))
             (let ((insts (car result)) (labels (cdr result)))
               (let ((next-inst (car text)))
                 (if (symbol? next-inst)
                     (cons insts
                           (cons (make-label-entry next-inst insts) labels))
                     (cons (cons (make-instruction next-inst) insts)
                           labels)))))))

which would be called by `assemble' as follows:

     (define (assemble controller-text machine)
       (let ((result (extract-labels controller-text)))
         (let ((insts (car result)) (labels (cdr result)))
           (update-insts! insts labels machine)
           insts)))

   You can consider our use of `receive' as demonstrating an elegant
way to return multiple values, or simply an excuse to show off a
programming trick.  An argument like `receive' that is the next
procedure to be invoked is called a "continuation."  Recall that we
also used continuations to implement the backtracking control structure
in the `amb' evaluator in section *Note 4-3-3::.


File: sicp.info,  Node: 5-2-3,  Next: 5-2-4,  Prev: 5-2-2,  Up: 5-2

5.2.3 Generating Execution Procedures for Instructions
------------------------------------------------------

The assembler calls `make-execution-procedure' to generate the execution
procedure for an instruction.  Like the `analyze' procedure in the
evaluator of section *Note 4-1-7::, this dispatches on the type of
instruction to generate the appropriate execution procedure.

     (define (make-execution-procedure inst labels machine
                                       pc flag stack ops)
       (cond ((eq? (car inst) 'assign)
              (make-assign inst machine labels ops pc))
             ((eq? (car inst) 'test)
              (make-test inst machine labels ops flag pc))
             ((eq? (car inst) 'branch)
              (make-branch inst machine labels flag pc))
             ((eq? (car inst) 'goto)
              (make-goto inst machine labels pc))
             ((eq? (car inst) 'save)
              (make-save inst machine stack pc))
             ((eq? (car inst) 'restore)
              (make-restore inst machine stack pc))
             ((eq? (car inst) 'perform)
              (make-perform inst machine labels ops pc))
             (else (error "Unknown instruction type -- ASSEMBLE"
                          inst))))

   For each type of instruction in the register-machine language, there
is a generator that builds an appropriate execution procedure.  The
details of these procedures determine both the syntax and meaning of
the individual instructions in the register-machine language.  We use
data abstraction to isolate the detailed syntax of register-machine
expressions from the general execution mechanism, as we did for
evaluators in section *Note 4-1-2::, by using syntax procedures to
extract and classify the parts of an instruction.

`Assign' instructions
.....................

The `make-assign' procedure handles `assign' instructions:

     (define (make-assign inst machine labels operations pc)
       (let ((target
              (get-register machine (assign-reg-name inst)))
             (value-exp (assign-value-exp inst)))
         (let ((value-proc
                (if (operation-exp? value-exp)
                    (make-operation-exp
                     value-exp machine labels operations)
                    (make-primitive-exp
                     (car value-exp) machine labels))))
           (lambda ()                ; execution procedure for `assign'
             (set-contents! target (value-proc))
             (advance-pc pc)))))

   `Make-assign' extracts the target register name (the second element
of the instruction) and the value expression (the rest of the list that
forms the instruction) from the `assign' instruction using the selectors

     (define (assign-reg-name assign-instruction)
       (cadr assign-instruction))

     (define (assign-value-exp assign-instruction)
       (cddr assign-instruction))

   The register name is looked up with `get-register' to produce the
target register object.  The value expression is passed to
`make-operation-exp' if the value is the result of an operation, and to
`make-primitive-exp' otherwise.  These procedures (shown below) parse
the value expression and produce an execution procedure for the value.
This is a procedure of no arguments, called `value-proc', which will be
evaluated during the simulation to produce the actual value to be
assigned to the register.  Notice that the work of looking up the
register name and parsing the value expression is performed just once,
at assembly time, not every time the instruction is simulated.  This
saving of work is the reason we use execution procedures, and
corresponds directly to the saving in work we obtained by separating
program analysis from execution in the evaluator of section *Note
4-1-7::.

   The result returned by `make-assign' is the execution procedure for
the `assign' instruction.  When this procedure is called (by the machine
model's `execute' procedure), it sets the contents of the target
register to the result obtained by executing `value-proc'.  Then it
advances the `pc' to the next instruction by running the procedure

     (define (advance-pc pc)
       (set-contents! pc (cdr (get-contents pc))))

   `Advance-pc' is the normal termination for all instructions except
`branch' and `goto'.

`Test', `branch', and `goto' instructions
.........................................

`Make-test' handles `test' instructions in a similar way.  It extracts
the expression that specifies the condition to be tested and generates
an execution procedure for it.  At simulation time, the procedure for
the condition is called, the result is assigned to the `flag' register,
and the `pc' is advanced:

     (define (make-test inst machine labels operations flag pc)
       (let ((condition (test-condition inst)))
         (if (operation-exp? condition)
             (let ((condition-proc
                    (make-operation-exp
                     condition machine labels operations)))
               (lambda ()
                 (set-contents! flag (condition-proc))
                 (advance-pc pc)))
             (error "Bad TEST instruction -- ASSEMBLE" inst))))

     (define (test-condition test-instruction)
       (cdr test-instruction))

   The execution procedure for a `branch' instruction checks the
contents of the `flag' register and either sets the contents of the
`pc' to the branch destination (if the branch is taken) or else just
advances the `pc' (if the branch is not taken).  Notice that the
indicated destination in a `branch' instruction must be a label, and
the `make-branch' procedure enforces this.  Notice also that the label
is looked up at assembly time, not each time the `branch' instruction
is simulated.

     (define (make-branch inst machine labels flag pc)
       (let ((dest (branch-dest inst)))
         (if (label-exp? dest)
             (let ((insts
                    (lookup-label labels (label-exp-label dest))))
               (lambda ()
                 (if (get-contents flag)
                     (set-contents! pc insts)
                     (advance-pc pc))))
             (error "Bad BRANCH instruction -- ASSEMBLE" inst))))

     (define (branch-dest branch-instruction)
       (cadr branch-instruction))

   A `goto' instruction is similar to a branch, except that the
destination may be specified either as a label or as a register, and
there is no condition to check--the `pc' is always set to the new
destination.

     (define (make-goto inst machine labels pc)
       (let ((dest (goto-dest inst)))
         (cond ((label-exp? dest)
                (let ((insts
                       (lookup-label labels
                                     (label-exp-label dest))))
                  (lambda () (set-contents! pc insts))))
               ((register-exp? dest)
                (let ((reg
                       (get-register machine
                                     (register-exp-reg dest))))
                  (lambda ()
                    (set-contents! pc (get-contents reg)))))
               (else (error "Bad GOTO instruction -- ASSEMBLE"
                            inst)))))

     (define (goto-dest goto-instruction)
       (cadr goto-instruction))

Other instructions
..................

The stack instructions `save' and `restore' simply use the stack with
the designated register and advance the `pc':

     (define (make-save inst machine stack pc)
       (let ((reg (get-register machine
                                (stack-inst-reg-name inst))))
         (lambda ()
           (push stack (get-contents reg))
           (advance-pc pc))))

     (define (make-restore inst machine stack pc)
       (let ((reg (get-register machine
                                (stack-inst-reg-name inst))))
         (lambda ()
           (set-contents! reg (pop stack))
           (advance-pc pc))))

     (define (stack-inst-reg-name stack-instruction)
       (cadr stack-instruction))

   The final instruction type, handled by `make-perform', generates an
execution procedure for the action to be performed.  At simulation
time, the action procedure is executed and the `pc' advanced.

     (define (make-perform inst machine labels operations pc)
       (let ((action (perform-action inst)))
         (if (operation-exp? action)
             (let ((action-proc
                    (make-operation-exp
                     action machine labels operations)))
               (lambda ()
                 (action-proc)
                 (advance-pc pc)))
             (error "Bad PERFORM instruction -- ASSEMBLE" inst))))

     (define (perform-action inst) (cdr inst))

Execution procedures for subexpressions
.......................................

The value of a `reg', `label', or `const' expression may be needed for
assignment to a register (`make-assign') or for input to an operation
(`make-operation-exp', below).  The following procedure generates
execution procedures to produce values for these expressions during the
simulation:

     (define (make-primitive-exp exp machine labels)
       (cond ((constant-exp? exp)
              (let ((c (constant-exp-value exp)))
                (lambda () c)))
             ((label-exp? exp)
              (let ((insts
                     (lookup-label labels
                                   (label-exp-label exp))))
                (lambda () insts)))
             ((register-exp? exp)
              (let ((r (get-register machine
                                     (register-exp-reg exp))))
                (lambda () (get-contents r))))
             (else
              (error "Unknown expression type -- ASSEMBLE" exp))))

   The syntax of `reg', `label', and `const' expressions is determined
by

     (define (register-exp? exp) (tagged-list? exp 'reg))

     (define (register-exp-reg exp) (cadr exp))

     (define (constant-exp? exp) (tagged-list? exp 'const))

     (define (constant-exp-value exp) (cadr exp))

     (define (label-exp? exp) (tagged-list? exp 'label))

     (define (label-exp-label exp) (cadr exp))

   `Assign', `perform', and `test' instructions may include the
application of a machine operation (specified by an `op' expression) to
some operands (specified by `reg' and `const' expressions).  The
following procedure produces an execution procedure for an "operation
expression"--a list containing the operation and operand expressions
from the instruction:

     (define (make-operation-exp exp machine labels operations)
       (let ((op (lookup-prim (operation-exp-op exp) operations))
             (aprocs
              (map (lambda (e)
                     (make-primitive-exp e machine labels))
                   (operation-exp-operands exp))))
         (lambda ()
           (apply op (map (lambda (p) (p)) aprocs)))))

   The syntax of operation expressions is determined by

     (define (operation-exp? exp)
       (and (pair? exp) (tagged-list? (car exp) 'op)))

     (define (operation-exp-op operation-exp)
       (cadr (car operation-exp)))

     (define (operation-exp-operands operation-exp)
       (cdr operation-exp))

   Observe that the treatment of operation expressions is very much
like the treatment of procedure applications by the
`analyze-application' procedure in the evaluator of section *Note
4-1-7:: in that we generate an execution procedure for each operand.
At simulation time, we call the operand procedures and apply the Scheme
procedure that simulates the operation to the resulting values.  The
simulation procedure is found by looking up the operation name in the
operation table for the machine:

     (define (lookup-prim symbol operations)
       (let ((val (assoc symbol operations)))
         (if val
             (cadr val)
             (error "Unknown operation -- ASSEMBLE" symbol))))

     *Exercise 5.9:* The treatment of machine operations above permits
     them to operate on labels as well as on constants and the contents
     of registers.  Modify the expression-processing procedures to
     enforce the condition that operations can be used only with
     registers and constants.

     *Exercise 5.10:* Design a new syntax for register-machine
     instructions and modify the simulator to use your new syntax.  Can
     you implement your new syntax without changing any part of the
     simulator except the syntax procedures in this section?

     *Exercise 5.11:* When we introduced `save' and `restore' in
     section *Note 5-1-4::, we didn't specify what would happen if you
     tried to restore a register that was not the last one saved, as in
     the sequence

          (save y)
          (save x)
          (restore y)

     There are several reasonable possibilities for the meaning of
     `restore':

       a. `(restore y)' puts into `y' the last value saved on the stack,
          regardless of what register that value came from.  This is
          the way our simulator behaves.  Show how to take advantage of
          this behavior to eliminate one instruction from the Fibonacci
          machine of section *Note 5-1-4:: (*Note Figure 5-12::).

       b. `(restore y)' puts into `y' the last value saved on the
          stack, but only if that value was saved from `y'; otherwise,
          it signals an error.  Modify the simulator to behave this
          way.  You will have to change `save' to put the register name
          on the stack along with the value.

       c. `(restore y)' puts into `y' the last value saved from `y'
          regardless of what other registers were saved after `y' and
          not restored.  Modify the simulator to behave this way.  You
          will have to associate a separate stack with each register.
          You should make the `initialize-stack' operation initialize
          all the register stacks.


     *Exercise 5.12:* The simulator can be used to help determine the
     data paths required for implementing a machine with a given
     controller.  Extend the assembler to store the following
     information in the machine model:

        * a list of all instructions, with duplicates removed, sorted
          by instruction type (`assign', `goto', and so on);

        * a list (without duplicates) of the registers used to hold
          entry points (these are the registers referenced by `goto'
          instructions);

        * a list (without duplicates) of the registers that are `save'd
          or `restore'd;

        * for each register, a list (without duplicates) of the sources
          from which it is assigned (for example, the sources for
          register `val' in the factorial machine of *Note Figure
          5-11:: are `(const 1)' and `((op *) (reg n) (reg val))').


     Extend the message-passing interface to the machine to provide
     access to this new information.  To test your analyzer, define the
     Fibonacci machine from *Note Figure 5-12:: and examine the lists
     you constructed.

     *Exercise 5.13:* Modify the simulator so that it uses the
     controller sequence to determine what registers the machine has
     rather than requiring a list of registers as an argument to
     `make-machine'.  Instead of pre-allocating the registers in
     `make-machine', you can allocate them one at a time when they are
     first seen during assembly of the instructions.


File: sicp.info,  Node: 5-2-4,  Prev: 5-2-3,  Up: 5-2

5.2.4 Monitoring Machine Performance
------------------------------------

Simulation is useful not only for verifying the correctness of a
proposed machine design but also for measuring the machine's
performance.  For example, we can install in our simulation program a
"meter" that measures the number of stack operations used in a
computation.  To do this, we modify our simulated stack to keep track
of the number of times registers are saved on the stack and the maximum
depth reached by the stack, and add a message to the stack's interface
that prints the statistics, as shown below.  We also add an operation
to the basic machine model to print the stack statistics, by
initializing `the-ops' in `make-new-machine' to

     (list (list 'initialize-stack
                 (lambda () (stack 'initialize)))
           (list 'print-stack-statistics
                 (lambda () (stack 'print-statistics))))

   Here is the new version of `make-stack':

     (define (make-stack)
       (let ((s '())
             (number-pushes 0)
             (max-depth 0)
             (current-depth 0))
         (define (push x)
           (set! s (cons x s))
           (set! number-pushes (+ 1 number-pushes))
           (set! current-depth (+ 1 current-depth))
           (set! max-depth (max current-depth max-depth)))
         (define (pop)
           (if (null? s)
               (error "Empty stack -- POP")
               (let ((top (car s)))
                 (set! s (cdr s))
                 (set! current-depth (- current-depth 1))
                 top)))
         (define (initialize)
           (set! s '())
           (set! number-pushes 0)
           (set! max-depth 0)
           (set! current-depth 0)
           'done)
         (define (print-statistics)
           (newline)
           (display (list 'total-pushes  '= number-pushes
                          'maximum-depth '= max-depth)))
         (define (dispatch message)
           (cond ((eq? message 'push) push)
                 ((eq? message 'pop) (pop))
                 ((eq? message 'initialize) (initialize))
                 ((eq? message 'print-statistics)
                  (print-statistics))
                 (else
                  (error "Unknown request -- STACK" message))))
         dispatch))

   *Note Exercise 5-15:: through *Note Exercise 5-19:: describe other
useful monitoring and debugging features that can be added to the
register-machine simulator.

     *Exercise 5.14:* Measure the number of pushes and the maximum
     stack depth required to compute n! for various small values of n
     using the factorial machine shown in *Note Figure 5-11::.  From
     your data determine formulas in terms of n for the total number of
     push operations and the maximum stack depth used in computing n!
     for any n > 1. Note that each of these is a linear function of n
     and is thus determined by two constants.  In order to get the
     statistics printed, you will have to augment the factorial machine
     with instructions to initialize the stack and print the
     statistics.  You may want to also modify the machine so that it
     repeatedly reads a value for n, computes the factorial, and prints
     the result (as we did for the GCD machine in *Note Figure 5-4::),
     so that you will not have to repeatedly invoke
     `get-register-contents', `set-register-contents!', and `start'.

     *Exercise 5.15:* Add counting "instruction counting" to the
     register machine simulation.  That is, have the machine model keep
     track of the number of instructions executed.  Extend the machine
     model's interface to accept a new message that prints the value of
     the instruction count and resets the count to zero.

     *Exercise 5.16:* Augment the simulator to provide for "instruction
     tracing".  That is, before each instruction is executed, the
     simulator should print the text of the instruction.  Make the
     machine model accept `trace-on' and `trace-off' messages to turn
     tracing on and off.

     *Exercise 5.17:* Extend the instruction tracing of *Note Exercise
     5-16:: so that before printing an instruction, the simulator
     prints any labels that immediately precede that instruction in the
     controller sequence.  Be careful to do this in a way that does not
     interfere with instruction counting (*Note Exercise 5-15::).  You
     will have to make the simulator retain the necessary label
     information.

     *Exercise 5.18:* Modify the `make-register' procedure of section
     *Note 5-2-1:: so that registers can be traced.  Registers should
     accept messages that turn tracing on and off.  When a register is
     traced, assigning a value to the register should print the name of
     the register, the old contents of the register, and the new
     contents being assigned.  Extend the interface to the machine
     model to permit you to turn tracing on and off for designated
     machine registers.

     *Exercise 5.19:* Alyssa P. Hacker wants a "breakpoint" feature in
     the simulator to help her debug her machine designs.  You have
     been hired to install this feature for her.  She wants to be able
     to specify a place in the controller sequence where the simulator
     will stop and allow her to examine the state of the machine.  You
     are to implement a procedure

          (set-breakpoint <MACHINE> <LABEL> <N>)

     that sets a breakpoint just before the nth instruction after the
     given label.  For example,

          (set-breakpoint gcd-machine 'test-b 4)

     installs a breakpoint in `gcd-machine' just before the assignment
     to register `a'.  When the simulator reaches the breakpoint it
     should print the label and the offset of the breakpoint and stop
     executing instructions.  Alyssa can then use
     `get-register-contents' and `set-register-contents!' to manipulate
     the state of the simulated machine.  She should then be able to
     continue execution by saying

          (proceed-machine <MACHINE>)

     She should also be able to remove a specific breakpoint by means of

          (cancel-breakpoint <MACHINE> <LABEL> <N>)

     or to remove all breakpoints by means of

          (cancel-all-breakpoints <MACHINE>)


File: sicp.info,  Node: 5-3,  Next: 5-4,  Prev: 5-2,  Up: Chapter 5

5.3 Storage Allocation and Garbage Collection
=============================================

In section *Note 5-4::, we will show how to implement a Scheme
evaluator as a register machine.  In order to simplify the discussion,
we will assume that our register machines can be equipped with a "list-structured
memory", in which the basic operations for manipulating list-structured
data are primitive.  Postulating the existence of such a memory is a
useful abstraction when one is focusing on the mechanisms of control in
a Scheme interpreter, but this does not reflect a realistic view of the
actual primitive data operations of contemporary computers.  To obtain
a more complete picture of how a Lisp system operates, we must
investigate how list structure can be represented in a way that is
compatible with conventional computer memories.

   There are two considerations in implementing list structure.  The
first is purely an issue of representation: how to represent the
"box-and-pointer" structure of Lisp pairs, using only the storage and
addressing capabilities of typical computer memories.  The second issue
concerns the management of memory as a computation proceeds.  The
operation of a Lisp system depends crucially on the ability to
continually create new data objects.  These include objects that are
explicitly created by the Lisp procedures being interpreted as well as
structures created by the interpreter itself, such as environments and
argument lists.  Although the constant creation of new data objects
would pose no problem on a computer with an infinite amount of rapidly
addressable memory, computer memories are available only in finite
sizes (more's the pity).  Lisp systems thus provide an "automatic
storage allocation" facility to support the illusion of an infinite
memory.  When a data object is no longer needed, the memory allocated
to it is automatically recycled and used to construct new data objects.
There are various techniques for providing such automatic storage
allocation.  The method we shall discuss in this section is called "garbage
collection".

* Menu:

* 5-3-1::            Memory as Vectors
* 5-3-2::            Maintaining the Illusion of Infinite Memory


File: sicp.info,  Node: 5-3-1,  Next: 5-3-2,  Prev: 5-3,  Up: 5-3

5.3.1 Memory as Vectors
-----------------------

A conventional computer memory can be thought of as an array of
cubbyholes, each of which can contain a piece of information.  Each
cubbyhole has a unique name, called its "address" or "location".
Typical memory systems provide two primitive operations: one that
fetches the data stored in a specified location and one that assigns
new data to a specified location.  Memory addresses can be incremented
to support sequential access to some set of the cubbyholes.  More
generally, many important data operations require that memory addresses
be treated as data, which can be stored in memory locations and
manipulated in machine registers.  The representation of list structure
is one application of such "address arithmetic".

