These 2011 nostalgia autists didn't expect that: Structure and Interpretation of Computer Programs
second edition
Harold Abelson and Gerald Jay Sussman with Julie Sussman
foreword by Alan J. Perlis
The MIT Press Cambridge, Massachusetts London, England
McGraw-Hill Book Company New York St. Louis San Francisco Montreal Toronto
This book is one of a series of texts written by faculty of the Electrical Engineering and Computer Science Department at the Massachusetts Institute of Technology. It was edited and produced by The MIT Press under a joint production-distribution arrangement with the McGraw-Hill Book Company.
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Creative Commons License Structure and Interpretation of Computer Programs by Harold Abelson and Gerald Jay Sussman with Julie Sussman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License by the MIT Press.
This book was set by the authors using the LATEX typesetting system and was printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Abelson, Harold Structure and interpretation of computer programs / Harold Abelson and Gerald Jay Sussman, with Julie Sussman. -- 2nd ed. p. cm. -- (Electrical engineering and computer science series) Includes bibliographical references and index. ISBN 0-262-01153-0 (MIT Press hardcover) ISBN 0-262-51087-1 (MIT Press paperback) ISBN 0-07-000484-6 (McGraw-Hill hardcover) 1. Electronic digital computers -- Programming. 2. LISP (Computer program language) I. Sussman, Gerald Jay. II. Sussman, Julie. III. Title. IV. Series: MIT electrical engineering and computer science series. QA76.6.A255 1996 005.13'3 -- dc20 96-17756
Fourth printing, 1999
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)
Contents
Foreword
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
1 Building Abstractions with Procedures 1.1 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 1.2 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 1.3 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
2 Building Abstractions with Data 2.1 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 2.2 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 2.3 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 2.4 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 2.5 Systems with Generic Operations 2.5.1 Generic Arithmetic Operations 2.5.2 Combining Data of Different Types 2.5.3 Example: Symbolic Algebra
3 Modularity, Objects, and State 3.1 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 3.2 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 3.3 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 3.4 Concurrency: Time Is of the Essence 3.4.1 The Nature of Time in Concurrent Systems 3.4.2 Mechanisms for Controlling Concurrency 3.5 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
4 Metalinguistic Abstraction 4.1 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 4.2 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 4.3 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 4.4 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
5 Computing with Register Machines 5.1 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 5.2 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 5.3 Storage Allocation and Garbage Collection 5.3.1 Memory as Vectors 5.3.2 Maintaining the Illusion of Infinite Memory 5.4 The Explicit-Control Evaluator 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 5.5 Compilation 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
References
List of Exercises
Index
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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.
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 exercise 2.47).
Figure 2.15: A frame is described by three vectors -- an origin and two edges.
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
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:26
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:
Exercise 2.47. Here are two possible constructors for frames:
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 figure 2.10, from lists of line segments as follows:27
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.28
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 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 (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:
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:
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.
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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 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.31 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 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 figure 2.9 by working at each of the levels described above. In particular:
a. Add some segments to the primitive wave painter of 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.)
6 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.
7 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.''
8 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.
9 Since nested applications of car and cdr are cumbersome to write, Lisp dialects provide abbreviations for them -- for instance,
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.
10 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 2.3, we will denote the empty list as '() and dispense with the variable nil entirely.
11 To define f and g using lambda we would write
(define f (lambda (x y . z) <body>)) (define g (lambda w <body>))
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:
13 The order of the first two clauses in the cond matters, since the empty list satisfies null? and also is not a pair.
14 This is, in fact, precisely the fringe procedure from exercise 2.28. Here we've renamed it to emphasize that it is part of a family of general sequence-manipulation procedures.
15 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.
16 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 an xn, then adding an-1xn-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.
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17 This definition uses the extended version of map described in footnote 12.
18 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 exercise 2.42) are adapted from Turner 1981. In section 3.5.3, we'll see how this approach generalizes to infinite sequences.
19 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).
20 The set S - x is the set of all elements of S, excluding x.
21 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.
22 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.
23 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.