   To model computer memory, we use a new kind of data structure called
a "vector".  Abstractly, a vector is a compound data object whose
individual elements can be accessed by means of an integer index in an
amount of time that is independent of the index.(1) In order to
describe memory operations, we use two primitive Scheme procedures for
manipulating vectors:

   * `(vector-ref <VECTOR> <N>)' returns the nth element of the vector.

   * `(vector-set! <VECTOR> <N> <VALUE>)' sets the nth element of the
     vector to the designated value.


   For example, if `v' is a vector, then `(vector-ref v 5)' gets the
fifth entry in the vector `v' and `(vector-set! v 5 7)' changes the
value of the fifth entry of the vector `v' to 7.(2)  For computer
memory, this access can be implemented through the use of address
arithmetic to combine a address "base address" that specifies the
beginning location of a vector in memory with an "index" that specifies
the offset of a particular element of the vector.

Representing Lisp data
......................

We can use vectors to implement the basic pair structures required for a
list-structured memory.  Let us imagine that computer memory is divided
into two vectors: `the-cars' and `the-cdrs'.  We will represent list
structure as follows: A pointer to a pair is an index into the two
vectors.  The `car' of the pair is the entry in `the-cars' with the
designated index, and the `cdr' of the pair is the entry in `the-cdrs'
with the designated index.  We also need a representation for objects
other than pairs (such as numbers and symbols) and a way to distinguish
one kind of data from another.  There are many methods of accomplishing
this, but they all reduce to using "typed pointers", that is, to
extending the notion of "pointer" to include information on data
type.(3) The data type enables the system to distinguish a pointer to a
pair (which consists of the "pair" data type and an index into the
memory vectors) from pointers to other kinds of data (which consist of
some other data type and whatever is being used to represent data of
that type).  Two data objects are considered to be the same (`eq?') if
their pointers are identical.(4) *Note Figure 5-14:: illustrates the
use of this method to represent the list `((1 2) 3 4)', whose
box-and-pointer diagram is also shown.  We use letter prefixes to
denote the data-type information.  Thus, a pointer to the pair with
index 5 is denoted `p5', the empty list is denoted by the pointer `e0',
and a pointer to the number 4 is denoted `n4'.  In the box-and-pointer
diagram, we have indicated at the lower left of each pair the vector
index that specifies where the `car' and `cdr' of the pair are stored.
The blank locations in `the-cars' and `the-cdrs' may contain parts of
other list structures (not of interest here).

     *Figure 5.14:* Box-and-pointer and memory-vector representations
     of the list `((1 2) 3 4)'.

                         +---+---+               +---+---+    +---+---+
          ((1 2) 3 4) -->| * | *-+-------------->| * | *-+--->| * | / |
                         +-|-+---+               +-|-+---+    +-|-+---+
                        1  |                    2  |         4  |
                           V                       V            V
                         +---+---+    +---+---+  +---+        +---+
                         | * | *-+--->| * | / |  | 3 |        | 4 |
                         +-|-+---+    +-|-+---+  +---+        +---+
                        5  |         7  |
                           V            V
                         +---+        +---+
                         | 1 |        | 2 |
                         +---+        +---+

             Index   0    1    2    3    4    5    6    7    8    ...
                   +----+----+----+----+----+----+----+----+----+----
          the-cars |    | p5 | n3 |    | n4 | n1 |    | n2 |    | ...
                   +----+----+----+----+----+----+----+----+----+----
          the-cdrs |    | p2 | p4 |    | e0 | p7 |    | e0 |    | ...
                   +----+----+----+----+----+----+----+----+----+----

   A pointer to a number, such as `n4', might consist of a type
indicating numeric data together with the actual representation of the
number 4.(5)  To deal with numbers that are too large to be represented
in the fixed amount of space allocated for a single pointer, we could
use a distinct "bignum" data type, for which the pointer designates a
list in which the parts of the number are stored.(6)

   A symbol might be represented as a typed pointer that designates a
sequence of the characters that form the symbol's printed
representation.  This sequence is constructed by the Lisp reader when
the character string is initially encountered in input.  Since we want
two instances of a symbol to be recognized as the "same" symbol by
`eq?' and we want `eq?' to be a simple test for equality of pointers,
we must ensure that if the reader sees the same character string twice,
it will use the same pointer (to the same sequence of characters) to
represent both occurrences.  To accomplish this, the reader maintains a
table, traditionally called the "obarray", of all the symbols it has
ever encountered.  When the reader encounters a character string and is
about to construct a symbol, it checks the obarray to see if it has ever
before seen the same character string.  If it has not, it uses the
characters to construct a new symbol (a typed pointer to a new
character sequence) and enters this pointer in the obarray.  If the
reader has seen the string before, it returns the symbol pointer stored
in the obarray.  This process of replacing character strings by unique
pointers is called "interning" symbols.

Implementing the primitive list operations
..........................................

Given the above representation scheme, we can replace each "primitive"
list operation of a register machine with one or more primitive vector
operations.  We will use two registers, `the-cars' and `the-cdrs', to
identify the memory vectors, and will assume that `vector-ref' and
`vector-set!' are available as primitive operations.  We also assume
that numeric operations on pointers (such as incrementing a pointer,
using a pair pointer to index a vector, or adding two numbers) use only
the index portion of the typed pointer.

   For example, we can make a register machine support the instructions

     (assign <REG_1> (op car) (reg <REG_2>))

     (assign <REG_1> (op cdr) (reg <REG_2>))

if we implement these, respectively, as

     (assign <REG_1> (op vector-ref) (reg the-cars) (reg <REG_2>))

     (assign <REG_1> (op vector-ref) (reg the-cdrs) (reg <REG_2>))

   The instructions

     (perform (op set-car!) (reg <REG_1>) (reg <REG_2>))

     (perform (op set-cdr!) (reg <REG_1>) (reg <REG_2>))

are implemented as

     (perform
      (op vector-set!) (reg the-cars) (reg <REG_1>) (reg <REG_2>))

     (perform
      (op vector-set!) (reg the-cdrs) (reg <REG_1>) (reg <REG_2>))

   `Cons' is performed by allocating an unused index and storing the
arguments to `cons' in `the-cars' and `the-cdrs' at that indexed vector
position.  We presume that there is a special register, `free', that
always holds a pair pointer containing the next available index, and
that we can increment the index part of that pointer to find the next
free location.(7)  For example, the instruction

     (assign <REG_1> (op cons) (reg <REG_2>) (reg <REG_3>))

is implemented as the following sequence of vector operations:(8)

     (perform
      (op vector-set!) (reg the-cars) (reg free) (reg <REG_2>))
     (perform
      (op vector-set!) (reg the-cdrs) (reg free) (reg <REG_3>))
     (assign <REG_1> (reg free))
     (assign free (op +) (reg free) (const 1))

   The `eq?' operation

     (op eq?) (reg <REG_1>) (reg <REG_2>)

simply tests the equality of all fields in the registers, and
predicates such as `pair?', `null?', `symbol?', and `number?' need only
check the type field.

Implementing stacks
...................

Although our register machines use stacks, we need do nothing special
here, since stacks can be modeled in terms of lists.  The stack can be
a list of the saved values, pointed to by a special register
`the-stack'.  Thus, ` (save <REG>)' can be implemented as

     (assign the-stack (op cons) (reg <REG>) (reg the-stack))

Similarly, `(restore <REG>)' can be implemented as

     (assign <REG> (op car) (reg the-stack))
     (assign the-stack (op cdr) (reg the-stack))

and `(perform (op initialize-stack))' can be implemented as

     (assign the-stack (const ()))

   These operations can be further expanded in terms of the vector
operations given above.  In conventional computer architectures,
however, it is usually advantageous to allocate the stack as a separate
vector.  Then pushing and popping the stack can be accomplished by
incrementing or decrementing an index into that vector.

     *Exercise 5.20:* Draw the box-and-pointer representation and the
     memory-vector representation (as in *Note Figure 5-14::) of the
     list structure produced by

          (define x (cons 1 2))
          (define y (list x x))

     with the `free' pointer initially `p1'.  What is the final value of
     `free' ?  What pointers represent the values of `x' and `y' ?

     *Exercise 5.21:* Implement register machines for the following
     procedures.  Assume that the list-structure memory operations are
     available as machine primitives.

       a. Recursive `count-leaves':

               (define (count-leaves tree)
                 (cond ((null? tree) 0)
                       ((not (pair? tree)) 1)
                       (else (+ (count-leaves (car tree))
                                (count-leaves (cdr tree))))))

       b. Recursive `count-leaves' with explicit counter:

               (define (count-leaves tree)
                 (define (count-iter tree n)
                   (cond ((null? tree) n)
                         ((not (pair? tree)) (+ n 1))
                         (else (count-iter (cdr tree)
                                           (count-iter (car tree) n)))))
                 (count-iter tree 0))

     *Exercise 5.22:* *Note Exercise 3-12:: of section *Note 3-3-1::
     presented an `append' procedure that appends two lists to form a
     new list and an `append!' procedure that splices two lists
     together.  Design a register machine to implement each of these
     procedures.  Assume that the list-structure memory operations are
     available as primitive operations.

   ---------- Footnotes ----------

   (1) We could represent memory as lists of items.  However, the
access time would then not be independent of the index, since accessing
the nth element of a list requires n - 1 `cdr' operations.

   (2) For completeness, we should specify a `make-vector' operation
that constructs vectors.  However, in the present application we will
use vectors only to model fixed divisions of the computer memory.

   (3) This is precisely the same "tagged data" idea we introduced in
*Note Chapter 2:: for dealing with generic operations.  Here, however,
the data types are included at the primitive machine level rather than
constructed through the use of lists.

   (4) Type information may be encoded in a variety of ways, depending
on the details of the machine on which the Lisp system is to be
implemented.  The execution efficiency of Lisp programs will be
strongly dependent on how cleverly this choice is made, but it is
difficult to formulate general design rules for good choices.  The most
straightforward way to implement typed pointers is to allocate a fixed
set of bits in each pointer to be a "type field" that encodes the data
type.  Important questions to be addressed in designing such a
representation include the following: How many type bits are required?
How large must the vector indices be?  How efficiently can the
primitive machine instructions be used to manipulate the type fields of
pointers?  Machines that include special hardware for the efficient
handling of type fields are said to have architectures "tagged
architectures".

   (5) This decision on the representation of numbers determines whether
`eq?', which tests equality of pointers, can be used to test for
equality of numbers.  If the pointer contains the number itself, then
equal numbers will have the same pointer.  But if the pointer contains
the index of a location where the number is stored, equal numbers will
be guaranteed to have equal pointers only if we are careful never to
store the same number in more than one location.

   (6) This is just like writing a number as a sequence of digits,
except that each "digit" is a number between 0 and the largest number
that can be stored in a single pointer.

   (7) There are other ways of finding free storage.  For example, we
could link together all the unused pairs into a "free list".  Our free
locations are consecutive (and hence can be accessed by incrementing a
pointer) because we are using a compacting garbage collector, as we
will see in section *Note 5-3-2::.

   (8) This is essentially the implementation of `cons' in terms of
`set-car!' and `set-cdr!', as described in section *Note 3-3-1::.  The
operation `get-new-pair' used in that implementation is realized here
by the `free' pointer.


File: sicp.info,  Node: 5-3-2,  Prev: 5-3-1,  Up: 5-3

5.3.2 Maintaining the Illusion of Infinite Memory
-------------------------------------------------

The representation method outlined in section *Note 5-3-1:: solves the
problem of implementing list structure, provided that we have an
infinite amount of memory.  With a real computer we will eventually run
out of free space in which to construct new pairs.(1)  However, most of
the pairs generated in a typical computation are used only to hold
intermediate results.  After these results are accessed, the pairs are
no longer needed--they are "garbage".  For instance, the computation

     (accumulate + 0 (filter odd? (enumerate-interval 0 n)))

constructs two lists: the enumeration and the result of filtering the
enumeration.  When the accumulation is complete, these lists are no
longer needed, and the allocated memory can be reclaimed.  If we can
arrange to collect all the garbage periodically, and if this turns out
to recycle memory at about the same rate at which we construct new
pairs, we will have preserved the illusion that there is an infinite
amount of memory.

   In order to recycle pairs, we must have a way to determine which
allocated pairs are not needed (in the sense that their contents can no
longer influence the future of the computation).  The method we shall
examine for accomplishing this is known as "garbage collection".
Garbage collection is based on the observation that, at any moment in a
Lisp interpretation, the only objects that can affect the future of the
computation are those that can be reached by some succession of `car'
and `cdr' operations starting from the pointers that are currently in
the machine registers.(2) Any memory cell that is not so accessible may
be recycled.

   There are many ways to perform garbage collection.  The method we
shall examine here is called "stop-and-copy".  The basic idea is to
divide memory into two halves: "working memory" and "free memory."
When `cons' constructs pairs, it allocates these in working memory.
When working memory is full, we perform garbage collection by locating
all the useful pairs in working memory and copying these into
consecutive locations in free memory.  (The useful pairs are located by
tracing all the `car' and `cdr' pointers, starting with the machine
registers.)  Since we do not copy the garbage, there will presumably be
additional free memory that we can use to allocate new pairs.  In
addition, nothing in the working memory is needed, since all the useful
pairs in it have been copied.  Thus, if we interchange the roles of
working memory and free memory, we can continue processing; new pairs
will be allocated in the new working memory (which was the old free
memory).  When this is full, we can copy the useful pairs into the new
free memory (which was the old working memory).(3)

Implementation of a stop-and-copy garbage collector
...................................................

We now use our register-machine language to describe the stop-and-copy
algorithm in more detail.  We will assume that there is a register
called `root' that contains a pointer to a structure that eventually
points at all accessible data.  This can be arranged by storing the
contents of all the machine registers in a pre-allocated list pointed
at by `root' just before starting garbage collection.(4) We also assume
that, in addition to the current working memory, there is free memory
available into which we can copy the useful data.  The current working
memory consists of vectors whose base addresses are in registers called
`the-cars' and `the-cdrs', and the free memory is in registers called
`new-cars' and `new-cdrs'.

   Garbage collection is triggered when we exhaust the free cells in
the current working memory, that is, when a `cons' operation attempts
to increment the `free' pointer beyond the end of the memory vector.
When the garbage-collection process is complete, the `root' pointer
will point into the new memory, all objects accessible from the `root'
will have been moved to the new memory, and the `free' pointer will
indicate the next place in the new memory where a new pair can be
allocated.  In addition, the roles of working memory and new memory
will have been interchanged--new pairs will be constructed in the new
memory, beginning at the place indicated by `free', and the (previous)
working memory will be available as the new memory for the next garbage
collection.  *Note Figure 5-15:: shows the arrangement of memory just
before and just after garbage collection.

     *Figure 5.15:* Reconfiguration of memory by the garbage-collection
     process.

                       Just before garbage collection

                   +------------------------------------+
          the-cars |                                    | working
                   | mixture of useful data and garbage | memory
          the-cdrs |                                    |
                   +------------------------------------+
                                                      ^
                                                      | free

                   +------------------------------------+
          new-cars |                                    | free
                   |            free memory             | memory
          new-cdrs |                                    |
                   +------------------------------------+

                       Just after garbage collection

                   +------------------------------------+
          new-cars |                                    | new
                   |          discarded memory          | free
          new-cdrs |                                    | memory
                   +------------------------------------+

                   +------------------+-----------------+
          the-cars |                  |                 | new
                   |   useful data    |    free area    | working
          the-cdrs |                  |                 | memory
                   +------------------+-----------------+
                                        ^
                                        | free

   The state of the garbage-collection process is controlled by
maintaining two pointers: `free' and `scan'.  These are initialized to
point to the beginning of the new memory.  The algorithm begins by
relocating the pair pointed at by `root' to the beginning of the new
memory.  The pair is copied, the `root' pointer is adjusted to point to
the new location, and the `free' pointer is incremented.  In addition,
the old location of the pair is marked to show that its contents have
been moved.  This marking is done as follows: In the `car' position, we
place a special tag that signals that this is an already-moved object.
(Such an object is traditionally called a "broken heart".)(5)  In the
`cdr' position we place a "forwarding address" that points at the
location to which the object has been moved.

   After relocating the root, the garbage collector enters its basic
cycle.  At each step in the algorithm, the `scan' pointer (initially
pointing at the relocated root) points at a pair that has been moved to
the new memory but whose `car' and `cdr' pointers still refer to
objects in the old memory.  These objects are each relocated, and the
`scan' pointer is incremented.  To relocate an object (for example, the
object indicated by the `car' pointer of the pair we are scanning) we
check to see if the object has already been moved (as indicated by the
presence of a broken-heart tag in the `car' position of the object).
If the object has not already been moved, we copy it to the place
indicated by `free', update `free', set up a broken heart at the
object's old location, and update the pointer to the object (in this
example, the `car' pointer of the pair we are scanning) to point to the
new location.  If the object has already been moved, its forwarding
address (found in the `cdr' position of the broken heart) is
substituted for the pointer in the pair being scanned.  Eventually, all
accessible objects will have been moved and scanned, at which point the
`scan' pointer will overtake the `free' pointer and the process will
terminate.

   We can specify the stop-and-copy algorithm as a sequence of
instructions for a register machine.  The basic step of relocating an
object is accomplished by a subroutine called
`relocate-old-result-in-new'.  This subroutine gets its argument, a
pointer to the object to be relocated, from a register named `old'.  It
relocates the designated object (incrementing `free' in the process),
puts a pointer to the relocated object into a register called `new',
and returns by branching to the entry point stored in the register
`relocate-continue'.  To begin garbage collection, we invoke this
subroutine to relocate the `root' pointer, after initializing `free'
and `scan'.  When the relocation of `root' has been accomplished, we
install the new pointer as the new `root' and enter the main loop of the
garbage collector.

     begin-garbage-collection
       (assign free (const 0))
       (assign scan (const 0))
       (assign old (reg root))
       (assign relocate-continue (label reassign-root))
       (goto (label relocate-old-result-in-new))
     reassign-root
       (assign root (reg new))
       (goto (label gc-loop))

   In the main loop of the garbage collector we must determine whether
there are any more objects to be scanned.  We do this by testing
whether the `scan' pointer is coincident with the `free' pointer.  If
the pointers are equal, then all accessible objects have been
relocated, and we branch to `gc-flip', which cleans things up so that
we can continue the interrupted computation.  If there are still pairs
to be scanned, we call the relocate subroutine to relocate the `car' of
the next pair (by placing the `car' pointer in `old').  The
`relocate-continue' register is set up so that the subroutine will
return to update the `car' pointer.

     gc-loop
       (test (op =) (reg scan) (reg free))
       (branch (label gc-flip))
       (assign old (op vector-ref) (reg new-cars) (reg scan))
       (assign relocate-continue (label update-car))
       (goto (label relocate-old-result-in-new))

   At `update-car', we modify the `car' pointer of the pair being
scanned, then proceed to relocate the `cdr' of the pair.  We return to
`update-cdr' when that relocation has been accomplished.  After
relocating and updating the `cdr', we are finished scanning that pair,
so we continue with the main loop.

     update-car
       (perform
        (op vector-set!) (reg new-cars) (reg scan) (reg new))
       (assign old (op vector-ref) (reg new-cdrs) (reg scan))
       (assign relocate-continue (label update-cdr))
       (goto (label relocate-old-result-in-new))

     update-cdr
       (perform
        (op vector-set!) (reg new-cdrs) (reg scan) (reg new))
       (assign scan (op +) (reg scan) (const 1))
       (goto (label gc-loop))

   The subroutine `relocate-old-result-in-new' relocates objects as
follows: If the object to be relocated (pointed at by `old') is not a
pair, then we return the same pointer to the object unchanged (in
`new').  (For example, we may be scanning a pair whose `car' is the
number 4.  If we represent the `car' by `n4', as described in section
*Note 5-3-1::, then we want the "relocated" `car' pointer to still be
`n4'.)  Otherwise, we must perform the relocation.  If the `car'
position of the pair to be relocated contains a broken-heart tag, then
the pair has in fact already been moved, so we retrieve the forwarding
address (from the `cdr' position of the broken heart) and return this
in `new'.  If the pointer in `old' points at a yet-unmoved pair, then
we move the pair to the first free cell in new memory (pointed at by
`free') and set up the broken heart by storing a broken-heart tag and
forwarding address at the old location.  `Relocate-old-result-in-new'
uses a register `oldcr' to hold the `car' or the `cdr' of the object
pointed at by `old'.(6)

     relocate-old-result-in-new
       (test (op pointer-to-pair?) (reg old))
       (branch (label pair))
       (assign new (reg old))
       (goto (reg relocate-continue))
     pair
       (assign oldcr (op vector-ref) (reg the-cars) (reg old))
       (test (op broken-heart?) (reg oldcr))
       (branch (label already-moved))
       (assign new (reg free)) ; new location for pair
       ;; Update `free' pointer.
       (assign free (op +) (reg free) (const 1))
       ;; Copy the `car' and `cdr' to new memory.
       (perform (op vector-set!)
                (reg new-cars) (reg new) (reg oldcr))
       (assign oldcr (op vector-ref) (reg the-cdrs) (reg old))
       (perform (op vector-set!)
                (reg new-cdrs) (reg new) (reg oldcr))
       ;; Construct the broken heart.
       (perform (op vector-set!)
                (reg the-cars) (reg old) (const broken-heart))
       (perform
        (op vector-set!) (reg the-cdrs) (reg old) (reg new))
       (goto (reg relocate-continue))
     already-moved
       (assign new (op vector-ref) (reg the-cdrs) (reg old))
       (goto (reg relocate-continue))

   At the very end of the garbage-collection process, we interchange
the role of old and new memories by interchanging pointers:
interchanging `the-cars' with `new-cars', and `the-cdrs' with
`new-cdrs'.  We will then be ready to perform another garbage
collection the next time memory runs out.

     gc-flip
       (assign temp (reg the-cdrs))
       (assign the-cdrs (reg new-cdrs))
       (assign new-cdrs (reg temp))
       (assign temp (reg the-cars))
       (assign the-cars (reg new-cars))
       (assign new-cars (reg temp))

   ---------- Footnotes ----------

   (1) This may not be true eventually, because memories may get large
enough so that it would be impossible to run out of free memory in the
lifetime of the computer.  For example, there are about 3*(10^13),
microseconds in a year, so if we were to `cons' once per microsecond we
would need about 10^15 cells of memory to build a machine that could
operate for 30 years without running out of memory.  That much memory
seems absurdly large by today's standards, but it is not physically
impossible.  On the other hand, processors are getting faster and a
future computer may have large numbers of processors operating in
parallel on a single memory, so it may be possible to use up memory
much faster than we have postulated.

   (2) We assume here that the stack is represented as a list as
described in section *Note 5-3-1::, so that items on the stack are
accessible via the pointer in the stack register.

   (3) This idea was invented and first implemented by Minsky, as part
of the implementation of Lisp for the PDP-1 at the MIT Research
Laboratory of Electronics.  It was further developed by Fenichel and
Yochelson (1969) for use in the Lisp implementation for the Multics
time-sharing system.  Later, Baker (1978) developed a "real-time"
version of the method, which does not require the computation to stop
during garbage collection.  Baker's idea was extended by Hewitt,
Lieberman, and Moon (see Lieberman and Hewitt 1983) to take advantage
of the fact that some structure is more volatile and other structure is
more permanent.

   An alternative commonly used garbage-collection technique is the "mark-sweep"
method.  This consists of tracing all the structure accessible from the
machine registers and marking each pair we reach.  We then scan all of
memory, and any location that is unmarked is "swept up" as garbage and
made available for reuse.  A full discussion of the mark-sweep method
can be found in Allen 1978.

   The Minsky-Fenichel-Yochelson algorithm is the dominant algorithm in
use for large-memory systems because it examines only the useful part
of memory.  This is in contrast to mark-sweep, in which the sweep phase
must check all of memory.  A second advantage of stop-and-copy is that
it is a "compacting" garbage collector.  That is, at the end of the
garbage-collection phase the useful data will have been moved to
consecutive memory locations, with all garbage pairs compressed out.
This can be an extremely important performance consideration in
machines with virtual memory, in which accesses to widely separated
memory addresses may require extra paging operations.

   (4) This list of registers does not include the registers used by
the storage-allocation system--`root', `the-cars', `the-cdrs', and the
other registers that will be introduced in this section.

   (5) The term _broken heart_ was coined by David Cressey, who wrote a
garbage collector for MDL, a dialect of Lisp developed at MIT during
the early 1970s.

   (6) The garbage collector uses the low-level predicate
`pointer-to-pair?' instead of the list-structure `pair?'  operation
because in a real system there might be various things that are treated
as pairs for garbage-collection purposes.  For example, in a Scheme
system that conforms to the IEEE standard a procedure object may be
implemented as a special kind of "pair" that doesn't satisfy the
`pair?' predicate.  For simulation purposes, `pointer-to-pair?' can be
implemented as `pair?'.


File: sicp.info,  Node: 5-4,  Next: 5-5,  Prev: 5-3,  Up: Chapter 5

5.4 The Explicit-Control Evaluator
==================================

In section *Note 5-1:: we saw how to transform simple Scheme programs
into descriptions of register machines.  We will now perform this
transformation on a more complex program, the metacircular evaluator of
sections *Note 4-1-1::-*Note 4-1-4::, which shows how the behavior of a
Scheme interpreter can be described in terms of the procedures `eval'
and `apply'.  The "explicit-control evaluator" that we develop in this
section shows how the underlying procedure-calling and argument-passing
mechanisms used in the evaluation process can be described in terms of
operations on registers and stacks.  In addition, the explicit-control
evaluator can serve as an implementation of a Scheme interpreter,
written in a language that is very similar to the native machine
language of conventional computers.  The evaluator can be executed by
the register-machine simulator of section *Note 5-2::.  Alternatively,
it can be used as a starting point for building a machine-language
implementation of a Scheme evaluator, or even a special-purpose machine
for evaluating Scheme expressions.  *Note Figure 5-16:: shows such a
hardware implementation: a silicon chip that acts as an evaluator for
Scheme.  The chip designers started with the data-path and controller
specifications for a register machine similar to the evaluator
described in this section and used design automation programs to
construct the integrated-circuit layout.(1)

Registers and operations
........................

In designing the explicit-control evaluator, we must specify the
operations to be used in our register machine.  We described the
metacircular evaluator in terms of abstract syntax, using procedures
such as `quoted?' and `make-procedure'.  In implementing the register
machine, we could expand these procedures into sequences of elementary
list-structure memory operations, and implement these operations on our
register machine.  However, this would make our evaluator very long,
obscuring the basic structure with details.  To clarify the
presentation, we will include as primitive operations of the register
machine the syntax procedures given in section *Note 4-1-2:: and the
procedures for representing environments and other run-time data given
in sections *Note 4-1-3:: and *Note 4-1-4::.  In order to completely
specify an evaluator that could be programmed in a low-level machine
language or implemented in hardware, we would replace these operations
by more elementary operations, using the list-structure implementation
we described in section *Note 5-3::.

     *Figure 5.16:* A silicon-chip implementation of an evaluator for
     Scheme.

     [This figure is missing.]


   Our Scheme evaluator register machine includes a stack and seven
registers: `exp', `env', `val', `continue', `proc', `argl', and `unev'.
`Exp' is used to hold the expression to be evaluated, and `env'
contains the environment in which the evaluation is to be performed.
At the end of an evaluation, `val' contains the value obtained by
evaluating the expression in the designated environment.  The
`continue' register is used to implement recursion, as explained in
section *Note 5-1-4::.  (The evaluator needs to call itself
recursively, since evaluating an expression requires evaluating its
subexpressions.)  The registers `proc', `argl', and `unev' are used in
evaluating combinations.

   We will not provide a data-path diagram to show how the registers and
operations of the evaluator are connected, nor will we give the
complete list of machine operations.  These are implicit in the
evaluator's controller, which will be presented in detail.

* Menu:

* 5-4-1::            The Core of the Explicit-Control Evaluator
* 5-4-2::            Sequence Evaluation and Tail Recursion
* 5-4-3::            Conditionals, Assignments, and Definitions
* 5-4-4::            Running the Evaluator

   ---------- Footnotes ----------

   (1) See Batali et al. 1982 for more information on the chip and the
method by which it was designed.


File: sicp.info,  Node: 5-4-1,  Next: 5-4-2,  Prev: 5-4,  Up: 5-4

5.4.1 The Core of the Explicit-Control Evaluator
------------------------------------------------

The central element in the evaluator is the sequence of instructions
beginning at `eval-dispatch'.  This corresponds to the `eval' procedure
of the metacircular evaluator described in section *Note 4-1-1::.  When
the controller starts at `eval-dispatch', it evaluates the expression
specified by `exp' in the environment specified by `env'.  When
evaluation is complete, the controller will go to the entry point
stored in `continue', and the `val' register will hold the value of the
expression.  As with the metacircular `eval', the structure of
`eval-dispatch' is a case analysis on the syntactic type of the
expression to be evaluated.(1)

     eval-dispatch
       (test (op self-evaluating?) (reg exp))
       (branch (label ev-self-eval))
       (test (op variable?) (reg exp))
       (branch (label ev-variable))
       (test (op quoted?) (reg exp))
       (branch (label ev-quoted))
       (test (op assignment?) (reg exp))
       (branch (label ev-assignment))
       (test (op definition?) (reg exp))
       (branch (label ev-definition))
       (test (op if?) (reg exp))
       (branch (label ev-if))
       (test (op lambda?) (reg exp))
       (branch (label ev-lambda))
       (test (op begin?) (reg exp))
       (branch (label ev-begin))
       (test (op application?) (reg exp))
       (branch (label ev-application))
       (goto (label unknown-expression-type))

Evaluating simple expressions
.............................

Numbers and strings (which are self-evaluating), variables, quotations,
and `lambda' expressions have no subexpressions to be evaluated.  For
these, the evaluator simply places the correct value in the `val'
register and continues execution at the entry point specified by
`continue'.  Evaluation of simple expressions is performed by the
following controller code:

     ev-self-eval
       (assign val (reg exp))
       (goto (reg continue))
     ev-variable
       (assign val (op lookup-variable-value) (reg exp) (reg env))
       (goto (reg continue))
     ev-quoted
       (assign val (op text-of-quotation) (reg exp))
       (goto (reg continue))
     ev-lambda
       (assign unev (op lambda-parameters) (reg exp))
       (assign exp (op lambda-body) (reg exp))
       (assign val (op make-procedure)
                   (reg unev) (reg exp) (reg env))
       (goto (reg continue))

   Observe how `ev-lambda' uses the `unev' and `exp' registers to hold
the parameters and body of the lambda expression so that they can be
passed to the `make-procedure' operation, along with the environment in
`env'.

Evaluating procedure applications
.................................

A procedure application is specified by a combination containing an
operator and operands.  The operator is a subexpression whose value is
a procedure, and the operands are subexpressions whose values are the
arguments to which the procedure should be applied.  The metacircular
`eval' handles applications by calling itself recursively to evaluate
each element of the combination, and then passing the results to
`apply', which performs the actual procedure application.  The
explicit-control evaluator does the same thing; these recursive calls
are implemented by `goto' instructions, together with use of the stack
to save registers that will be restored after the recursive call
returns.  Before each call we will be careful to identify which
registers must be saved (because their values will be needed later).(2)

   We begin the evaluation of an application by evaluating the operator
to produce a procedure, which will later be applied to the evaluated
operands.  To evaluate the operator, we move it to the `exp' register
and go to `eval-dispatch'.  The environment in the `env' register is
already the correct one in which to evaluate the operator.  However, we
save `env' because we will need it later to evaluate the operands.  We
also extract the operands into `unev' and save this on the stack.  We
set up `continue' so that `eval-dispatch' will resume at
`ev-appl-did-operator' after the operator has been evaluated.  First,
however, we save the old value of `continue', which tells the controller
where to continue after the application.

     ev-application
       (save continue)
       (save env)
       (assign unev (op operands) (reg exp))
       (save unev)
       (assign exp (op operator) (reg exp))
       (assign continue (label ev-appl-did-operator))
       (goto (label eval-dispatch))

   Upon returning from evaluating the operator subexpression, we
proceed to evaluate the operands of the combination and to accumulate
the resulting arguments in a list, held in `argl'.  First we restore
the unevaluated operands and the environment.  We initialize `argl' to
an empty list.  Then we assign to the `proc' register the procedure
that was produced by evaluating the operator.  If there are no
operands, we go directly to `apply-dispatch'.  Otherwise we save `proc'
on the stack and start the argument-evaluation loop:(3)

     ev-appl-did-operator
       (restore unev)                  ; the operands
       (restore env)
       (assign argl (op empty-arglist))
       (assign proc (reg val))         ; the operator
       (test (op no-operands?) (reg unev))
       (branch (label apply-dispatch))
       (save proc)

   Each cycle of the argument-evaluation loop evaluates an operand from
the list in `unev' and accumulates the result into `argl'.  To evaluate
an operand, we place it in the `exp' register and go to `eval-dispatch',
after setting `continue' so that execution will resume with the
argument-accumulation phase.  But first we save the arguments
accumulated so far (held in `argl'), the environment (held in `env'),
and the remaining operands to be evaluated (held in `unev').  A special
case is made for the evaluation of the last operand, which is handled at
`ev-appl-last-arg'.

     ev-appl-operand-loop
       (save argl)
       (assign exp (op first-operand) (reg unev))
       (test (op last-operand?) (reg unev))
       (branch (label ev-appl-last-arg))
       (save env)
       (save unev)
       (assign continue (label ev-appl-accumulate-arg))
       (goto (label eval-dispatch))

   When an operand has been evaluated, the value is accumulated into
the list held in `argl'.  The operand is then removed from the list of
unevaluated operands in `unev', and the argument-evaluation continues.

     ev-appl-accumulate-arg
       (restore unev)
       (restore env)
       (restore argl)
       (assign argl (op adjoin-arg) (reg val) (reg argl))
       (assign unev (op rest-operands) (reg unev))
       (goto (label ev-appl-operand-loop))

   Evaluation of the last argument is handled differently.  There is no
need to save the environment or the list of unevaluated operands before
going to `eval-dispatch', since they will not be required after the
last operand is evaluated.  Thus, we return from the evaluation to a
special entry point `ev-appl-accum-last-arg', which restores the
argument list, accumulates the new argument, restores the saved
procedure, and goes off to perform the application.(4)

     ev-appl-last-arg
       (assign continue (label ev-appl-accum-last-arg))
       (goto (label eval-dispatch))
     ev-appl-accum-last-arg
       (restore argl)
       (assign argl (op adjoin-arg) (reg val) (reg argl))
       (restore proc)
       (goto (label apply-dispatch))

   The details of the argument-evaluation loop determine the order in
which the interpreter evaluates the operands of a combination (e.g.,
left to right or right to left--see *Note Exercise 3-8::).  This order
is not determined by the metacircular evaluator, which inherits its
control structure from the underlying Scheme in which it is
implemented.(5) Because the `first-operand' selector (used in
`ev-appl-operand-loop' to extract successive operands from `unev') is
implemented as `car' and the `rest-operands' selector is implemented as
`cdr', the explicit-control evaluator will evaluate the operands of a
combination in left-to-right order.

Procedure application
.....................

The entry point `apply-dispatch' corresponds to the `apply' procedure
of the metacircular evaluator.  By the time we get to `apply-dispatch',
the `proc' register contains the procedure to apply and `argl' contains
the list of evaluated arguments to which it must be applied.  The saved
value of `continue' (originally passed to `eval-dispatch' and saved at
`ev-application'), which tells where to return with the result of the
procedure application, is on the stack.  When the application is
complete, the controller transfers to the entry point specified by the
saved `continue', with the result of the application in `val'.  As with
the metacircular `apply', there are two cases to consider.  Either the
procedure to be applied is a primitive or it is a compound procedure.

     apply-dispatch
       (test (op primitive-procedure?) (reg proc))
       (branch (label primitive-apply))
       (test (op compound-procedure?) (reg proc))
       (branch (label compound-apply))
       (goto (label unknown-procedure-type))

   We assume that each primitive is implemented so as to obtain its
arguments from `argl' and place its result in `val'.  To specify how
the machine handles primitives, we would have to provide a sequence of
controller instructions to implement each primitive and arrange for
`primitive-apply' to dispatch to the instructions for the primitive
identified by the contents of `proc'.  Since we are interested in the
structure of the evaluation process rather than the details of the
primitives, we will instead just use an `apply-primitive-procedure'
operation that applies the procedure in `proc' to the arguments in
`argl'.  For the purpose of simulating the evaluator with the simulator
of section *Note 5-2:: we use the procedure
`apply-primitive-procedure', which calls on the underlying Scheme system
to perform the application, just as we did for the metacircular
evaluator in section *Note 4-1-4::.  After computing the value of the
primitive application, we restore `continue' and go to the designated
entry point.

     primitive-apply
       (assign val (op apply-primitive-procedure)
                   (reg proc)
                   (reg argl))
       (restore continue)
       (goto (reg continue))

   To apply a compound procedure, we proceed just as with the
metacircular evaluator.  We construct a frame that binds the
procedure's parameters to the arguments, use this frame to extend the
environment carried by the procedure, and evaluate in this extended
environment the sequence of expressions that forms the body of the
procedure.  `Ev-sequence', described below in section *Note 5-4-2::,
handles the evaluation of the sequence.

     compound-apply
       (assign unev (op procedure-parameters) (reg proc))
       (assign env (op procedure-environment) (reg proc))
       (assign env (op extend-environment)
                   (reg unev) (reg argl) (reg env))
       (assign unev (op procedure-body) (reg proc))
       (goto (label ev-sequence))

   `Compound-apply' is the only place in the interpreter where the `env'
register is ever assigned a new value.  Just as in the metacircular
evaluator, the new environment is constructed from the environment
carried by the procedure, together with the argument list and the
corresponding list of variables to be bound.

   ---------- Footnotes ----------

   (1) In our controller, the dispatch is written as a sequence of
`test' and `branch' instructions.  Alternatively, it could have been
written in a data-directed style (and in a real system it probably
would have been) to avoid the need to perform sequential tests and to
facilitate the definition of new expression types.  A machine designed
to run Lisp would probably include a `dispatch-on-type' instruction
that would efficiently execute such data-directed dispatches.

   (2) This is an important but subtle point in translating algorithms
from a procedural language, such as Lisp, to a register-machine
language.  As an alternative to saving only what is needed, we could
save all the registers (except `val') before each recursive call. This
is called a "framed-stack" discipline.  This would work but might save
more registers than necessary; this could be an important consideration
in a system where stack operations are expensive.  Saving registers
whose contents will not be needed later may also hold onto useless data
that could otherwise be garbage-collected, freeing space to be reused.

   (3) We add to the evaluator data-structure procedures in section
*Note 4-1-3:: the following two procedures for manipulating argument
lists:

     (define (empty-arglist) '())

     (define (adjoin-arg arg arglist)
       (append arglist (list arg)))

   We also use an additional syntax procedure to test for the last
operand in a combination:

     (define (last-operand? ops)
       (null? (cdr ops)))

   (4) The optimization of treating the last operand specially is known
as "evlis tail recursion" (see Wand 1980).  We could be somewhat more
efficient in the argument evaluation loop if we made evaluation of the
first operand a special case too.  This would permit us to postpone
initializing `argl' until after evaluating the first operand, so as to
avoid saving `argl' in this case.  The compiler in section *Note 5-5::
performs this optimization.  (Compare the `construct-arglist' procedure
of section *Note 5-5-3::.)

   (5) The order of operand evaluation in the metacircular evaluator is
determined by the order of evaluation of the arguments to `cons' in the
procedure `list-of-values' of section *Note 4-1-1:: (see *Note Exercise
4-1::).


File: sicp.info,  Node: 5-4-2,  Next: 5-4-3,  Prev: 5-4-1,  Up: 5-4

5.4.2 Sequence Evaluation and Tail Recursion
--------------------------------------------

The portion of the explicit-control evaluator at `ev-sequence' is
analogous to the metacircular evaluator's `eval-sequence' procedure.  It
handles sequences of expressions in procedure bodies or in explicit
`begin' expressions.

   Explicit `begin' expressions are evaluated by placing the sequence of
expressions to be evaluated in `unev', saving `continue' on the stack,
and jumping to `ev-sequence'.

     ev-begin
       (assign unev (op begin-actions) (reg exp))
       (save continue)
       (goto (label ev-sequence))

   The implicit sequences in procedure bodies are handled by jumping to
`ev-sequence' from `compound-apply', at which point `continue' is
already on the stack, having been saved at `ev-application'.