25 Rotate180 rotates a painter by 180 degrees (see exercise 2.50). Instead of rotate180 we could say (compose flip-vert flip-horiz), using the compose procedure from exercise 1.42.
26 Frame-coord-map uses the vector operations described in 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.
27 Segments->painter uses the representation for line segments described in exercise 2.48 below. It also uses the for-each procedure described in exercise 2.23.
28 For example, the rogers painter of 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 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 2.1.3 also showed how pairs could be implemented as procedures. Painters are our second example of a procedural representation for data.
29 Rotate90 is a pure rotation only for square frames, because it also stretches and shrinks the image to fit into the rotated frame.
30 The diamond-shaped images in figures 2.10 and 2.11 were created with squash-inwards applied to wave and rogers.
31 Section 3.3.4 describes one such language.
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.
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.32
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.33
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:34
(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.35 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:
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.36
Exercise 2.55. Eva Lu Ator types to the interpreter the expression
(car ''abracadabra)
To her surprise, the interpreter prints back quote. Explain.
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 ax2 + 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 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:
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:
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.
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Anonymous2021-03-16 9:20
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?:
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
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.
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
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?
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Anonymous2021-03-16 9:21
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.37
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 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:
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 (n). The number of steps required by adjoin-set, which uses this operation, also grows as (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 (n2) 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 (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 (n2) 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 cdrs 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:
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 (n) growth rather than (n2) -- 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 (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. 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 }.
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 (log n).38 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:39
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 (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 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 (log n) steps.40
Figure 2.17: Unbalanced tree produced by adjoining 1 through 7 in sequence.
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Anonymous2021-03-16 9:21
Exercise 2.63. Each of the following two procedures converts a binary tree to a list.
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 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))))
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 exercises 2.63 and 2.64 to give (n) implementations of union-set and intersection-set for sets implemented as (balanced) binary trees.41
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
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.
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 27, or 128, possible different characters. In general, if we want to distinguish n different symbols, we will need to use log2 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
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.
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 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.
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Anonymous2021-03-16 9:22
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.42
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 figure 2.18 was generated:
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:
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 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:
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 2.4 and 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.
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 consing 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 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:
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 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.
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 (exercise 2.69) to generate a corresponding Huffman tree, and use encode (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, ..., 2n-1. Sketch the tree for n=5; for n=10. In such a tree (for general n) how many bits are required to encode the most frequent symbol? the least frequent symbol?
Exercise 2.72. Consider the encoding procedure that you designed in 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 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.
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32 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 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.''
33 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.
34 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))).
35 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 chapter 3 (section 3.1.3).
36 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.
37 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'').
38 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 1.2.4 and the half-interval search method of section 1.3.3.
39 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.
40 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.
41 Exercises 2.63-2.65 are due to Paul Hilfinger.
42 See Hamming 1980 for a discussion of the mathematical properties of Huffman codes.
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 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 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 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 figure 2.19, contains two different kinds of abstraction barriers. The ``horizontal'' abstraction barriers play the same role as the ones in 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.
In section 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 2.5.3, we'll show how to use generic arithmetic in a system that performs symbolic algebra.
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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).43 Section 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 figure 2.20.) From this point of view, the complex number z = x + iy (where i2 = - 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:
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 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:
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.
Figure 2.20: Complex numbers as points in the plane.
To design such a system, we can follow the same data-abstraction strategy we followed in designing the rational-number package in section 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 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:
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
which relate the real and imaginary parts (x, y) to the magnitude and the angle (r, A).44 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.
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 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:
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 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:
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:
To implement the complex-number arithmetic operations, we can use the same procedures add-complex, sub-complex, mul-complex, and div-complex from section 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
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))
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Figure 2.21: Structure of the generic complex-arithmetic system.
The resulting complex-number system has the structure shown in 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 2.5.
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. On the other hand, implementing the dispatch as in section 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 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 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 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.
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 chapter 3 (section 3.3.3, 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 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.45 The interface also defines the constructors to be used by the external system.46 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 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:47
(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:
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 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 2.3.2 described a program that performs symbolic differentiation:
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
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 (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?