   The entries at `ev-sequence' and `ev-sequence-continue' form a loop
that successively evaluates each expression in a sequence.  The list of
unevaluated expressions is kept in `unev'.  Before evaluating each
expression, we check to see if there are additional expressions to be
evaluated in the sequence.  If so, we save the rest of the unevaluated
expressions (held in `unev') and the environment in which these must be
evaluated (held in `env') and call `eval-dispatch' to evaluate the
expression.  The two saved registers are restored upon the return from
this evaluation, at `ev-sequence-continue'.

   The final expression in the sequence is handled differently, at the
entry point `ev-sequence-last-exp'.  Since there are no more
expressions to be evaluated after this one, we need not save `unev' or
`env' before going to `eval-dispatch'.  The value of the whole sequence
is the value of the last expression, so after the evaluation of the
last expression there is nothing left to do except continue at the
entry point currently held on the stack (which was saved by
`ev-application' or `ev-begin'.)  Rather than setting up `continue' to
arrange for `eval-dispatch' to return here and then restoring
`continue' from the stack and continuing at that entry point, we
restore `continue' from the stack before going to `eval-dispatch', so
that `eval-dispatch' will continue at that entry point after evaluating
the expression.

     ev-sequence
       (assign exp (op first-exp) (reg unev))
       (test (op last-exp?) (reg unev))
       (branch (label ev-sequence-last-exp))
       (save unev)
       (save env)
       (assign continue (label ev-sequence-continue))
       (goto (label eval-dispatch))
     ev-sequence-continue
       (restore env)
       (restore unev)
       (assign unev (op rest-exps) (reg unev))
       (goto (label ev-sequence))
     ev-sequence-last-exp
       (restore continue)
       (goto (label eval-dispatch))

Tail recursion
..............

In *Note Chapter 1:: we said that the process described by a procedure
such as

     (define (sqrt-iter guess x)
       (if (good-enough? guess x)
           guess
           (sqrt-iter (improve guess x)
                      x)))

is an iterative process.  Even though the procedure is syntactically
recursive (defined in terms of itself), it is not logically necessary
for an evaluator to save information in passing from one call to
`sqrt-iter' to the next.(1) An evaluator that can execute a procedure
such as `sqrt-iter' without requiring increasing storage as the
procedure continues to call itself is called a "tail-recursive"
evaluator.  The metacircular implementation of the evaluator in *Note
Chapter 4:: does not specify whether the evaluator is tail-recursive,
because that evaluator inherits its mechanism for saving state from the
underlying Scheme.  With the explicit-control evaluator, however, we
can trace through the evaluation process to see when procedure calls
cause a net accumulation of information on the stack.

   Our evaluator is tail-recursive, because in order to evaluate the
final expression of a sequence we transfer directly to `eval-dispatch'
without saving any information on the stack.  Hence, evaluating the
final expression in a sequence--even if it is a procedure call (as in
`sqrt-iter', where the `if' expression, which is the last expression in
the procedure body, reduces to a call to `sqrt-iter')--will not cause
any information to be accumulated on the stack.(2)

   If we did not think to take advantage of the fact that it was
unnecessary to save information in this case, we might have implemented
`eval-sequence' by treating all the expressions in a sequence in the
same way--saving the registers, evaluating the expression, returning to
restore the registers, and repeating this until all the expressions
have been evaluated:(3)

     ev-sequence
       (test (op no-more-exps?) (reg unev))
       (branch (label ev-sequence-end))
       (assign exp (op first-exp) (reg unev))
       (save unev)
       (save env)
       (assign continue (label ev-sequence-continue))
       (goto (label eval-dispatch))
     ev-sequence-continue
       (restore env)
       (restore unev)
       (assign unev (op rest-exps) (reg unev))
       (goto (label ev-sequence))
     ev-sequence-end
       (restore continue)
       (goto (reg continue))

   This may seem like a minor change to our previous code for
evaluation of a sequence: The only difference is that we go through the
save-restore cycle for the last expression in a sequence as well as for
the others.  The interpreter will still give the same value for any
expression.  But this change is fatal to the tail-recursive
implementation, because we must now return after evaluating the final
expression in a sequence in order to undo the (useless) register saves.
These extra saves will accumulate during a nest of procedure calls.
Consequently, processes such as `sqrt-iter' will require space
proportional to the number of iterations rather than requiring constant
space.  This difference can be significant.  For example, with tail
recursion, an infinite loop can be expressed using only the
procedure-call mechanism:

     (define (count n)
       (newline)
       (display n)
       (count (+ n 1)))

   Without tail recursion, such a procedure would eventually run out of
stack space, and expressing a true iteration would require some control
mechanism other than procedure call.

   ---------- Footnotes ----------

   (1) We saw in section *Note 5-1:: how to implement such a process
with a register machine that had no stack; the state of the process was
stored in a fixed set of registers.

   (2) This implementation of tail recursion in `ev-sequence' is one
variety of a well-known optimization technique used by many compilers.
In compiling a procedure that ends with a procedure call, one can
replace the call by a jump to the called procedure's entry point.
Building this strategy into the interpreter, as we have done in this
section, provides the optimization uniformly throughout the language.

   (3) We can define `no-more-exps?' as follows:

     (define (no-more-exps? seq) (null? seq))


File: sicp.info,  Node: 5-4-3,  Next: 5-4-4,  Prev: 5-4-2,  Up: 5-4

5.4.3 Conditionals, Assignments, and Definitions
------------------------------------------------

As with the metacircular evaluator, special forms are handled by
selectively evaluating fragments of the expression.  For an `if'
expression, we must evaluate the predicate and decide, based on the
value of predicate, whether to evaluate the consequent or the
alternative.

   Before evaluating the predicate, we save the `if' expression itself
so that we can later extract the consequent or alternative.  We also
save the environment, which we will need later in order to evaluate the
consequent or the alternative, and we save `continue', which we will
need later in order to return to the evaluation of the expression that
is waiting for the value of the `if'.

     ev-if
       (save exp)                    ; save expression for later
       (save env)
       (save continue)
       (assign continue (label ev-if-decide))
       (assign exp (op if-predicate) (reg exp))
       (goto (label eval-dispatch))  ; evaluate the predicate

   When we return from evaluating the predicate, we test whether it was
true or false and, depending on the result, place either the consequent
or the alternative in `exp' before going to `eval-dispatch'.  Notice
that restoring `env' and `continue' here sets up `eval-dispatch' to
have the correct environment and to continue at the right place to
receive the value of the `if' expression.

     ev-if-decide
       (restore continue)
       (restore env)
       (restore exp)
       (test (op true?) (reg val))
       (branch (label ev-if-consequent))

     ev-if-alternative
       (assign exp (op if-alternative) (reg exp))
       (goto (label eval-dispatch))
     ev-if-consequent
       (assign exp (op if-consequent) (reg exp))
       (goto (label eval-dispatch))

Assignments and definitions
...........................

Assignments are handled by `ev-assignment', which is reached from
`eval-dispatch' with the assignment expression in `exp'.  The code at
`ev-assignment' first evaluates the value part of the expression and
then installs the new value in the environment.  `Set-variable-value!'
is assumed to be available as a machine operation.

     ev-assignment
       (assign unev (op assignment-variable) (reg exp))
       (save unev)                   ; save variable for later
       (assign exp (op assignment-value) (reg exp))
       (save env)
       (save continue)
       (assign continue (label ev-assignment-1))
       (goto (label eval-dispatch))  ; evaluate the assignment value
     ev-assignment-1
       (restore continue)
       (restore env)
       (restore unev)
       (perform
        (op set-variable-value!) (reg unev) (reg val) (reg env))
       (assign val (const ok))
       (goto (reg continue))

   Definitions are handled in a similar way:

     ev-definition
       (assign unev (op definition-variable) (reg exp))
       (save unev)                   ; save variable for later
       (assign exp (op definition-value) (reg exp))
       (save env)
       (save continue)
       (assign continue (label ev-definition-1))
       (goto (label eval-dispatch))  ; evaluate the definition value
     ev-definition-1
       (restore continue)
       (restore env)
       (restore unev)
       (perform
        (op define-variable!) (reg unev) (reg val) (reg env))
       (assign val (const ok))
       (goto (reg continue))

     *Exercise 5.23:* Extend the evaluator to handle derived
     expressions such as `cond', `let', and so on (section *Note
     4-1-2::).  You may "cheat" and assume that the syntax transformers
     such as `cond->if' are available as machine operations.(1)

     *Exercise 5.24:* Implement `cond' as a new basic special form
     without reducing it to `if'.  You will have to construct a loop
     that tests the predicates of successive `cond' clauses until you
     find one that is true, and then use `ev-sequence' to evaluate the
     actions of the clause.

     *Exercise 5.25:* Modify the evaluator so that it uses normal-order
     evaluation, based on the lazy evaluator of section *Note 4-2::.

   ---------- Footnotes ----------

   (1) This isn't really cheating.  In an actual implementation built
from scratch, we would use our explicit-control evaluator to interpret
a Scheme program that performs source-level transformations like
`cond->if' in a syntax phase that runs before execution.


File: sicp.info,  Node: 5-4-4,  Prev: 5-4-3,  Up: 5-4

5.4.4 Running the Evaluator
---------------------------

With the implementation of the explicit-control evaluator we come to
the end of a development, begun in *Note Chapter 1::, in which we have
explored successively more precise models of the evaluation process.
We started with the relatively informal substitution model, then
extended this in *Note Chapter 3:: to the environment model, which
enabled us to deal with state and change.  In the metacircular
evaluator of *Note Chapter 4::, we used Scheme itself as a language for
making more explicit the environment structure constructed during
evaluation of an expression.  Now, with register machines, we have
taken a close look at the evaluator's mechanisms for storage
management, argument passing, and control.  At each new level of
description, we have had to raise issues and resolve ambiguities that
were not apparent at the previous, less precise treatment of
evaluation.  To understand the behavior of the explicit-control
evaluator, we can simulate it and monitor its performance.

   We will install a driver loop in our evaluator machine.  This plays
the role of the `driver-loop' procedure of section *Note 4-1-4::.  The
evaluator will repeatedly print a prompt, read an expression, evaluate
the expression by going to `eval-dispatch', and print the result.  The
following instructions form the beginning of the explicit-control
evaluator's controller sequence:(1)

     read-eval-print-loop
       (perform (op initialize-stack))
       (perform
        (op prompt-for-input) (const ";;; EC-Eval input:"))
       (assign exp (op read))
       (assign env (op get-global-environment))
       (assign continue (label print-result))
       (goto (label eval-dispatch))
     print-result
       (perform
        (op announce-output) (const ";;; EC-Eval value:"))
       (perform (op user-print) (reg val))
       (goto (label read-eval-print-loop))

   When we encounter an error in a procedure (such as the "unknown
procedure type error" indicated at `apply-dispatch'), we print an error
message and return to the driver loop.(2)

     unknown-expression-type
       (assign val (const unknown-expression-type-error))
       (goto (label signal-error))

     unknown-procedure-type
       (restore continue)    ; clean up stack (from `apply-dispatch')
       (assign val (const unknown-procedure-type-error))
       (goto (label signal-error))

     signal-error
       (perform (op user-print) (reg val))
       (goto (label read-eval-print-loop))

   For the purposes of the simulation, we initialize the stack each
time through the driver loop, since it might not be empty after an
error (such as an undefined variable) interrupts an evaluation.(3)

   If we combine all the code fragments presented in sections *Note
5-4-1::-*Note 5-4-4::, we can create an evaluator machine model that we
can run using the register-machine simulator of section *Note 5-2::.

     (define eceval
       (make-machine
        '(exp env val proc argl continue unev)
        eceval-operations
       '(
         read-eval-print-loop
           <_entire machine controller as given above_>
        )))

   We must define Scheme procedures to simulate the operations used as
primitives by the evaluator.  These are the same procedures we used for
the metacircular evaluator in section *Note 4-1::, together with the
few additional ones defined in footnotes throughout section *Note 5-4::.

     (define eceval-operations
       (list (list 'self-evaluating? self-evaluating)
             <_complete list of operations for eceval machine_>))

   Finally, we can initialize the global environment and run the
evaluator:

     (define the-global-environment (setup-environment))

     (start eceval)

     ;;; EC-Eval input:
     (define (append x y)
       (if (null? x)
           y
           (cons (car x)
                 (append (cdr x) y))))
     ;;; EC-Eval value:
     ok

     ;;; EC-Eval input:
     (append '(a b c) '(d e f))
     ;;; EC-Eval value:
     (a b c d e f)

   Of course, evaluating expressions in this way will take much longer
than if we had directly typed them into Scheme, because of the multiple
levels of simulation involved.  Our expressions are evaluated by the
explicit-control-evaluator machine, which is being simulated by a Scheme
program, which is itself being evaluated by the Scheme interpreter.

Monitoring the performance of the evaluator
...........................................

Simulation can be a powerful tool to guide the implementation of
evaluators.  Simulations make it easy not only to explore variations of
the register-machine design but also to monitor the performance of the
simulated evaluator.  For example, one important factor in performance
is how efficiently the evaluator uses the stack.  We can observe the
number of stack operations required to evaluate various expressions by
defining the evaluator register machine with the version of the
simulator that collects statistics on stack use (section *Note
5-2-4::), and adding an instruction at the evaluator's `print-result'
entry point to print the statistics:

     print-result
       (perform (op print-stack-statistics)); added instruction
       (perform
        (op announce-output) (const ";;; EC-Eval value:"))
       ... ; same as before

   Interactions with the evaluator now look like this:

     ;;; EC-Eval input:
     (define (factorial n)
       (if (= n 1)
           1
           (* (factorial (- n 1)) n)))
     (total-pushes = 3 maximum-depth = 3)
     ;;; EC-Eval value:
     ok

     ;;; EC-Eval input:
     (factorial 5)
     (total-pushes = 144 maximum-depth = 28)
     ;;; EC-Eval value:
     120

   Note that the driver loop of the evaluator reinitializes the stack
at the start of each interaction, so that the statistics printed will
refer only to stack operations used to evaluate the previous expression.

     *Exercise 5.26:* Use the monitored stack to explore the
     tail-recursive property of the evaluator (section *Note 5-4-2::).
     Start the evaluator and define the iterative `factorial' procedure
     from section *Note 1-2-1:::

          (define (factorial n)
            (define (iter product counter)
              (if (> counter n)
                  product
                  (iter (* counter product)
                        (+ counter 1))))
            (iter 1 1))

     Run the procedure with some small values of n.  Record the maximum
     stack depth and the number of pushes required to compute n! for
     each of these values.

       a. You will find that the maximum depth required to evaluate n!
          is independent of n.  What is that depth?

       b. Determine from your data a formula in terms of n for the
          total number of push operations used in evaluating n! for any
          n >= 1.  Note that the number of operations used is a linear
          function of n and is thus determined by two constants.


     *Exercise 5.27:* For comparison with *Note Exercise 5-26::,
     explore the behavior of the following procedure for computing
     factorials recursively:

          (define (factorial n)
            (if (= n 1)
                1
                (* (factorial (- n 1)) n)))

     By running this procedure with the monitored stack, determine, as
     a function of n, the maximum depth of the stack and the total
     number of pushes used in evaluating n! for n >= 1.  (Again, these
     functions will be linear.)  Summarize your experiments by filling
     in the following table with the appropriate expressions in terms
     of n:

                         Maximum depth       Number of pushes

          Recursive
          factorial

          Iterative
          factorial

     The maximum depth is a measure of the amount of space used by the
     evaluator in carrying out the computation, and the number of
     pushes correlates well with the time required.

     *Exercise 5.28:* Modify the definition of the evaluator by
     changing `eval-sequence' as described in section *Note 5-4-2:: so
     that the evaluator is no longer tail-recursive.  Rerun your
     experiments from *Note Exercise 5-26:: and *Note Exercise 5-27::
     to demonstrate that both versions of the `factorial' procedure now
     require space that grows linearly with their input.

     *Exercise 5.29:* Monitor the stack operations in the
     tree-recursive Fibonacci computation:

          (define (fib n)
            (if (< n 2)
                n
                (+ (fib (- n 1)) (fib (- n 2)))))

       a. Give a formula in terms of n for the maximum depth of the
          stack required to compute _Fib_(n) for n >= 2.  Hint: In
          section *Note 1-2-2:: we argued that the space used by this
          process grows linearly with n.

       b. Give a formula for the total number of pushes used to compute
          _Fib_(n) for n >= 2.  You should find that the number of
          pushes (which correlates well with the time used) grows
          exponentially with n.  Hint: Let S(n) be the number of pushes
          used in computing _Fib_(n).  You should be able to argue that
          there is a formula that expresses S(n) in terms of S(n - 1),
          S(n - 2), and some fixed "overhead" constant k that is
          independent of n.  Give the formula, and say what k is.  Then
          show that S(n) can be expressed as a _Fib_(n + 1) + b and
          give the values of a and b.


     *Exercise 5.30:* Our evaluator currently catches and signals only
     two kinds of errors--unknown expression types and unknown
     procedure types.  Other errors will take us out of the evaluator
     read-eval-print loop.  When we run the evaluator using the
     register-machine simulator, these errors are caught by the
     underlying Scheme system.  This is analogous to the computer
     crashing when a user program makes an error.(4)  It is a large
     project to make a real error system work, but it is well worth the
     effort to understand what is involved here.

       a. Errors that occur in the evaluation process, such as an
          attempt to access an unbound variable, could be caught by
          changing the lookup operation to make it return a
          distinguished condition code, which cannot be a possible
          value of any user variable.  The evaluator can test for this
          condition code and then do what is necessary to go to
          `signal-error'.  Find all of the places in the evaluator
          where such a change is necessary and fix them.  This is lots
          of work.

       b. Much worse is the problem of handling errors that are
          signaled by applying primitive procedures, such as an attempt
          to divide by zero or an attempt to extract the `car' of a
          symbol.  In a professionally written high-quality system,
          each primitive application is checked for safety as part of
          the primitive.  For example, every call to `car' could first
          check that the argument is a pair.  If the argument is not a
          pair, the application would return a distinguished condition
          code to the evaluator, which would then report the failure.
          We could arrange for this in our register-machine simulator by
          making each primitive procedure check for applicability and
          returning an appropriate distinguished condition code on
          failure. Then the `primitive-apply' code in the evaluator can
          check for the condition code and go to `signal-error' if
          necessary.  Build this structure and make it work.  This is a
          major project.


   ---------- Footnotes ----------

   (1) We assume here that `read' and the various printing operations
are available as primitive machine operations, which is useful for our
simulation, but completely unrealistic in practice.  These are actually
extremely complex operations.  In practice, they would be implemented
using low-level input-output operations such as transferring single
characters to and from a device.

   To support the `get-global-environment' operation we define

     (define the-global-environment (setup-environment))

     (define (get-global-environment)
       the-global-environment)

   (2) There are other errors that we would like the interpreter to
handle, but these are not so simple.  See *Note Exercise 5-30::.

   (3) We could perform the stack initialization only after errors, but
doing it in the driver loop will be convenient for monitoring the
evaluator's performance, as described below.

   (4) Regrettably, this is the normal state of affairs in conventional
compiler-based language systems such as C.  In UNIX(tm) the system
"dumps core," and in DOS/Windows(tm) it becomes catatonic.  The
Macintosh(tm) displays a picture of an exploding bomb and offers you
the opportunity to reboot the computer--if you're lucky.


File: sicp.info,  Node: 5-5,  Prev: 5-4,  Up: Chapter 5

5.5 Compilation
===============

The explicit-control evaluator of section *Note 5-4:: is a register
machine whose controller interprets Scheme programs.  In this section
we will see how to run Scheme programs on a register machine whose
controller is not a Scheme interpreter.

   The explicit-control evaluator machine is universal--it can carry
out any computational process that can be described in Scheme.  The
evaluator's controller orchestrates the use of its data paths to
perform the desired computation.  Thus, the evaluator's data paths are
universal: They are sufficient to perform any computation we desire,
given an appropriate controller.(1)

   Commercial general-purpose computers are register machines organized
around a collection of registers and operations that constitute an
efficient and convenient universal set of data paths.  The controller
for a general-purpose machine is an interpreter for a register-machine
language like the one we have been using.  This language is called the "native
language" of the machine, or simply "machine language".  Programs
written in machine language are sequences of instructions that use the
machine's data paths.  For example, the explicit-control evaluator's
instruction sequence can be thought of as a machine-language program
for a general-purpose computer rather than as the controller for a
specialized interpreter machine.

   There are two common strategies for bridging the gap between
higher-level languages and register-machine languages.  The
explicit-control evaluator illustrates the strategy of interpretation.
An interpreter written in the native language of a machine configures
the machine to execute programs written in a language (called the "source
language") that may differ from the native language of the machine
performing the evaluation.  The primitive procedures of the source
language are implemented as a library of subroutines written in the
native language of the given machine.  A program to be interpreted
(called the "source program") is represented as a data structure.  The
interpreter traverses this data structure, analyzing the source
program.  As it does so, it simulates the intended behavior of the
source program by calling appropriate primitive subroutines from the
library.

   In this section, we explore the alternative strategy of "compilation".
A compiler for a given source language and machine translates a source
program into an equivalent program (called the "object program")
written in the machine's native language.  The compiler that we
implement in this section translates programs written in Scheme into
sequences of instructions to be executed using the explicit-control
evaluator machine's data paths.(2)

   Compared with interpretation, compilation can provide a great
increase in the efficiency of program execution, as we will explain
below in the overview of the compiler.  On the other hand, an
interpreter provides a more powerful environment for interactive
program development and debugging, because the source program being
executed is available at run time to be examined and modified.  In
addition, because the entire library of primitives is present, new
programs can be constructed and added to the system during debugging.

   In view of the complementary advantages of compilation and
interpretation, modern program-development environments pursue a mixed
strategy.  Lisp interpreters are generally organized so that
interpreted procedures and compiled procedures can call each other.
This enables a programmer to compile those parts of a program that are
assumed to be debugged, thus gaining the efficiency advantage of
compilation, while retaining the interpretive mode of execution for
those parts of the program that are in the flux of interactive
development and debugging.  In section *Note 5-5-7::, after we have
implemented the compiler, we will show how to interface it with our
interpreter to produce an integrated interpreter-compiler development
system.

An overview of the compiler
...........................

Our compiler is much like our interpreter, both in its structure and in
the function it performs.  Accordingly, the mechanisms used by the
compiler for analyzing expressions will be similar to those used by the
interpreter.  Moreover, to make it easy to interface compiled and
interpreted code, we will design the compiler to generate code that
obeys the same conventions of register usage as the interpreter: The
environment will be kept in the `env' register, argument lists will be
accumulated in `argl', a procedure to be applied will be in `proc',
procedures will return their answers in `val', and the location to
which a procedure should return will be kept in `continue'.  In
general, the compiler translates a source program into an object
program that performs essentially the same register operations as would
the interpreter in evaluating the same source program.

   This description suggests a strategy for implementing a rudimentary
compiler: We traverse the expression in the same way the interpreter
does.  When we encounter a register instruction that the interpreter
would perform in evaluating the expression, we do not execute the
instruction but instead accumulate it into a sequence.  The resulting
sequence of instructions will be the object code.  Observe the
efficiency advantage of compilation over interpretation.  Each time the
interpreter evaluates an expression--for example, `(f 84 96)'--it
performs the work of classifying the expression (discovering that this
is a procedure application) and testing for the end of the operand list
(discovering that there are two operands).  With a compiler, the
expression is analyzed only once, when the instruction sequence is
generated at compile time.  The object code produced by the compiler
contains only the instructions that evaluate the operator and the two
operands, assemble the argument list, and apply the procedure (in
`proc') to the arguments (in `argl').

   This is the same kind of optimization we implemented in the
analyzing evaluator of section *Note 4-1-7::.  But there are further
opportunities to gain efficiency in compiled code.  As the interpreter
runs, it follows a process that must be applicable to any expression in
the language.  In contrast, a given segment of compiled code is meant
to execute some particular expression.  This can make a big difference,
for example in the use of the stack to save registers.  When the
interpreter evaluates an expression, it must be prepared for any
contingency.  Before evaluating a subexpression, the interpreter saves
all registers that will be needed later, because the subexpression
might require an arbitrary evaluation.  A compiler, on the other hand,
can exploit the structure of the particular expression it is processing
to generate code that avoids unnecessary stack operations.

   As a case in point, consider the combination `(f 84 96)'.  Before the
interpreter evaluates the operator of the combination, it prepares for
this evaluation by saving the registers containing the operands and the
environment, whose values will be needed later.  The interpreter then
evaluates the operator to obtain the result in `val', restores the
saved registers, and finally moves the result from `val' to `proc'.
However, in the particular expression we are dealing with, the operator
is the symbol `f', whose evaluation is accomplished by the machine
operation `lookup-variable-value', which does not alter any registers.
The compiler that we implement in this section will take advantage of
this fact and generate code that evaluates the operator using the
instruction

     (assign proc (op lookup-variable-value) (const f) (reg env))

   This code not only avoids the unnecessary saves and restores but
also assigns the value of the lookup directly to `proc', whereas the
interpreter would obtain the result in `val' and then move this to
`proc'.

   A compiler can also optimize access to the environment.  Having
analyzed the code, the compiler can in many cases know in which frame a
particular variable will be located and access that frame directly,
rather than performing the `lookup-variable-value' search.  We will
discuss how to implement such variable access in section *Note 5-5-6::.
Until then, however, we will focus on the kind of register and stack
optimizations described above.  There are many other optimizations that
can be performed by a compiler, such as coding primitive operations "in
line" instead of using a general `apply' mechanism (see *Note Exercise
5-38::); but we will not emphasize these here.  Our main goal in this
section is to illustrate the compilation process in a simplified (but
still interesting) context.

* Menu:

* 5-5-1::            Structure of the Compiler
* 5-5-2::            Compiling Expressions
* 5-5-3::            Compiling Combinations
* 5-5-4::            Combining Instruction Sequences
* 5-5-5::            An Example of Compiled Code
* 5-5-6::            Lexical Addressing
* 5-5-7::            Interfacing Compiled Code to the Evaluator

   ---------- Footnotes ----------

   (1) This is a theoretical statement.  We are not claiming that the
evaluator's data paths are a particularly convenient or efficient set of
data paths for a general-purpose computer.  For example, they are not
very good for implementing high-performance floating-point calculations
or calculations that intensively manipulate bit vectors.

   (2) Actually, the machine that runs compiled code can be simpler
than the interpreter machine, because we won't use the `exp' and `unev'
registers.  The interpreter used these to hold pieces of unevaluated
expressions.  With the compiler, however, these expressions get built
into the compiled code that the register machine will run.  For the same
reason, we don't need the machine operations that deal with expression
syntax.  But compiled code will use a few additional machine operations
(to represent compiled procedure objects) that didn't appear in the
explicit-control evaluator machine.


File: sicp.info,  Node: 5-5-1,  Next: 5-5-2,  Prev: 5-5,  Up: 5-5

5.5.1 Structure of the Compiler
-------------------------------

In section *Note 4-1-7:: we modified our original metacircular
interpreter to separate analysis from execution.  We analyzed each
expression to produce an execution procedure that took an environment
as argument and performed the required operations.  In our compiler, we
will do essentially the same analysis.  Instead of producing execution
procedures, however, we will generate sequences of instructions to be
run by our register machine.

   The procedure `compile' is the top-level dispatch in the compiler.
It corresponds to the `eval' procedure of section *Note 4-1-1::, the
`analyze' procedure of section *Note 4-1-7::, and the `eval-dispatch'
entry point of the explicit-control-evaluator in section *Note 5-4-1::.
The compiler, like the interpreters, uses the expression-syntax
procedures defined in section *Note 4-1-2::.(1)  `Compile' performs a
case analysis on the syntactic type of the expression to be compiled.
For each type of expression, it dispatches to a specialized "code
generator":

     (define (compile exp target linkage)
       (cond ((self-evaluating? exp)
              (compile-self-evaluating exp target linkage))
             ((quoted? exp) (compile-quoted exp target linkage))
             ((variable? exp)
              (compile-variable exp target linkage))
             ((assignment? exp)
              (compile-assignment exp target linkage))
             ((definition? exp)
              (compile-definition exp target linkage))
             ((if? exp) (compile-if exp target linkage))
             ((lambda? exp) (compile-lambda exp target linkage))
             ((begin? exp)
              (compile-sequence (begin-actions exp)
                                target
                                linkage))
             ((cond? exp) (compile (cond->if exp) target linkage))
             ((application? exp)
              (compile-application exp target linkage))
             (else
              (error "Unknown expression type -- COMPILE" exp))))

Targets and linkages
....................

`Compile' and the code generators that it calls take two arguments in
addition to the expression to compile.  There is a "target", which
specifies the register in which the compiled code is to return the
value of the expression.  There is also a "linkage descriptor", which
describes how the code resulting from the compilation of the expression
should proceed when it has finished its execution.  The linkage
descriptor can require that the code do one of the following three
things:

   * continue at the next instruction in sequence (this is specified by
     the linkage descriptor `next'),

   * return from the procedure being compiled (this is specified by the
     linkage descriptor `return'), or

   * jump to a named entry point (this is specified by using the
     designated label as the linkage descriptor).


   For example, compiling the expression `5' (which is self-evaluating)
with a target of the `val' register and a linkage of `next' should
produce the instruction

     (assign val (const 5))

Compiling the same expression with a linkage of `return' should produce
the instructions

     (assign val (const 5))
     (goto (reg continue))

   In the first case, execution will continue with the next instruction
in the sequence. In the second case, we will return from a procedure
call.  In both cases, the value of the expression will be placed into
the target `val' register.

Instruction sequences and stack usage
.....................................

Each code generator returns an "instruction sequence" containing the
object code it has generated for the expression.  Code generation for a
compound expression is accomplished by combining the output from
simpler code generators for component expressions, just as evaluation
of a compound expression is accomplished by evaluating the component
expressions.

   The simplest method for combining instruction sequences is a
procedure called `append-instruction-sequences'.  It takes as arguments
any number of instruction sequences that are to be executed
sequentially; it appends them and returns the combined sequence.  That
is, if <SEQ_1> and <SEQ_2> are sequences of instructions, then
evaluating

     (append-instruction-sequences <SEQ_1> <SEQ_2>)

produces the sequence

     <SEQ_1>
     <SEQ_2>

   Whenever registers might need to be saved, the compiler's code
generators use `preserving', which is a more subtle method for
combining instruction sequences.  `Preserving' takes three arguments: a
set of registers and two instruction sequences that are to be executed
sequentially.  It appends the sequences in such a way that the contents
of each register in the set is preserved over the execution of the
first sequence, if this is needed for the execution of the second
sequence.  That is, if the first sequence modifies the register and the
second sequence actually needs the register's original contents, then
`preserving' wraps a `save' and a `restore' of the register around the
first sequence before appending the sequences.  Otherwise, `preserving'
simply returns the appended instruction sequences.  Thus, for example,

     (preserving (list <REG_1> <REG_2>) <SEQ_1> <SEQ_2>)

produces one of the following four sequences of instructions, depending
on how <SEQ_1> and <SEQ_2> use <REG_1> and <REG_2>:

     <seq_1> | (save <reg_1>)    | (save <reg_2>)    | (save <reg_2>)
     <seq_2> | <seq_1>           | <seq_1>           | (save <reg_1>)
             | (restore <reg_1>) | (restore <reg_2>) | <seq_1>
             | <seq_2>           | <seq_2>           | (restore <reg_1>)
             |                   |                   | (restore <reg_2>)
             |                   |                   | <seq_2>

   By using `preserving' to combine instruction sequences the compiler
avoids unnecessary stack operations.  This also isolates the details of
whether or not to generate `save' and `restore' instructions within the
`preserving' procedure, separating them from the concerns that arise in
writing each of the individual code generators.  In fact no `save' or
`restore' instructions are explicitly produced by the code generators.

   In principle, we could represent an instruction sequence simply as a
list of instructions.  `Append-instruction-sequences' could then combine
instruction sequences by performing an ordinary list `append'.  However,
`preserving' would then be a complex operation, because it would have to
analyze each instruction sequence to determine how the sequence uses its
registers.  `Preserving' would be inefficient as well as complex,
because it would have to analyze each of its instruction sequence
arguments, even though these sequences might themselves have been
constructed by calls to `preserving', in which case their parts would
have already been analyzed.  To avoid such repetitious analysis we will
associate with each instruction sequence some information about its
register use.  When we construct a basic instruction sequence we will
provide this information explicitly, and the procedures that combine
instruction sequences will derive register-use information for the
combined sequence from the information associated with the component
sequences.

   An instruction sequence will contain three pieces of information:

   * the set of registers that must be initialized before the
     instructions in the sequence are executed (these registers are
     said to be "needed" by the sequence),

   * the set of registers whose values are modified by the instructions
     in the sequence, and

   * the actual instructions (also called "statements") in the sequence.


   We will represent an instruction sequence as a list of its three
parts.  The constructor for instruction sequences is thus

     (define (make-instruction-sequence needs modifies statements)
       (list needs modifies statements))

   For example, the two-instruction sequence that looks up the value of
the variable `x' in the current environment, assigns the result to
`val', and then returns, requires registers `env' and `continue' to have
been initialized, and modifies register `val'.  This sequence would
therefore be constructed as

     (make-instruction-sequence '(env continue) '(val)
      '((assign val
                (op lookup-variable-value) (const x) (reg env))
        (goto (reg continue))))

   We sometimes need to construct an instruction sequence with no
statements:

     (define (empty-instruction-sequence)
       (make-instruction-sequence '() '() '()))

   The procedures for combining instruction sequences are shown in
section *Note 5-5-4::.

     *Exercise 5.31:* In evaluating a procedure application, the
     explicit-control evaluator always saves and restores the `env'
     register around the evaluation of the operator, saves and restores
     `env' around the evaluation of each operand (except the final
     one), saves and restores `argl' around the evaluation of each
     operand, and saves and restores `proc' around the evaluation of
     the operand sequence.  For each of the following combinations, say
     which of these `save' and `restore' operations are superfluous and
     thus could be eliminated by the compiler's `preserving' mechanism:

          (f 'x 'y)

          ((f) 'x 'y)

          (f (g 'x) y)

          (f (g 'x) 'y)

     *Exercise 5.32:* Using the `preserving' mechanism, the compiler
     will avoid saving and restoring `env' around the evaluation of the
     operator of a combination in the case where the operator is a
     symbol.  We could also build such optimizations into the
     evaluator.  Indeed, the explicit-control evaluator of section
     *Note 5-4:: already performs a similar optimization, by treating
     combinations with no operands as a special case.

       a. Extend the explicit-control evaluator to recognize as a
          separate class of expressions combinations whose operator is
          a symbol, and to take advantage of this fact in evaluating
          such expressions.

       b. Alyssa P. Hacker suggests that by extending the evaluator to
          recognize more and more special cases we could incorporate
          all the compiler's optimizations, and that this would
          eliminate the advantage of compilation altogether.  What do
          you think of this idea?


   ---------- Footnotes ----------

   (1) Notice, however, that our compiler is a Scheme program, and the
syntax procedures that it uses to manipulate expressions are the actual
Scheme procedures used with the metacircular evaluator.  For the
explicit-control evaluator, in contrast, we assumed that equivalent
syntax operations were available as operations for the register
machine.  (Of course, when we simulated the register machine in Scheme,
we used the actual Scheme procedures in our register machine
simulation.)


File: sicp.info,  Node: 5-5-2,  Next: 5-5-3,  Prev: 5-5-1,  Up: 5-5

5.5.2 Compiling Expressions
---------------------------

In this section and the next we implement the code generators to which
the `compile' procedure dispatches.

Compiling linkage code
......................

In general, the output of each code generator will end with
instructions--generated by the procedure `compile-linkage'--that
implement the required linkage.  If the linkage is `return' then we must
generate the instruction `(goto (reg continue))'.  This needs the
`continue' register and does not modify any registers.  If the linkage
is `next', then we needn't include any additional instructions.
Otherwise, the linkage is a label, and we generate a `goto' to that
label, an instruction that does not need or modify any registers.(1)

     (define (compile-linkage linkage)
       (cond ((eq? linkage 'return)
              (make-instruction-sequence '(continue) '()
               '((goto (reg continue)))))
             ((eq? linkage 'next)
              (empty-instruction-sequence))
             (else
              (make-instruction-sequence '() '()
               `((goto (label ,linkage)))))))

   The linkage code is appended to an instruction sequence by
`preserving' the `continue' register, since a `return' linkage will
require the `continue' register: If the given instruction sequence
modifies `continue' and the linkage code needs it, `continue' will be
saved and restored.

     (define (end-with-linkage linkage instruction-sequence)
       (preserving '(continue)
        instruction-sequence
        (compile-linkage linkage)))

Compiling simple expressions
............................

The code generators for self-evaluating expressions, quotations, and
variables construct instruction sequences that assign the required
value to the target register and then proceed as specified by the
linkage descriptor.

     (define (compile-self-evaluating exp target linkage)
       (end-with-linkage linkage
        (make-instruction-sequence '() (list target)
         `((assign ,target (const ,exp))))))

     (define (compile-quoted exp target linkage)
       (end-with-linkage linkage
        (make-instruction-sequence '() (list target)
         `((assign ,target (const ,(text-of-quotation exp)))))))

     (define (compile-variable exp target linkage)
       (end-with-linkage linkage
        (make-instruction-sequence '(env) (list target)
         `((assign ,target
                   (op lookup-variable-value)
                   (const ,exp)
                   (reg env))))))

   All these assignment instructions modify the target register, and
the one that looks up a variable needs the `env' register.

   Assignments and definitions are handled much as they are in the
interpreter.  We recursively generate code that computes the value to
be assigned to the variable, and append to it a two-instruction
sequence that actually sets or defines the variable and assigns the
value of the whole expression (the symbol `ok') to the target register.
The recursive compilation has target `val' and linkage `next' so that
the code will put its result into `val' and continue with the code that
is appended after it.  The appending is done preserving `env', since
the environment is needed for setting or defining the variable and the
code for the variable value could be the compilation of a complex
expression that might modify the registers in arbitrary ways.

     (define (compile-assignment exp target linkage)
       (let ((var (assignment-variable exp))
             (get-value-code
              (compile (assignment-value exp) 'val 'next)))
         (end-with-linkage linkage
          (preserving '(env)
           get-value-code
           (make-instruction-sequence '(env val) (list target)
            `((perform (op set-variable-value!)
                       (const ,var)
                       (reg val)
                       (reg env))
              (assign ,target (const ok))))))))

     (define (compile-definition exp target linkage)
       (let ((var (definition-variable exp))
             (get-value-code
              (compile (definition-value exp) 'val 'next)))
         (end-with-linkage linkage
          (preserving '(env)
           get-value-code
           (make-instruction-sequence '(env val) (list target)
            `((perform (op define-variable!)
                       (const ,var)
                       (reg val)
                       (reg env))
              (assign ,target (const ok))))))))

   The appended two-instruction sequence requires `env' and `val' and
modifies the target.  Note that although we preserve `env' for this
sequence, we do not preserve `val', because the `get-value-code' is
designed to explicitly place its result in `val' for use by this
sequence.  (In fact, if we did preserve `val', we would have a bug,
because this would cause the previous contents of `val' to be restored
right after the `get-value-code' is run.)

Compiling conditional expressions
.................................

The code for an `if' expression compiled with a given target and linkage
has the form

     <_compilation of predicate, target `val', linkage `next'_>
      (test (op false?) (reg val))
      (branch (label false-branch))
     true-branch
      <_compilation of consequent with given target and given linkage or `after-if'_>
     false-branch
      <_compilation of alternative with given target and linkage_>
     after-if

   To generate this code, we compile the predicate, consequent, and
alternative, and combine the resulting code with instructions to test
the predicate result and with newly generated labels to mark the true
and false branches and the end of the conditional.(2) In this
arrangement of code, we must branch around the true branch if the test
is false.  The only slight complication is in how the linkage for the
true branch should be handled.  If the linkage for the conditional is
`return' or a label, then the true and false branches will both use
this same linkage.  If the linkage is `next', the true branch ends with
a jump around the code for the false branch to the label at the end of
the conditional.

     (define (compile-if exp target linkage)
       (let ((t-branch (make-label 'true-branch))
             (f-branch (make-label 'false-branch))
             (after-if (make-label 'after-if)))
         (let ((consequent-linkage
                (if (eq? linkage 'next) after-if linkage)))
           (let ((p-code (compile (if-predicate exp) 'val 'next))
                 (c-code
                  (compile
                   (if-consequent exp) target consequent-linkage))
                 (a-code
                  (compile (if-alternative exp) target linkage)))
             (preserving '(env continue)
              p-code
              (append-instruction-sequences
               (make-instruction-sequence '(val) '()
                `((test (op false?) (reg val))
                  (branch (label ,f-branch))))
               (parallel-instruction-sequences
                (append-instruction-sequences t-branch c-code)
                (append-instruction-sequences f-branch a-code))
               after-if))))))

   `Env' is preserved around the predicate code because it could be
needed by the true and false branches, and `continue' is preserved
because it could be needed by the linkage code in those branches.  The
code for the true and false branches (which are not executed
sequentially) is appended using a special combiner
`parallel-instruction-sequences' described in section *Note 5-5-4::.