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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 figure 2.22. The style of programming we used in section 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:48
(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 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 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?
43 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.
44 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.
45 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.
46 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.
47 Apply-generic uses the dotted-tail notation described in 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.
48 One limitation of this organization is it permits only generic procedures of one argument.
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Anonymous2021-03-16 9:26
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 2.1.1, and the complex-number arithmetic that we implemented in section 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.
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.
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 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):
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 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 2.4.1.
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 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.
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 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:''
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 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 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.
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Anonymous2021-03-16 9:27
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.
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):49
;; 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:
(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 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.50 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 n2 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.51 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.
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Anonymous2021-03-16 9:28
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 figure 2.25.
Figure 2.25: A tower of types.
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. (Exercises 2.83 and 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. 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.
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. 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.52
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:
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.
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Anonymous2021-03-16 9:28
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 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 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 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 operations53 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 exercise 2.79. Finally, use drop to rewrite apply-generic from 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.
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
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).54 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,
is a simple polynomial in x, and
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 5y2 + 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.55 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 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 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:56
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:
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 2.5.2, then we also are automatically able to handle operations on polynomials of different coefficient types, such as
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
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.57
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Anonymous2021-03-16 9:29
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 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,
is a dense polynomial, whereas
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 1.58 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:59
where =zero? is as defined in exercise 2.80. (See also 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 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,
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.
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!)
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Anonymous2021-03-16 9:30
Extended exercise: Rational functions
We can extend our generic arithmetic system to include rational functions. These are ``fractions'' whose numerator and denominator are polynomials, such as
The system should be able to add, subtract, multiply, and divide rational functions, and to perform such computations as
(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
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.60 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 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
Exercise 2.95. Define P1, P2, and P3 to be the polynomials
Now define Q1 to be the product of P1 and P2 and Q2 to be the product of P1 and P3, and use greatest-common-divisor (exercise 2.94) to compute the GCD of Q1 and Q2. Note that the answer is not the same as P1. This example introduces noninteger operations into the computation, causing difficulties with the GCD algorithm.61 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 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 O1 be the order of P (i.e., the order of the largest term of P) and let O2 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 c1+O1 -O2, 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 exercise 2.95.
b. The GCD now has integer coefficients, but they are larger than those of P1. 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 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 + O1 - O2, where O2 is the order of the GCD and O1 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:
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.62
49 We also have to supply an almost identical procedure to handle the types (scheme-number complex).
50 See exercise 2.82 for generalizations.
51 If we are clever, we can usually get by with fewer than n2 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.
52 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 chapter 3 avoids these issues entirely. Readers familiar with object-oriented programming will notice that we have much to say in 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.
53 A real number can be projected to an integer using the round primitive, which returns the closest integer to its argument.
54 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.
55 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.
56 This operation is very much like the ordered union-set operation we developed in 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.
57 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
which requires adding the coefficient y + 1 to the coefficient 2.
58 In these polynomial examples, we assume that we have implemented the generic arithmetic system using the type mechanism suggested in 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.
59 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 (exercise 2.61).
60 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.
61 In an implementation like MIT Scheme, this produces a polynomial that is indeed a divisor of Q1 and Q2, 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.
62 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 chapter 1. Zippel's book (1993) describes this method, together with other ways to compute polynomial GCDs.
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Anonymous2021-03-16 9:31
Chapter 3
Modularity, Objects, and State
M
(Even while it changes, it stands still.)
Heraclitus
Plus ça change, plus c'est la même 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 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.
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 assignment operator to enable us to change the value associated with a name.
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:
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:
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 <exp1> <exp2> ... <expk>)
causes the expressions <exp1> through <expk> to be evaluated in sequence and the value of the final expression <expk> 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:
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 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 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 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
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:
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 2.4.3, although here we are using it in conjunction with the ability to modify local variables.
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.
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Anonymous2021-03-16 9:31
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:
Exercise 3.4. Modify the make-account procedure of 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.
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 x1 and form
x2 = (rand-update x1) x3 = (rand-update x2)
then the sequence of values x1, x2, x3, ..., will have the desired statistical properties.6
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.