   Note that `cond' is a derived expression, so all that the compiler
needs to do handle it is to apply the `cond->if' transformer (from
section *Note 4-1-2::) and compile the resulting `if' expression.

Compiling sequences
...................

The compilation of sequences (from procedure bodies or explicit `begin'
expressions) parallels their evaluation.  Each expression of the
sequence is compiled--the last expression with the linkage specified
for the sequence, and the other expressions with linkage `next' (to
execute the rest of the sequence).  The instruction sequences for the
individual expressions are appended to form a single instruction
sequence, such that `env' (needed for the rest of the sequence) and
`continue' (possibly needed for the linkage at the end of the sequence)
are preserved.

     (define (compile-sequence seq target linkage)
       (if (last-exp? seq)
           (compile (first-exp seq) target linkage)
           (preserving '(env continue)
            (compile (first-exp seq) target 'next)
            (compile-sequence (rest-exps seq) target linkage))))

Compiling `lambda' expressions
..............................

`Lambda' expressions construct procedures.  The object code for a
`lambda' expression must have the form

     <_construct procedure object and assign it to target register_>
     <LINKAGE>

   When we compile the `lambda' expression, we also generate the code
for the procedure body.  Although the body won't be executed at the
time of procedure construction, it is convenient to insert it into the
object code right after the code for the `lambda'.  If the linkage for
the `lambda' expression is a label or `return', this is fine.  But if
the linkage is `next', we will need to skip around the code for the
procedure body by using a linkage that jumps to a label that is
inserted after the body.  The object code thus has the form

     <_construct procedure object and assign it to target register_>
      <_code for given linkage_> _or_ `(goto (label after-lambda))'
      <_compilation of procedure body_>
     after-lambda

   `Compile-lambda' generates the code for constructing the procedure
object followed by the code for the procedure body.  The procedure
object will be constructed at run time by combining the current
environment (the environment at the point of definition) with the entry
point to the compiled procedure body (a newly generated label).(3)

     (define (compile-lambda exp target linkage)
       (let ((proc-entry (make-label 'entry))
             (after-lambda (make-label 'after-lambda)))
         (let ((lambda-linkage
                (if (eq? linkage 'next) after-lambda linkage)))
           (append-instruction-sequences
            (tack-on-instruction-sequence
             (end-with-linkage lambda-linkage
              (make-instruction-sequence '(env) (list target)
               `((assign ,target
                         (op make-compiled-procedure)
                         (label ,proc-entry)
                         (reg env)))))
             (compile-lambda-body exp proc-entry))
            after-lambda))))

   `Compile-lambda' uses the special combiner
`tack-on-instruction-sequence' (section *Note 5-5-4::) rather than
`append-instruction-sequences' to append the procedure body to the
`lambda' expression code, because the body is not part of the sequence
of instructions that will be executed when the combined sequence is
entered; rather, it is in the sequence only because that was a
convenient place to put it.

   `Compile-lambda-body' constructs the code for the body of the
procedure.  This code begins with a label for the entry point.  Next
come instructions that will cause the run-time evaluation environment
to switch to the correct environment for evaluating the procedure
body--namely, the definition environment of the procedure, extended to
include the bindings of the formal parameters to the arguments with
which the procedure is called.  After this comes the code for the
sequence of expressions that makes up the procedure body.  The sequence
is compiled with linkage `return' and target `val' so that it will end
by returning from the procedure with the procedure result in `val'.

     (define (compile-lambda-body exp proc-entry)
       (let ((formals (lambda-parameters exp)))
         (append-instruction-sequences
          (make-instruction-sequence '(env proc argl) '(env)
           `(,proc-entry
             (assign env (op compiled-procedure-env) (reg proc))
             (assign env
                     (op extend-environment)
                     (const ,formals)
                     (reg argl)
                     (reg env))))
          (compile-sequence (lambda-body exp) 'val 'return))))

   ---------- Footnotes ----------

   (1) This procedure uses a feature of Lisp called "backquote" (or "quasiquote")
that is handy for constructing lists.  Preceding a list with a
backquote symbol is much like quoting it, except that anything in the
list that is flagged with a comma is evaluated.

   For example, if the value of `linkage' is the symbol
`branch25', then the expression
``((goto (label ,linkage)))'
evaluates to the list
`((goto (label branch25)))'.
Similarly, if the value of `x' is the list `(a b c)', then
``(1 2 ,(car x))'
evaluates to the list
`(1 2 a)'.

   (2) We can't just use the labels `true-branch', `false-branch', and
`after-if' as shown above, because there might be more than one `if' in
the program.  The compiler uses the procedure `make-label' to generate
labels.  `Make-label' takes a symbol as argument and returns a new
symbol that begins with the given symbol.  For example, successive
calls to `(make-label 'a)' would return `a1', `a2', and so on.
`Make-label' can be implemented similarly to the generation of unique
variable names in the query language, as follows:

     (define label-counter 0)

     (define (new-label-number)
       (set! label-counter (+ 1 label-counter))
       label-counter)

     (define (make-label name)
       (string->symbol
         (string-append (symbol->string name)
                        (number->string (new-label-number)))))

   (3) [Footnote 38] We need machine operations to implement a data
structure for representing compiled procedures, analogous to the
structure for compound procedures described in section *Note 4-1-3:::

     (define (make-compiled-procedure entry env)
       (list 'compiled-procedure entry env))

     (define (compiled-procedure? proc)
       (tagged-list? proc 'compiled-procedure))

     (define (compiled-procedure-entry c-proc) (cadr c-proc))

     (define (compiled-procedure-env c-proc) (caddr c-proc))


File: sicp.info,  Node: 5-5-3,  Next: 5-5-4,  Prev: 5-5-2,  Up: 5-5

5.5.3 Compiling Combinations
----------------------------

The essence of the compilation process is the compilation of procedure
applications.  The code for a combination compiled with a given target
and linkage has the form

     <_compilation of operator, target `proc', linkage `next'_>
     <_evaluate operands and construct argument list in `argl'_>
     <_compilation of procedure call with given target and linkage_>

   The registers `env', `proc', and `argl' may have to be saved and
restored during evaluation of the operator and operands.  Note that
this is the only place in the compiler where a target other than `val'
is specified.

   The required code is generated by `compile-application'.  This
recursively compiles the operator, to produce code that puts the
procedure to be applied into `proc', and compiles the operands, to
produce code that evaluates the individual operands of the application.
The instruction sequences for the operands are combined (by
`construct-arglist') with code that constructs the list of arguments in
`argl', and the resulting argument-list code is combined with the
procedure code and the code that performs the procedure call (produced
by `compile-procedure-call').  In appending the code sequences, the
`env' register must be preserved around the evaluation of the operator
(since evaluating the operator might modify `env', which will be needed
to evaluate the operands), and the `proc' register must be preserved
around the construction of the argument list (since evaluating the
operands might modify `proc', which will be needed for the actual
procedure application).  `Continue' must also be preserved throughout,
since it is needed for the linkage in the procedure call.

     (define (compile-application exp target linkage)
       (let ((proc-code (compile (operator exp) 'proc 'next))
             (operand-codes
              (map (lambda (operand) (compile operand 'val 'next))
                   (operands exp))))
         (preserving '(env continue)
          proc-code
          (preserving '(proc continue)
           (construct-arglist operand-codes)
           (compile-procedure-call target linkage)))))

   The code to construct the argument list will evaluate each operand
into `val' and then `cons' that value onto the argument list being
accumulated in `argl'.  Since we `cons' the arguments onto `argl' in
sequence, we must start with the last argument and end with the first,
so that the arguments will appear in order from first to last in the
resulting list.  Rather than waste an instruction by initializing `argl'
to the empty list to set up for this sequence of evaluations, we make
the first code sequence construct the initial `argl'.  The general form
of the argument-list construction is thus as follows:

     <_compilation of last operand, targeted to `val'_>
     (assign argl (op list) (reg val))
     <_compilation of next operand, targeted to `val'_>
     (assign argl (op cons) (reg val) (reg argl))
     ...
     <_compilation of first operand, targeted to `val'_>
     (assign argl (op cons) (reg val) (reg argl))

   `Argl' must be preserved around each operand evaluation except the
first (so that arguments accumulated so far won't be lost), and `env'
must be preserved around each operand evaluation except the last (for
use by subsequent operand evaluations).

   Compiling this argument code is a bit tricky, because of the special
treatment of the first operand to be evaluated and the need to preserve
`argl' and `env' in different places.  The `construct-arglist'
procedure takes as arguments the code that evaluates the individual
operands.  If there are no operands at all, it simply emits the
instruction

     (assign argl (const ()))

   Otherwise, `construct-arglist' creates code that initializes `argl'
with the last argument, and appends code that evaluates the rest of the
arguments and adjoins them to `argl' in succession.  In order to process
the arguments from last to first, we must reverse the list of operand
code sequences from the order supplied by `compile-application'.

     (define (construct-arglist operand-codes)
       (let ((operand-codes (reverse operand-codes)))
         (if (null? operand-codes)
             (make-instruction-sequence '() '(argl)
              '((assign argl (const ()))))
             (let ((code-to-get-last-arg
                    (append-instruction-sequences
                     (car operand-codes)
                     (make-instruction-sequence '(val) '(argl)
                      '((assign argl (op list) (reg val)))))))
               (if (null? (cdr operand-codes))
                   code-to-get-last-arg
                   (preserving '(env)
                    code-to-get-last-arg
                    (code-to-get-rest-args
                     (cdr operand-codes))))))))

     (define (code-to-get-rest-args operand-codes)
       (let ((code-for-next-arg
              (preserving '(argl)
               (car operand-codes)
               (make-instruction-sequence '(val argl) '(argl)
                '((assign argl
                   (op cons) (reg val) (reg argl)))))))
         (if (null? (cdr operand-codes))
             code-for-next-arg
             (preserving '(env)
              code-for-next-arg
              (code-to-get-rest-args (cdr operand-codes))))))

Applying procedures
...................

After evaluating the elements of a combination, the compiled code must
apply the procedure in `proc' to the arguments in `argl'.  The code
performs essentially the same dispatch as the `apply' procedure in the
metacircular evaluator of section *Note 4-1-1:: or the `apply-dispatch'
entry point in the explicit-control evaluator of section *Note 5-4-1::.
It checks whether the procedure to be applied is a primitive procedure
or a compiled procedure.  For a primitive procedure, it uses
`apply-primitive-procedure'; we will see shortly how it handles compiled
procedures.  The procedure-application code has the following form:

     (test (op primitive-procedure?) (reg proc))
      (branch (label primitive-branch))
     compiled-branch
      <_code to apply compiled procedure with given target and appropriate linkage_>
     primitive-branch
      (assign <TARGET>
              (op apply-primitive-procedure)
              (reg proc)
              (reg argl))
      <LINKAGE>
     after-call

   Observe that the compiled branch must skip around the primitive
branch.  Therefore, if the linkage for the original procedure call was
`next', the compound branch must use a linkage that jumps to a label
that is inserted after the primitive branch.  (This is similar to the
linkage used for the true branch in `compile-if'.)

     (define (compile-procedure-call target linkage)
       (let ((primitive-branch (make-label 'primitive-branch))
             (compiled-branch (make-label 'compiled-branch))
             (after-call (make-label 'after-call)))
         (let ((compiled-linkage
                (if (eq? linkage 'next) after-call linkage)))
           (append-instruction-sequences
            (make-instruction-sequence '(proc) '()
             `((test (op primitive-procedure?) (reg proc))
               (branch (label ,primitive-branch))))
            (parallel-instruction-sequences
             (append-instruction-sequences
              compiled-branch
              (compile-proc-appl target compiled-linkage))
             (append-instruction-sequences
              primitive-branch
              (end-with-linkage linkage
               (make-instruction-sequence '(proc argl)
                                          (list target)
                `((assign ,target
                          (op apply-primitive-procedure)
                          (reg proc)
                          (reg argl)))))))
            after-call))))

   The primitive and compound branches, like the true and false
branches in `compile-if', are appended using
`parallel-instruction-sequences' rather than the ordinary
`append-instruction-sequences', because they will not be executed
sequentially.

Applying compiled procedures
............................

The code that handles procedure application is the most subtle part of
the compiler, even though the instruction sequences it generates are
very short.  A compiled procedure (as constructed by `compile-lambda')
has an entry point, which is a label that designates where the code for
the procedure starts.  The code at this entry point computes a result
in `val' and returns by executing the instruction `(goto (reg
continue))'.  Thus, we might expect the code for a compiled-procedure
application (to be generated by `compile-proc-appl') with a given
target and linkage to look like this if the linkage is a label

     (assign continue (label proc-return))
      (assign val (op compiled-procedure-entry) (reg proc))
      (goto (reg val))
     proc-return
      (assign <TARGET> (reg val))   ; included if target is not `val'
      (goto (label <LINKAGE>))   ; linkage code

or like this if the linkage is `return'.

     (save continue)
      (assign continue (label proc-return))
      (assign val (op compiled-procedure-entry) (reg proc))
      (goto (reg val))
     proc-return
      (assign <TARGET> (reg val))   ; included if target is not `val'
      (restore continue)
      (goto (reg continue))   ; linkage code

   This code sets up `continue' so that the procedure will return to a
label `proc-return' and jumps to the procedure's entry point.  The code
at `proc-return' transfers the procedure's result from `val' to the
target register (if necessary) and then jumps to the location specified
by the linkage.  (The linkage is always `return' or a label, because
`compile-procedure-call' replaces a `next' linkage for the
compound-procedure branch by an `after-call' label.)

   In fact, if the target is not `val', that is exactly the code our
compiler will generate.(1)  Usually, however, the target is `val' (the
only time the compiler specifies a different register is when targeting
the evaluation of an operator to `proc'), so the procedure result is
put directly into the target register and there is no need to return to
a special location that copies it.  Instead, we simplify the code by
setting up `continue' so that the procedure will "return" directly to
the place specified by the caller's linkage:

     <_set up `continue' for linkage_>
     (assign val (op compiled-procedure-entry) (reg proc))
     (goto (reg val))

   If the linkage is a label, we set up `continue' so that the
procedure will return to that label.  (That is, the `(goto (reg
continue))' the procedure ends with becomes equivalent to the `(goto
(label <LINKAGE>))' at `proc-return' above.)

     (assign continue (label <LINKAGE>))
     (assign val (op compiled-procedure-entry) (reg proc))
     (goto (reg val))

   If the linkage is `return', we don't need to set up `continue' at
all: It already holds the desired location.  (That is, the `(goto (reg
continue))' the procedure ends with goes directly to the place where the
`(goto (reg continue))' at `proc-return' would have gone.)

     (assign val (op compiled-procedure-entry) (reg proc))
     (goto (reg val))

   With this implementation of the `return' linkage, the compiler
generates tail-recursive code.  Calling a procedure as the final step
in a procedure body does a direct transfer, without saving any
information on the stack.

   Suppose instead that we had handled the case of a procedure call
with a linkage of `return' and a target of `val' as shown above for a
non-`val' target.  This would destroy tail recursion.  Our system would
still give the same value for any expression.  But each time we called
a procedure, we would save `continue' and return after the call to undo
the (useless) save.  These extra saves would accumulate during a nest
of procedure calls.(2)

   `Compile-proc-appl' generates the above procedure-application code by
considering four cases, depending on whether the target for the call is
`val' and whether the linkage is `return'.  Observe that the
instruction sequences are declared to modify all the registers, since
executing the procedure body can change the registers in arbitrary
ways.(3) Also note that the code sequence for the case with target
`val' and linkage `return' is declared to need `continue': Even though
`continue' is not explicitly used in the two-instruction sequence, we
must be sure that `continue' will have the correct value when we enter
the compiled procedure.

     (define (compile-proc-appl target linkage)
       (cond ((and (eq? target 'val) (not (eq? linkage 'return)))
              (make-instruction-sequence '(proc) all-regs
                `((assign continue (label ,linkage))
                  (assign val (op compiled-procedure-entry)
                              (reg proc))
                  (goto (reg val)))))
             ((and (not (eq? target 'val))
                   (not (eq? linkage 'return)))
              (let ((proc-return (make-label 'proc-return)))
                (make-instruction-sequence '(proc) all-regs
                 `((assign continue (label ,proc-return))
                   (assign val (op compiled-procedure-entry)
                               (reg proc))
                   (goto (reg val))
                   ,proc-return
                   (assign ,target (reg val))
                   (goto (label ,linkage))))))
             ((and (eq? target 'val) (eq? linkage 'return))
              (make-instruction-sequence '(proc continue) all-regs
               '((assign val (op compiled-procedure-entry)
                             (reg proc))
                 (goto (reg val)))))
             ((and (not (eq? target 'val)) (eq? linkage 'return))
              (error "return linkage, target not val -- COMPILE"
                     target))))

   ---------- Footnotes ----------

   (1) Actually, we signal an error when the target is not `val' and
the linkage is `return', since the only place we request `return'
linkages is in compiling procedures, and our convention is that
procedures return their values in `val'.

   (2) Making a compiler generate tail-recursive code might seem like a
straightforward idea.  But most compilers for common languages,
including C and Pascal, do not do this, and therefore these languages
cannot represent iterative processes in terms of procedure call alone.
The difficulty with tail recursion in these languages is that their
implementations use the stack to store procedure arguments and local
variables as well as return addresses.  The Scheme implementations
described in this book store arguments and variables in memory to be
garbage-collected.  The reason for using the stack for variables and
arguments is that it avoids the need for garbage collection in languages
that would not otherwise require it, and is generally believed to be
more efficient.  Sophisticated Lisp compilers can, in fact, use the
stack for arguments without destroying tail recursion.  (See Hanson
1990 for a description.)  There is also some debate about whether stack
allocation is actually more efficient than garbage collection in the
first place, but the details seem to hinge on fine points of computer
architecture.  (See Appel 1987 and Miller and Rozas 1994 for opposing
views on this issue.)

   (3) The variable `all-regs' is bound to the list of names of all the
registers:

     (define all-regs '(env proc val argl continue))


File: sicp.info,  Node: 5-5-4,  Next: 5-5-5,  Prev: 5-5-3,  Up: 5-5

5.5.4 Combining Instruction Sequences
-------------------------------------

This section describes the details on how instruction sequences are
represented and combined.  Recall from section *Note 5-5-1:: that an
instruction sequence is represented as a list of the registers needed,
the registers modified, and the actual instructions.  We will also
consider a label (symbol) to be a degenerate case of an instruction
sequence, which doesn't need or modify any registers.  So to determine
the registers needed and modified by instruction sequences we use the
selectors

     (define (registers-needed s)
       (if (symbol? s) '() (car s)))

     (define (registers-modified s)
       (if (symbol? s) '() (cadr s)))

     (define (statements s)
       (if (symbol? s) (list s) (caddr s)))

and to determine whether a given sequence needs or modifies a given
register we use the predicates

     (define (needs-register? seq reg)
       (memq reg (registers-needed seq)))

     (define (modifies-register? seq reg)
       (memq reg (registers-modified seq)))

   In terms of these predicates and selectors, we can implement the
various instruction sequence combiners used throughout the compiler.

   The basic combiner is `append-instruction-sequences'.  This takes as
arguments an arbitrary number of instruction sequences that are to be
executed sequentially and returns an instruction sequence whose
statements are the statements of all the sequences appended together.
The subtle point is to determine the registers that are needed and
modified by the resulting sequence.  It modifies those registers that
are modified by any of the sequences; it needs those registers that
must be initialized before the first sequence can be run (the registers
needed by the first sequence), together with those registers needed by
any of the other sequences that are not initialized (modified) by
sequences preceding it.

   The sequences are appended two at a time by `append-2-sequences'.
This takes two instruction sequences `seq1' and `seq2' and returns the
instruction sequence whose statements are the statements of `seq1'
followed by the statements of `seq2', whose modified registers are those
registers that are modified by either `seq1' or `seq2', and whose
needed registers are the registers needed by `seq1' together with those
registers needed by `seq2' that are not modified by `seq1'.  (In terms
of set operations, the new set of needed registers is the union of the
set of registers needed by `seq1' with the set difference of the
registers needed by `seq2' and the registers modified by `seq1'.)  Thus,
`append-instruction-sequences' is implemented as follows:

     (define (append-instruction-sequences . seqs)
       (define (append-2-sequences seq1 seq2)
         (make-instruction-sequence
          (list-union (registers-needed seq1)
                      (list-difference (registers-needed seq2)
                                       (registers-modified seq1)))
          (list-union (registers-modified seq1)
                      (registers-modified seq2))
          (append (statements seq1) (statements seq2))))
       (define (append-seq-list seqs)
         (if (null? seqs)
             (empty-instruction-sequence)
             (append-2-sequences (car seqs)
                                 (append-seq-list (cdr seqs)))))
       (append-seq-list seqs))

   This procedure uses some simple operations for manipulating sets
represented as lists, similar to the (unordered) set representation
described in section *Note 2-3-3:::

     (define (list-union s1 s2)
       (cond ((null? s1) s2)
             ((memq (car s1) s2) (list-union (cdr s1) s2))
             (else (cons (car s1) (list-union (cdr s1) s2)))))

     (define (list-difference s1 s2)
       (cond ((null? s1) '())
             ((memq (car s1) s2) (list-difference (cdr s1) s2))
             (else (cons (car s1)
                         (list-difference (cdr s1) s2)))))

   `Preserving', the second major instruction sequence combiner, takes
a list of registers `regs' and two instruction sequences `seq1' and
`seq2' that are to be executed sequentially.  It returns an instruction
sequence whose statements are the statements of `seq1' followed by the
statements of `seq2', with appropriate `save' and `restore'
instructions around `seq1' to protect the registers in `regs' that are
modified by `seq1' but needed by `seq2'.  To accomplish this,
`preserving' first creates a sequence that has the required `save's
followed by the statements of `seq1' followed by the required
`restore's.  This sequence needs the registers being saved and restored
in addition to the registers needed by `seq1', and modifies the
registers modified by `seq1' except for the ones being saved and
restored.  This augmented sequence and `seq2' are then appended in the
usual way.  The following procedure implements this strategy
recursively, walking down the list of registers to be preserved:(1)

     (define (preserving regs seq1 seq2)
       (if (null? regs)
           (append-instruction-sequences seq1 seq2)
           (let ((first-reg (car regs)))
             (if (and (needs-register? seq2 first-reg)
                      (modifies-register? seq1 first-reg))
                 (preserving (cdr regs)
                  (make-instruction-sequence
                   (list-union (list first-reg)
                               (registers-needed seq1))
                   (list-difference (registers-modified seq1)
                                    (list first-reg))
                   (append `((save ,first-reg))
                           (statements seq1)
                           `((restore ,first-reg))))
                  seq2)
                 (preserving (cdr regs) seq1 seq2)))))

   Another sequence combiner, `tack-on-instruction-sequence', is used by
`compile-lambda' to append a procedure body to another sequence.
Because the procedure body is not "in line" to be executed as part of
the combined sequence, its register use has no impact on the register
use of the sequence in which it is embedded.  We thus ignore the
procedure body's sets of needed and modified registers when we tack it
onto the other sequence.

     (define (tack-on-instruction-sequence seq body-seq)
       (make-instruction-sequence
        (registers-needed seq)
        (registers-modified seq)
        (append (statements seq) (statements body-seq))))

   `Compile-if' and `compile-procedure-call' use a special combiner
called `parallel-instruction-sequences' to append the two alternative
branches that follow a test.  The two branches will never be executed
sequentially; for any particular evaluation of the test, one branch or
the other will be entered.  Because of this, the registers needed by
the second branch are still needed by the combined sequence, even if
these are modified by the first branch.

     (define (parallel-instruction-sequences seq1 seq2)
       (make-instruction-sequence
        (list-union (registers-needed seq1)
                    (registers-needed seq2))
        (list-union (registers-modified seq1)
                    (registers-modified seq2))
        (append (statements seq1) (statements seq2))))

   ---------- Footnotes ----------

   (1) Note that `preserving' calls `append' with three arguments.
Though the definition of `append' shown in this book accepts only two
arguments, Scheme standardly provides an `append' procedure that takes
an arbitrary number of arguments.


File: sicp.info,  Node: 5-5-5,  Next: 5-5-6,  Prev: 5-5-4,  Up: 5-5

5.5.5 An Example of Compiled Code
---------------------------------

Now that we have seen all the elements of the compiler, let us examine
an example of compiled code to see how things fit together.  We will
compile the definition of a recursive `factorial' procedure by calling
`compile':

     (compile
      '(define (factorial n)
         (if (= n 1)
             1
             (* (factorial (- n 1)) n)))
      'val
      'next)

   We have specified that the value of the `define' expression should be
placed in the `val' register.  We don't care what the compiled code does
after executing the `define', so our choice of `next' as the linkage
descriptor is arbitrary.

   `Compile' determines that the expression is a definition, so it calls
`compile-definition' to compile code to compute the value to be assigned
(targeted to `val'), followed by code to install the definition,
followed by code to put the value of the `define' (which is the symbol
`ok') into the target register, followed finally by the linkage code.
`Env' is preserved around the computation of the value, because it is
needed in order to install the definition.  Because the linkage is
`next', there is no linkage code in this case.  The skeleton of the
compiled code is thus

     <_save `env' if modified by code to compute value_>
       <_compilation of definition value, target `val', linkage `next'_>
       <_restore `env' if saved above_>
       (perform (op define-variable!)
                (const factorial)
                (reg val)
                (reg env))
       (assign val (const ok))

   The expression that is to be compiled to produce the value for the
variable `factorial' is a `lambda' expression whose value is the
procedure that computes factorials.  `Compile' handles this by calling
`compile-lambda', which compiles the procedure body, labels it as a new
entry point, and generates the instruction that will combine the
procedure body at the new entry point with the run-time environment and
assign the result to `val'.  The sequence then skips around the
compiled procedure code, which is inserted at this point.  The
procedure code itself begins by extending the procedure's definition
environment by a frame that binds the formal parameter `n' to the
procedure argument.  Then comes the actual procedure body.  Since this
code for the value of the variable doesn't modify the `env' register,
the optional `save' and `restore' shown above aren't generated.  (The
procedure code at `entry2' isn't executed at this point, so its use of
`env' is irrelevant.)  Therefore, the skeleton for the compiled code
becomes

     (assign val (op make-compiled-procedure)
                   (label entry2)
                   (reg env))
       (goto (label after-lambda1))
     entry2
       (assign env (op compiled-procedure-env) (reg proc))
       (assign env (op extend-environment)
                   (const (n))
                   (reg argl)
                   (reg env))
       <_compilation of procedure body_>
     after-lambda1
       (perform (op define-variable!)
                (const factorial)
                (reg val)
                (reg env))
       (assign val (const ok))

   A procedure body is always compiled (by `compile-lambda-body') as a
sequence with target `val' and linkage `return'.  The sequence in this
case consists of a single `if' expression:

     (if (= n 1)
         1
         (* (factorial (- n 1)) n))

   `Compile-if' generates code that first computes the predicate
(targeted to `val'), then checks the result and branches around the
true branch if the predicate is false.  `Env' and `continue' are
preserved around the predicate code, since they may be needed for the
rest of the `if' expression.  Since the `if' expression is the final
expression (and only expression) in the sequence making up the
procedure body, its target is `val' and its linkage is `return', so the
true and false branches are both compiled with target `val' and linkage
`return'.  (That is, the value of the conditional, which is the value
computed by either of its branches, is the value of the procedure.)

     <_save `continue', `env' if modified by predicate and needed by branches_>
       <_compilation of predicate, target `val', linkage `next'_>
       <_restore `continue', `env' if saved above_>
       (test (op false?) (reg val))
       (branch (label false-branch4))
     true-branch5
       <_compilation of true branch, target `val', linkage `return'_>
     false-branch4
       <_compilation of false branch, target `val', linkage `return'_>
     after-if3

   The predicate `(= n 1)' is a procedure call.  This looks up the
operator (the symbol `=') and places this value in `proc'.  It then
assembles the arguments `1' and the value of `n' into `argl'.  Then it
tests whether `proc' contains a primitive or a compound procedure, and
dispatches to a primitive branch or a compound branch accordingly.  Both
branches resume at the `after-call' label.  The requirements to preserve
registers around the evaluation of the operator and operands don't
result in any saving of registers, because in this case those
evaluations don't modify the registers in question.

     (assign proc
               (op lookup-variable-value) (const =) (reg env))
       (assign val (const 1))
       (assign argl (op list) (reg val))
       (assign val (op lookup-variable-value) (const n) (reg env))
       (assign argl (op cons) (reg val) (reg argl))
       (test (op primitive-procedure?) (reg proc))
       (branch (label primitive-branch17))
     compiled-branch16
       (assign continue (label after-call15))
       (assign val (op compiled-procedure-entry) (reg proc))
       (goto (reg val))
     primitive-branch17
       (assign val (op apply-primitive-procedure)
                   (reg proc)
                   (reg argl))
     after-call15

   The true branch, which is the constant 1, compiles (with target
`val' and linkage `return') to

     (assign val (const 1))
       (goto (reg continue))

   The code for the false branch is another a procedure call, where the
procedure is the value of the symbol `*', and the arguments are `n' and
the result of another procedure call (a call to `factorial').  Each of
these calls sets up `proc' and `argl' and its own primitive and compound
branches.  *Note Figure 5-17:: shows the complete compilation of the
definition of the `factorial' procedure.  Notice that the possible
`save' and `restore' of `continue' and `env' around the predicate, shown
above, are in fact generated, because these registers are modified by
the procedure call in the predicate and needed for the procedure call
and the `return' linkage in the branches.

     *Exercise 5.33:* Consider the following definition of a factorial
     procedure, which is slightly different from the one given above:

          (define (factorial-alt n)
            (if (= n 1)
                1
                (* n (factorial-alt (- n 1)))))

     Compile this procedure and compare the resulting code with that
     produced for `factorial'.  Explain any differences you find.  Does
     either program execute more efficiently than the other?

     *Exercise 5.34:* Compile the iterative factorial procedure

          (define (factorial n)
            (define (iter product counter)
              (if (> counter n)
                  product
                  (iter (* counter product)
                        (+ counter 1))))
            (iter 1 1))

     Annotate the resulting code, showing the essential difference
     between the code for iterative and recursive versions of
     `factorial' that makes one process build up stack space and the
     other run in constant stack space.

     *Figure 5.17:* Compilation of the definition of the `factorial'
     procedure

          ;; construct the procedure and skip over code for the procedure body
            (assign val
                    (op make-compiled-procedure) (label entry2) (reg env))
            (goto (label after-lambda1))

          entry2     ; calls to `factorial' will enter here
            (assign env (op compiled-procedure-env) (reg proc))
            (assign env
                    (op extend-environment) (const (n)) (reg argl) (reg env))
          ;; begin actual procedure body
            (save continue)
            (save env)

          ;; compute `(= n 1)'
            (assign proc (op lookup-variable-value) (const =) (reg env))
            (assign val (const 1))
            (assign argl (op list) (reg val))
            (assign val (op lookup-variable-value) (const n) (reg env))
            (assign argl (op cons) (reg val) (reg argl))
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch17))
          compiled-branch16
            (assign continue (label after-call15))
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch17
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))

          after-call15   ; `val' now contains result of `(= n 1)'
            (restore env)
            (restore continue)
            (test (op false?) (reg val))
            (branch (label false-branch4))
          true-branch5  ; return 1
            (assign val (const 1))
            (goto (reg continue))

          false-branch4
          ;; compute and return `(* (factorial (- n 1)) n)'
            (assign proc (op lookup-variable-value) (const *) (reg env))
            (save continue)
            (save proc)   ; save `*' procedure
            (assign val (op lookup-variable-value) (const n) (reg env))
            (assign argl (op list) (reg val))
            (save argl)   ; save partial argument list for `*'

          ;; compute `(factorial (- n 1))', which is the other argument for `*'
            (assign proc
                    (op lookup-variable-value) (const factorial) (reg env))
            (save proc)  ; save `factorial' procedure

          ;; compute `(- n 1)', which is the argument for `factorial'
            (assign proc (op lookup-variable-value) (const -) (reg env))
            (assign val (const 1))
            (assign argl (op list) (reg val))
            (assign val (op lookup-variable-value) (const n) (reg env))
            (assign argl (op cons) (reg val) (reg argl))
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch8))
          compiled-branch7
            (assign continue (label after-call6))
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch8
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))

          after-call6   ; `val' now contains result of `(- n 1)'
            (assign argl (op list) (reg val))
            (restore proc) ; restore `factorial'
          ;; apply `factorial'
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch11))
          compiled-branch10
            (assign continue (label after-call9))
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch11
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))

          after-call9      ; `val' now contains result of `(factorial (- n 1))'
            (restore argl) ; restore partial argument list for `*'
            (assign argl (op cons) (reg val) (reg argl))
            (restore proc) ; restore `*'
            (restore continue)
          ;; apply `*' and return its value
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch14))
          compiled-branch13
          ;; note that a compound procedure here is called tail-recursively
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch14
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
            (goto (reg continue))
          after-call12
          after-if3
          after-lambda1
          ;; assign the procedure to the variable `factorial'
            (perform
             (op define-variable!) (const factorial) (reg val) (reg env))
            (assign val (const ok))


     *Exercise 5.35:* What expression was compiled to produce the code
     shown in *Note Figure 5-18::?

     *Figure 5.18:* An example of compiler output.  See *Note Exercise
     5-35::.

          (assign val (op make-compiled-procedure) (label entry16)
                                                     (reg env))
            (goto (label after-lambda15))
          entry16
            (assign env (op compiled-procedure-env) (reg proc))
            (assign env
                    (op extend-environment) (const (x)) (reg argl) (reg env))
            (assign proc (op lookup-variable-value) (const +) (reg env))
            (save continue)
            (save proc)
            (save env)
            (assign proc (op lookup-variable-value) (const g) (reg env))
            (save proc)
            (assign proc (op lookup-variable-value) (const +) (reg env))
            (assign val (const 2))
            (assign argl (op list) (reg val))
            (assign val (op lookup-variable-value) (const x) (reg env))
            (assign argl (op cons) (reg val) (reg argl))
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch19))
          compiled-branch18
            (assign continue (label after-call17))
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch19
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
          after-call17
            (assign argl (op list) (reg val))
            (restore proc)
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch22))
          compiled-branch21
            (assign continue (label after-call20))
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch22
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))

          after-call20
            (assign argl (op list) (reg val))
            (restore env)
            (assign val (op lookup-variable-value) (const x) (reg env))
            (assign argl (op cons) (reg val) (reg argl))
            (restore proc)
            (restore continue)
            (test (op primitive-procedure?) (reg proc))
            (branch (label primitive-branch25))
          compiled-branch24
            (assign val (op compiled-procedure-entry) (reg proc))
            (goto (reg val))
          primitive-branch25
            (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
            (goto (reg continue))
          after-call23
          after-lambda15
            (perform (op define-variable!) (const f) (reg val) (reg env))
            (assign val (const ok))


     *Exercise 5.36:* What order of evaluation does our compiler
     produce for operands of a combination?  Is it left-to-right,
     right-to-left, or some other order?  Where in the compiler is this
     order determined?  Modify the compiler so that it produces some
     other order of evaluation.  (See the discussion of order of
     evaluation for the explicit-control evaluator in section *Note
     5-4-1::.)  How does changing the order of operand evaluation
     affect the efficiency of the code that constructs the argument
     list?

     *Exercise 5.37:* One way to understand the compiler's `preserving'
     mechanism for optimizing stack usage is to see what extra
     operations would be generated if we did not use this idea.  Modify
     `preserving' so that it always generates the `save' and `restore'
     operations.  Compile some simple expressions and identify the
     unnecessary stack operations that are generated.  Compare the code
     to that generated with the `preserving' mechanism intact.

     *Exercise 5.38:* Our compiler is clever about avoiding unnecessary
     stack operations, but it is not clever at all when it comes to
     compiling calls to the primitive procedures of the language in
     terms of the primitive operations supplied by the machine.  For
     example, consider how much code is compiled to compute `(+ a 1)':
     The code sets up an argument list in `argl', puts the primitive
     addition procedure (which it finds by looking up the symbol `+' in
     the environment) into `proc', and tests whether the procedure is
     primitive or compound.  The compiler always generates code to
     perform the test, as well as code for primitive and compound
     branches (only one of which will be executed).  We have not shown
     the part of the controller that implements primitives, but we
     presume that these instructions make use of primitive arithmetic
     operations in the machine's data paths.  Consider how much less
     code would be generated if the compiler could "open-code"
     primitives--that is, if it could generate code to directly use
     these primitive machine operations.  The expression `(+ a 1)'
     might be compiled into something as simple as (1)

          (assign val (op lookup-variable-value) (const a) (reg env))
          (assign val (op +) (reg val) (const 1))

     In this exercise we will extend our compiler to support open
     coding of selected primitives.  Special-purpose code will be
     generated for calls to these primitive procedures instead of the
     general procedure-application code.  In order to support this, we
     will augment our machine with special argument registers `arg1'
     and `arg2'.  The primitive arithmetic operations of the machine
     will take their inputs from `arg1' and `arg2'. The results may be
     put into `val', `arg1', or `arg2'.

     The compiler must be able to recognize the application of an
     open-coded primitive in the source program.  We will augment the
     dispatch in the `compile' procedure to recognize the names of
     these primitives in addition to the reserved words (the special
     forms) it currently recognizes.(2) For each special form our
     compiler has a code generator.  In this exercise we will construct
     a family of code generators for the open-coded primitives.

       a. The open-coded primitives, unlike the special forms, all need
          their operands evaluated.  Write a code generator
          `spread-arguments' for use by all the open-coding code
          generators.  `Spread-arguments' should take an operand list
          and compile the given operands targeted to successive
          argument registers.  Note that an operand may contain a call
          to an open-coded primitive, so argument registers will have
          to be preserved during operand evaluation.

       b. For each of the primitive procedures `=', `*', `-', and `+',
          write a code generator that takes a combination with that
          operator, together with a target and a linkage descriptor,
          and produces code to spread the arguments into the registers
          and then perform the operation targeted to the given target
          with the given linkage.  You need only handle expressions
          with two operands.  Make `compile' dispatch to these code
          generators.

       c. Try your new compiler on the `factorial' example.  Compare
          the resulting code with the result produced without open
          coding.

       d. Extend your code generators for `+' and `*' so that they can
          handle expressions with arbitrary numbers of operands.  An
          expression with more than two operands will have to be
          compiled into a sequence of operations, each with only two
          inputs.


   ---------- Footnotes ----------

   (1) We have used the same symbol `+' here to denote both the
source-language procedure and the machine operation.  In general there
will not be a one-to-one correspondence between primitives of the
source language and primitives of the machine.

   (2) Making the primitives into reserved words is in general a bad
idea, since a user cannot then rebind these names to different
procedures.  Moreover, if we add reserved words to a compiler that is
in use, existing programs that define procedures with these names will
stop working.  See *Note Exercise 5-44:: for ideas on how to avoid this
problem.


File: sicp.info,  Node: 5-5-6,  Next: 5-5-7,  Prev: 5-5-5,  Up: 5-5

5.5.6 Lexical Addressing
------------------------

One of the most common optimizations performed by compilers is the
optimization of variable lookup.  Our compiler, as we have implemented
it so far, generates code that uses the `lookup-variable-value'
operation of the evaluator machine.  This searches for a variable by
comparing it with each variable that is currently bound, working frame
by frame outward through the run-time environment.  This search can be
expensive if the frames are deeply nested or if there are many
variables.  For example, consider the problem of looking up the value
of `x' while evaluating the expression `(* x y z)' in an application of
the procedure that is returned by

     (let ((x 3) (y 4))
       (lambda (a b c d e)
         (let ((y (* a b x))
               (z (+ c d x)))
           (* x y z))))

   Since a `let' expression is just syntactic sugar for a `lambda'
combination, this expression is equivalent to

     ((lambda (x y)
        (lambda (a b c d e)
          ((lambda (y z) (* x y z))
           (* a b x)
           (+ c d x))))
      3
      4)

   Each time `lookup-variable-value' searches for `x', it must determine
that the symbol `x' is not `eq?' to `y' or `z' (in the first frame),
nor to `a', `b', `c', `d', or `e' (in the second frame).  We will
assume, for the moment, that our programs do not use `define'--that
variables are bound only with `lambda'.  Because our language is
lexically scoped, the run-time environment for any expression will have
a structure that parallels the lexical structure of the program in
which the expression appears.(1) Thus, the compiler can know, when it
analyzes the above expression, that each time the procedure is applied
the variable `x' in `(* x y z)' will be found two frames out from the
current frame and will be the first variable in that frame.