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 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 using the fact that 6/2 is the probability that two integers chosen at random will have no factors in common; that is, that their greatest common divisor will be 1.7 To obtain the approximation to , 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 6/2, and from this we obtain our approximation to .
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.
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:
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< 32. 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 . Use your estimate-integral to produce an estimate of 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 1.2.6, which returns a nonnegative number less than its input.8
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.
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 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 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!:
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 to9
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 3.2 we will see how environments play this role of ``place'' in our computational model.
Name:
Anonymous2021-03-16 9:32
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:
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:
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
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.10
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 1.2.1:
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:
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
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.11 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 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 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
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 exercise 3.3 to accommodate this new feature.
Exercise 3.8. When we defined the evaluation model in section 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.
1 Actually, this is not quite true. One exception was the random-number generator in section 1.2.6. Another exception involved the operation/type tables we introduced in section 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 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 3.2. (See also exercise 3.10.)
6 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.
7 This theorem is due to E. Cesàro. See section 4.5.2 of Knuth 1981 for a discussion and a proof.
8 MIT Scheme provides such a procedure. If random is given an exact integer (as in section 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.
9 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.
10 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 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).
11 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.
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Anonymous2021-03-16 9:33
3.2 The Environment Model of Evaluation
When we introduced compound procedures in chapter 1, we used the substitution model of evaluation (section 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 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 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.
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.
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 1.1.3:
To evaluate a combination:
1. Evaluate the subexpressions of the combination.12
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.
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.
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, 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 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.
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.13 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 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. 3.2.2 Applying Simple Procedures
When we introduced the substitution model in section 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. 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.
In 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 figure 3.4.
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 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 chapter 5.
Name:
Anonymous2021-03-16 9:33
Exercise 3.9. In section 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)))))
Show the environment structures created by evaluating (factorial 6) using each version of the factorial procedure.14
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 3.1.1 created by calling the procedure
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.
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. Figure 3.7 shows the resulting environment structure.
Figure 3.7: Result of evaluating (define W1 (make-withdraw 100)).
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:
The resulting environment structure is shown in 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.
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. Figure 3.9 shows the situation after the call to W1.
Figure 3.9: Environments after the call to W1.
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 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.15 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.
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:
Recall from section 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?
3.2.4 Internal Definitions
Section 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:
Now we can use the environment model to see why these internal definitions behave as desired. 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.
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
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, 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 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 3.1.1:
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?
12 Assignment introduces a subtlety into step 1 of the evaluation rule. As shown in 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.
13 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.
14 The environment model will not clarify our claim in section 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 5.4.
15 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 chapter 4, the code is in fact shared.
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Anonymous2021-03-16 9:34
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 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 mutable data objects.
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 2.2. We also present some examples of simulations in which complex systems are modeled as collections of objects with local state.
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).
Figure 3.13: Effect of (set-car! x y) on the lists in figure 3.12.
Figure 3.14: Effect of (define z (cons y (cdr x))) on the lists in figure 3.12.
Figure 3.15: Effect of (set-cdr! x y) on the lists in figure 3.12.
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!.16
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 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 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.17
Compare figure 3.13 with 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 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 figure 3.12 is shown in 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.18
(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 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 consing 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 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 exercise 3.12:
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 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 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).
Figure 3.17: The list z2 formed by (cons (list 'a 'b) (list 'a 'b)).
In contrast to figure 3.16, 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.19
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:
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 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 3.16 and 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.20
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:
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.
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Anonymous2021-03-16 9:35
Exercise 3.17. Devise a correct version of the count-pairs procedure of 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 cdrs would go into an infinite loop. Exercise 3.13 constructed such lists.
Exercise 3.19. Redo 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 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 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.21
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 exercise 3.11.)
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 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). 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.
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 Figure 3.18: Queue operations.
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 (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 (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. Figure 3.19 illustrates this representation.
Figure 3.19: Implementation of a queue as a list with front and rear pointers.
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:
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 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 figure 3.19.
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 figure 3.21):22
Figure 3.21: Result of using (delete-queue! q) on the queue of figure 3.20.
(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.23 All operations should be accomplished in (1) steps.
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Anonymous2021-03-16 9:35
3.3.3 Representing Tables
When we studied various ways of representing sets in chapter 2, we mentioned in section 2.3.3 the task of maintaining a table of records indexed by identifying keys. In the implementation of data-directed programming in section 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 cars 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*. 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.
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.24 Lookup then checks to see that the resulting record returned by assoc is not false, and returns the value (the cdr) of the record.
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 consing 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.25
(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. 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.
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.
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:
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:
Using make-table, we could implement the get and put operations used in section 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 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 exercise 2.66 of 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 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)?
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Anonymous2021-03-16 9:36
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 figure 3.24. An and-gate, also shown in 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.
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 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.
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 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. Figure 3.26, for example, shows a full-adder composed of two half-adders and an or-gate.26 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 exercise 3.30.)
Figure 3.26: A full-adder circuit.
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 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:
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.
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. 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 A1, A2, A3, ..., An and B1, B2, B3, ..., Bn are the two binary numbers to be added (each Ak and Bk is a 0 or a 1). The circuit generates S1, S2, S3, ..., Sn, 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 Ak, the Bk, and the Sk -- 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.
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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 3.1.1:
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:
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 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:27
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:
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:
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:
Next we connect the wires in a half-adder circuit (as in 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:
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
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 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 3.3.2.
The agenda itself is a one-dimensional table of time segments. It differs from the tables described in section 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 0:28
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.
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:29
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).
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
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.31
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
Such a constraint can be thought of as a network consisting of primitive adder, multiplier, and constant constraints (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.
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 figure 3.28 using the primitive constraint constructors adder, multiplier, and constant. Just as with the digital-circuit simulator of section 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 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.)
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:
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:
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 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):
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 27 in section 3.3.4) are used in conjunction with the dispatch:
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.
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.
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.
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.32 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:
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:
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 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:
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.33
16 Set-car! and set-cdr! return implementation-dependent values. Like set!, they should be used only for their effect.
17 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 5.3.2 that Lisp memory-management systems include a garbage collector, which identifies and recycles the memory space used by unneeded pairs.
18 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 5.3.1.
19 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.
20 The subtleties of dealing with sharing of mutable data objects reflect the underlying issues of ``sameness'' and ``change'' that were raised in section 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.
21 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.
22 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.
23 Be careful not to make the interpreter try to print a structure that contains cycles. (See exercise 3.13.)
24 Because assoc uses equal?, it can recognize keys that are symbols, numbers, or list structure.
25 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.
26 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 Cin 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 Cout, which is the carry bit to be propagated to the left.
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.
28 The agenda is a headed list, like the tables in section 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).
29 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>.
30 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.
31 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.
32 The setter might not be a constraint. In our temperature example, we used user as the setter.
33 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.
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Anonymous2021-03-16 9:39
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 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 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.34 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.
3.4.1 The Nature of Time in Concurrent Systems
On the surface, time seems straightforward. It is an ordering imposed on events.35 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 3.1.1:
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 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 $75.36
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.
Name:
Anonymous2021-03-16 9:39
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.37 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.
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. 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.38 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.
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 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 exercise 3.38).39
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 figure 3.29 to explain how these values can occur.
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:
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 <p1> <p2> ... <pk>)
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.40
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 -- P1, which sets x to x times x, and P2, 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 P1 and P2: 101: P1 sets x to 100 and then P2 increments x to 101. 121: P2 increments x to 11 and then P1 sets x to x times x. 110: P2 changes x from 10 to 11 between the two times that P1 accesses the value of x during the evaluation of (* x x). 11: P2 accesses x, then P1 sets x to 100, then P2 sets x. 100: P1 accesses x (twice), then P2 sets x to 11, then P1 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 P1 and P2 cannot be interleaved.
Here is a version of the make-account procedure from section 3.1.1, where the deposits and withdrawals have been serialized:
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 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):
(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?
Name:
Anonymous2021-03-16 9:41
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.