   We can exploit this fact by inventing a new kind of variable-lookup
operation, `lexical-address-lookup', that takes as arguments an
environment and a "lexical address" that consists of two numbers: a number
"frame number", which specifies how many frames to pass over, and a "displacement
number", which specifies how many variables to pass over in that frame.
`Lexical-address-lookup' will produce the value of the variable stored
at that lexical address relative to the current environment.  If we add
the `lexical-address-lookup' operation to our machine, we can make the
compiler generate code that references variables using this operation,
rather than `lookup-variable-value'.  Similarly, our compiled code can
use a new `lexical-address-set!'  operation instead of
`set-variable-value!'.

   In order to generate such code, the compiler must be able to
determine the lexical address of a variable it is about to compile a
reference to.  The lexical address of a variable in a program depends
on where one is in the code.  For example, in the following program,
the address of `x' in expression <E1> is (2,0)--two frames back and the
first variable in the frame.  At that point `y' is at address (0,0) and
`c' is at address (1,2).  In expression <E2>, `x' is at (1,0), `y' is
at (1,1), and `c' is at (0,2).

     ((lambda (x y)
        (lambda (a b c d e)
          ((lambda (y z) <E1>)
           <E2>
           (+ c d x))))
      3
      4)

   One way for the compiler to produce code that uses lexical
addressing is to maintain a data structure called a "compile-time
environment".  This keeps track of which variables will be at which
positions in which frames in the run-time environment when a particular
variable-access operation is executed.  The compile-time environment is
a list of frames, each containing a list of variables.  (There will of
course be no values bound to the variables, since values are not
computed at compile time.)  The compile-time environment becomes an
additional argument to `compile' and is passed along to each code
generator.  The top-level call to `compile' uses an empty compile-time
environment.  When a `lambda' body is compiled, `compile-lambda-body'
extends the compile-time environment by a frame containing the
procedure's parameters, so that the sequence making up the body is
compiled with that extended environment.  At each point in the
compilation, `compile-variable' and `compile-assignment' use the
compile-time environment in order to generate the appropriate lexical
addresses.

   *Note Exercise 5-39:: through *Note Exercise 5-43:: describe how to
complete this sketch of the lexical-addressing strategy in order to
incorporate lexical lookup into the compiler.  *Note Exercise 5-44::
describes another use for the compile-time environment.

     *Exercise 5.39:* Write a procedure `lexical-address-lookup' that
     implements the new lookup operation.  It should take two
     arguments--a lexical address and a run-time environment--and
     return the value of the variable stored at the specified lexical
     address.  `Lexical-address-lookup' should signal an error if the
     value of the variable is the symbol `*unassigned*'.(2) Also write
     a procedure `lexical-address-set!' that implements the operation
     that changes the value of the variable at a specified lexical
     address.

     *Exercise 5.40:* Modify the compiler to maintain the compile-time
     environment as described above.  That is, add a
     compile-time-environment argument to `compile' and the various code
     generators, and extend it in `compile-lambda-body'.

     *Exercise 5.41:* Write a procedure `find-variable' that takes as
     arguments a variable and a compile-time environment and returns
     the lexical address of the variable with respect to that
     environment.  For example, in the program fragment that is shown
     above, the compile-time environment during the compilation of
     expression <E1> is `((y z) (a b c d e) (x y))'.  `Find-variable'
     should produce

          (find-variable 'c '((y z) (a b c d e) (x y)))
          (1 2)

          (find-variable 'x '((y z) (a b c d e) (x y)))
          (2 0)

          (find-variable 'w '((y z) (a b c d e) (x y)))
          not-found

     *Exercise 5.42:* Using `find-variable' from *Note Exercise 5-41::,
     rewrite `compile-variable' and `compile-assignment' to output
     lexical-address instructions.  In cases where `find-variable'
     returns `not-found' (that is, where the variable is not in the
     compile-time environment), you should have the code generators use
     the evaluator operations, as before, to search for the binding.
     (The only place a variable that is not found at compile time can
     be is in the global environment, which is part of the run-time
     environment but is not part of the compile-time environment.(3)
     Thus, if you wish, you may have the evaluator operations look
     directly in the global environment, which can be obtained with the
     operation `(op get-global-environment)', instead of having them
     search the whole run-time environment found in `env'.)  Test the
     modified compiler on a few simple cases, such as the nested
     `lambda' combination at the beginning of this section.

     *Exercise 5.43:* We argued in section *Note 4-1-6:: that internal
     definitions for block structure should not be considered "real"
     `define's.  Rather, a procedure body should be interpreted as if
     the internal variables being defined were installed as ordinary
     `lambda' variables initialized to their correct values using
     `set!'.  Section *Note 4-1-6:: and *Note Exercise 4-16:: showed
     how to modify the metacircular interpreter to accomplish this by
     scanning out internal definitions.  Modify the compiler to perform
     the same transformation before it compiles a procedure body.

     *Exercise 5.44:* In this section we have focused on the use of the
     compile-time environment to produce lexical addresses.  But there
     are other uses for compile-time environments.  For instance, in
     *Note Exercise 5-38:: we increased the efficiency of compiled code
     by open-coding primitive procedures.  Our implementation treated
     the names of open-coded procedures as reserved words.  If a
     program were to rebind such a name, the mechanism described in
     *Note Exercise 5-38:: would still open-code it as a primitive,
     ignoring the new binding.  For example, consider the procedure

          (lambda (+ * a b x y)
            (+ (* a x) (* b y)))

     which computes a linear combination of `x' and `y'.  We might call
     it with arguments `+matrix', `*matrix', and four matrices, but the
     open-coding compiler would still open-code the `+' and the `*' in
     `(+ (* a x) (* b y))' as primitive `+' and `*'.  Modify the
     open-coding compiler to consult the compile-time environment in
     order to compile the correct code for expressions involving the
     names of primitive procedures.  (The code will work correctly as
     long as the program does not `define' or `set!' these names.)

   ---------- Footnotes ----------

   (1) This is not true if we allow internal definitions, unless we
scan them out.  See *Note Exercise 5-43::.

   (2) This is the modification to variable lookup required if we
implement the scanning method to eliminate internal definitions (*Note
Exercise 5-43::).  We will need to eliminate these definitions in order
for lexical addressing to work.

   (3) Lexical addresses cannot be used to access variables in the
global environment, because these names can be defined and redefined
interactively at any time.  With internal definitions scanned out, as
in *Note Exercise 5-43::, the only definitions the compiler sees are
those at top level, which act on the global environment.  Compilation
of a definition does not cause the defined name to be entered in the
compile-time environment.


File: sicp.info,  Node: 5-5-7,  Prev: 5-5-6,  Up: 5-5

5.5.7 Interfacing Compiled Code to the Evaluator
------------------------------------------------

We have not yet explained how to load compiled code into the evaluator
machine or how to run it.  We will assume that the
explicit-control-evaluator machine has been defined as in section *Note
5-4-4::, with the additional operations specified in footnote *Note
Footnote 38::.  We will implement a procedure `compile-and-go' that
compiles a Scheme expression, loads the resulting object code into the
evaluator machine, and causes the machine to run the code in the
evaluator global environment, print the result, and enter the
evaluator's driver loop.  We will also modify the evaluator so that
interpreted expressions can call compiled procedures as well as
interpreted ones.  We can then put a compiled procedure into the
machine and use the evaluator to call it:

     (compile-and-go
      '(define (factorial n)
         (if (= n 1)
             1
             (* (factorial (- n 1)) n))))
      ;;; EC-Eval value:
     ok

      ;;; EC-Eval input:
     (factorial 5)
     ;;; EC-Eval value:
     120

   To allow the evaluator to handle compiled procedures (for example,
to evaluate the call to `factorial' above), we need to change the code
at `apply-dispatch' (section *Note 5-4-1::) so that it recognizes
compiled procedures (as distinct from compound or primitive procedures)
and transfers control directly to the entry point of the compiled
code:(1)

     apply-dispatch
       (test (op primitive-procedure?) (reg proc))
       (branch (label primitive-apply))
       (test (op compound-procedure?) (reg proc))
       (branch (label compound-apply))
       (test (op compiled-procedure?) (reg proc))
       (branch (label compiled-apply))
       (goto (label unknown-procedure-type))

     compiled-apply
       (restore continue)
       (assign val (op compiled-procedure-entry) (reg proc))
       (goto (reg val))

   Note the restore of `continue' at `compiled-apply'.  Recall that the
evaluator was arranged so that at `apply-dispatch', the continuation
would be at the top of the stack.  The compiled code entry point, on
the other hand, expects the continuation to be in `continue', so
`continue' must be restored before the compiled code is executed.

   To enable us to run some compiled code when we start the evaluator
machine, we add a `branch' instruction at the beginning of the
evaluator machine, which causes the machine to go to a new entry point
if the `flag' register is set.(2)

     (branch (label external-entry))      ; branches if `flag' is set
     read-eval-print-loop
       (perform (op initialize-stack))
       ...

   `External-entry' assumes that the machine is started with `val'
containing the location of an instruction sequence that puts a result
into `val' and ends with `(goto (reg continue))'.  Starting at this
entry point jumps to the location designated by `val', but first assigns
`continue' so that execution will return to `print-result', which
prints the value in `val' and then goes to the beginning of the
evaluator's read-eval-print loop.(3)

     external-entry
       (perform (op initialize-stack))
       (assign env (op get-global-environment))
       (assign continue (label print-result))
       (goto (reg val))

   Now we can use the following procedure to compile a procedure
definition, execute the compiled code, and run the read-eval-print loop
so we can try the procedure.  Because we want the compiled code to
return to the location in `continue' with its result in `val', we
compile the expression with a target of `val' and a linkage of
`return'.  In order to transform the object code produced by the
compiler into executable instructions for the evaluator register
machine, we use the procedure `assemble' from the register-machine
simulator (section *Note 5-2-2::).  We then initialize the `val'
register to point to the list of instructions, set the `flag' so that
the evaluator will go to `external-entry', and start the evaluator.

     (define (compile-and-go expression)
       (let ((instructions
              (assemble (statements
                         (compile expression 'val 'return))
                        eceval)))
         (set! the-global-environment (setup-environment))
         (set-register-contents! eceval 'val instructions)
         (set-register-contents! eceval 'flag true)
         (start eceval)))

   If we have set up stack monitoring, as at the end of section *Note
5-4-4::, we can examine the stack usage of compiled code:

     (compile-and-go
      '(define (factorial n)
         (if (= n 1)
             1
             (* (factorial (- n 1)) n))))

     (total-pushes = 0 maximum-depth = 0)
      ;;; EC-Eval value:
     ok

      ;;; EC-Eval input:
     (factorial 5)
     (total-pushes = 31 maximum-depth = 14)
     ;;; EC-Eval value:
     120

   Compare this example with the evaluation of `(factorial 5)' using the
interpreted version of the same procedure, shown at the end of section
*Note 5-4-4::.  The interpreted version required 144 pushes and a
maximum stack depth of 28.  This illustrates the optimization that
results from our compilation strategy.

Interpretation and compilation
..............................

With the programs in this section, we can now experiment with the
alternative execution strategies of interpretation and compilation.(4)
An interpreter raises the machine to the level of the user program; a
compiler lowers the user program to the level of the machine language.
We can regard the Scheme language (or any programming language) as a
coherent family of abstractions erected on the machine language.
Interpreters are good for interactive program development and debugging
because the steps of program execution are organized in terms of these
abstractions, and are therefore more intelligible to the programmer.
Compiled code can execute faster, because the steps of program
execution are organized in terms of the machine language, and the
compiler is free to make optimizations that cut across the higher-level
abstractions.(5)

   The alternatives of interpretation and compilation also lead to
different strategies for porting languages to new computers. Suppose
that we wish to implement Lisp for a new machine.  One strategy is to
begin with the explicit-control evaluator of section *Note 5-4:: and
translate its instructions to instructions for the new machine.  A
different strategy is to begin with the compiler and change the code
generators so that they generate code for the new machine.  The second
strategy allows us to run any Lisp program on the new machine by first
compiling it with the compiler running on our original Lisp system, and
linking it with a compiled version of the run-time library.(6) Better
yet, we can compile the compiler itself, and run this on the new
machine to compile other Lisp programs.(7)  Or we can compile one of
the interpreters of section *Note 4-1:: to produce an interpreter that
runs on the new machine.

     *Exercise 5.45:* By comparing the stack operations used by
     compiled code to the stack operations used by the evaluator for the
     same computation, we can determine the extent to which the
     compiler optimizes use of the stack, both in speed (reducing the
     total number of stack operations) and in space (reducing the
     maximum stack depth).  Comparing this optimized stack use to the
     performance of a special-purpose machine for the same computation
     gives some indication of the quality of the compiler.

       a. *Note Exercise 5-27:: asked you to determine, as a function
          of n, the number of pushes and the maximum stack depth needed
          by the evaluator to compute n!  using the recursive factorial
          procedure given above.  *Note Exercise 5-14:: asked you to do
          the same measurements for the special-purpose factorial
          machine shown in *Note Figure 5-11::. Now perform the same
          analysis using the compiled `factorial' procedure.

          Take the ratio of the number of pushes in the compiled
          version to the number of pushes in the interpreted version,
          and do the same for the maximum stack depth.  Since the
          number of operations and the stack depth used to compute n!
          are linear in n, these ratios should approach constants as n
          becomes large.  What are these constants?  Similarly, find
          the ratios of the stack usage in the special-purpose machine
          to the usage in the interpreted version.

          Compare the ratios for special-purpose versus interpreted
          code to the ratios for compiled versus interpreted code.  You
          should find that the special-purpose machine does much better
          than the compiled code, since the hand-tailored controller
          code should be much better than what is produced by our
          rudimentary general-purpose compiler.

       b. Can you suggest improvements to the compiler that would help
          it generate code that would come closer in performance to the
          hand-tailored version?


     *Exercise 5.46:* Carry out an analysis like the one in *Note
     Exercise 5-45:: to determine the effectiveness of compiling the
     tree-recursive Fibonacci procedure

          (define (fib n)
            (if (< n 2)
                n
                (+ (fib (- n 1)) (fib (- n 2)))))

     compared to the effectiveness of using the special-purpose
     Fibonacci machine of *Note Figure 5-12::.  (For measurement of the
     interpreted performance, see *Note Exercise 5-29::.)  For
     Fibonacci, the time resource used is not linear in n; hence the
     ratios of stack operations will not approach a limiting value that
     is independent of n.

     *Exercise 5.47:* This section described how to modify the
     explicit-control evaluator so that interpreted code can call
     compiled procedures.  Show how to modify the compiler so that
     compiled procedures can call not only primitive procedures and
     compiled procedures, but interpreted procedures as well.  This
     requires modifying `compile-procedure-call' to handle the case of
     compound (interpreted) procedures.  Be sure to handle all the same
     `target' and `linkage' combinations as in `compile-proc-appl'.  To
     do the actual procedure application, the code needs to jump to the
     evaluator's `compound-apply' entry point.  This label cannot be
     directly referenced in object code (since the assembler requires
     that all labels referenced by the code it is assembling be defined
     there), so we will add a register called `compapp' to the
     evaluator machine to hold this entry point, and add an instruction
     to initialize it:

          (assign compapp (label compound-apply))
            (branch (label external-entry))      ; branches if `flag' is set
          read-eval-print-loop
            ...

     To test your code, start by defining a procedure `f' that calls a
     procedure `g'.  Use `compile-and-go' to compile the definition of
     `f' and start the evaluator.  Now, typing at the evaluator, define
     `g' and try to call `f'.

     *Exercise 5.48:* The `compile-and-go' interface implemented in
     this section is awkward, since the compiler can be called only
     once (when the evaluator machine is started).  Augment the
     compiler-interpreter interface by providing a `compile-and-run'
     primitive that can be called from within the explicit-control
     evaluator as follows:

          ;;; EC-Eval input:
          (compile-and-run
           '(define (factorial n)
              (if (= n 1)
                  1
                  (* (factorial (- n 1)) n))))
          ;;; EC-Eval value:
          ok

          ;;; EC-Eval input:
          (factorial 5)
          ;;; EC-Eval value:
          120

     *Exercise 5.49:* As an alternative to using the explicit-control
     evaluator's read-eval-print loop, design a register machine that
     performs a read-compile-execute-print loop.  That is, the machine
     should run a loop that reads an expression, compiles it, assembles
     and executes the resulting code, and prints the result.  This is
     easy to run in our simulated setup, since we can arrange to call
     the procedures `compile' and `assemble' as "register-machine
     operations."

     *Exercise 5.50:* Use the compiler to compile the metacircular
     evaluator of section *Note 4-1:: and run this program using the
     register-machine simulator.  (To compile more than one definition
     at a time, you can package the definitions in a `begin'.)  The
     resulting interpreter will run very slowly because of the multiple
     levels of interpretation, but getting all the details to work is
     an instructive exercise.

     *Exercise 5.51:* Develop a rudimentary implementation of Scheme in
     C (or some other low-level language of your choice) by translating
     the explicit-control evaluator of section *Note 5-4:: into C.  In
     order to run this code you will need to also provide appropriate
     storage-allocation routines and other run-time support.

     *Exercise 5.52:* As a counterpoint to exercise *Note Exercise
     5-51::, modify the compiler so that it compiles Scheme procedures
     into sequences of C instructions.  Compile the metacircular
     evaluator of section *Note 4-1:: to produce a Scheme interpreter
     written in C.

   ---------- Footnotes ----------

   (1) Of course, compiled procedures as well as interpreted procedures
are compound (nonprimitive).  For compatibility with the terminology
used in the explicit-control evaluator, in this section we will use
"compound" to mean interpreted (as opposed to compiled).

   (2) Now that the evaluator machine starts with a `branch', we must
always initialize the `flag' register before starting the evaluator
machine.  To start the machine at its ordinary read-eval-print loop, we
could use

     (define (start-eceval)
       (set! the-global-environment (setup-environment))
       (set-register-contents! eceval 'flag false)
       (start eceval))

   (3) Since a compiled procedure is an object that the system may try
to print, we also modify the system print operation `user-print' (from
section *Note 4-1-4::) so that it will not attempt to print the
components of a compiled procedure:

     (define (user-print object)
       (cond ((compound-procedure? object)
              (display (list 'compound-procedure
                             (procedure-parameters object)
                             (procedure-body object)
                             '<procedure-env>)))
             ((compiled-procedure? object)
              (display '<compiled-procedure>))
             (else (display object))))

   (4) We can do even better by extending the compiler to allow
compiled code to call interpreted procedures.  See *Note Exercise
5-47::.

   (5) Independent of the strategy of execution, we incur significant
overhead if we insist that errors encountered in execution of a user
program be detected and signaled, rather than being allowed to kill the
system or produce wrong answers.  For example, an out-of-bounds array
reference can be detected by checking the validity of the reference
before performing it.  The overhead of checking, however, can be many
times the cost of the array reference itself, and a programmer should
weigh speed against safety in determining whether such a check is
desirable.  A good compiler should be able to produce code with such
checks, should avoid redundant checks, and should allow programmers to
control the extent and type of error checking in the compiled code.

   Compilers for popular languages, such as C and C++, put hardly any
error-checking operations into running code, so as to make things run
as fast as possible.  As a result, it falls to programmers to
explicitly provide error checking.  Unfortunately, people often neglect
to do this, even in critical applications where speed is not a
constraint.  Their programs lead fast and dangerous lives.  For
example, the notorious "Worm" that paralyzed the Internet in 1988
exploited the UNIX(tm) operating system's failure to check whether the
input buffer has overflowed in the finger daemon. (See Spafford 1989.)

   (6) Of course, with either the interpretation or the compilation
strategy we must also implement for the new machine storage allocation,
input and output, and all the various operations that we took as
"primitive" in our discussion of the evaluator and compiler.  One
strategy for minimizing work here is to write as many of these
operations as possible in Lisp and then compile them for the new
machine.  Ultimately, everything reduces to a small kernel (such as
garbage collection and the mechanism for applying actual machine
primitives) that is hand-coded for the new machine.

   (7) This strategy leads to amusing tests of correctness of the
compiler, such as checking whether the compilation of a program on the
new machine, using the compiled compiler, is identical with the
compilation of the program on the original Lisp system.  Tracking down
the source of differences is fun but often frustrating, because the
results are extremely sensitive to minuscule details.


File: sicp.info,  Node: References,  Next: Index,  Prev: Chapter 5,  Up: Top

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