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:41
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.42 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 3.1.1, except that a serializer is provided to protect the balance variable, and the serializer is exported via message passing:
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:43
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:
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 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.
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:
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.44 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.
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.45 To release the mutex, we set the cell contents to 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:
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 figure 3.29. (See 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.46 Alternatively, multiprocessing computers provide instructions that support atomic operations directly in hardware.47
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 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 exercises 3.48 and 3.49.)48
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.)
Name:
Anonymous2021-03-16 9:41
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.49
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.50
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.51 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.
34 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.
35 To quote some graffiti seen on a Cambridge building wall: ``Time is a device that was invented to keep everything from happening at once.''
36 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.
37 The factorial program in section 3.1.3 illustrates this for a single sequential process.
38 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.
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 4.3 we will study a language for expressing nondeterministic computations.
40 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.
41 We have simplified exchange by exploiting the fact that our deposit message accepts negative amounts. (This is a serious bug in our banking system!)
42 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 exercise 3.43.
43 Exercise 3.45 investigates why deposits and withdrawals are no longer automatically serialized by the account.
44 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 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.
45 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.
46 In MIT Scheme for a single processor, which uses a time-slicing model, test-and-set! can be implemented as follows:
Without-interrupts disables time-slicing interrupts while its procedure argument is being executed.
47 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.
48 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.
49 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).
50 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.
51 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.
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Anonymous2021-03-16 9:42
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.52
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 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.
3.5.1 Streams Are Delayed Lists
As we saw in section 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:53
(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 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
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
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?.54 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 chapter 2, such as list-ref, map, and for-each:55
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 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 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.56
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Anonymous2021-03-16 9:43
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:
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 2.2.3):
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 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
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)))
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
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 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 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.
Delay is then defined so that (delay <exp>) is equivalent to
(memo-proc (lambda () <exp>))
and force is as defined previously.58
Exercise 3.50. Complete the following definition, which generalizes stream-map to allow procedures that take multiple arguments, analogous to map in section 2.2.3, footnote 12.
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?59
(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.
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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 sieve of Eratosthenes.60 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:
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 figure 3.31.61 The input stream feeds into an ``unconser'' 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 consed 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.
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:62
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:
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:
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:
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 n:
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 n).63
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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:
Exercise 3.55. Define a procedure partial-sums that takes as argument a stream S and returns the stream whose elements are S0, S0 + S1, S0 + S1 + S2, .... 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:
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 3.5.1.64
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 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
represented as infinite streams. We will represent the series a0 + a1 x + a2 x2 + a3 x3 + ··· as the stream whose elements are the coefficients a0, a1, a2, a3, ....
a. The integral of the series a0 + a1 x + a2 x2 + a3 x3 + ··· is the series
where c is any constant. Define a procedure integrate-series that takes as input a stream a0, a1, a2, ... representing a power series and returns the stream a0, (1/2)a1, (1/3)a2, ... 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 ex is its own derivative. This implies that ex and the integral of ex are the same series, except for the constant term, which is e0 = 1. Accordingly, we can generate the series for ex as
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:
Exercise 3.60. With power series represented as streams of coefficients as in exercise 3.59, adding series is implemented by add-streams. Complete the definition of the following procedure for multiplying series:
You can test your procedure by verifying that sin2 x + cos2 x = 1, using the series from exercise 3.59.
Exercise 3.61. Let S be a power series (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 + SR where SR is the part of S after the constant term. Then we can solve for X as follows:
In other words, X is the power series whose constant term is 1 and whose higher-order terms are given by the negative of SR 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 exercise 3.60.
Exercise 3.62. Use the results of exercises 3.60 and 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 exercise 3.59 to generate the power series for tangent.
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 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 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 1:65
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 exercise 3.64.)
Another iteration that we can treat in the same way is to generate an approximation to , based upon the alternating series that we saw in section 1.3.1:
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 exercise 3.55) and scale the result by 4:
This gives us a stream of better and better approximations to , although the approximations converge rather slowly. Eight terms of the sequence bound the value of 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 Sn is the nth term of the original sum sequence, then the accelerated sequence has terms
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)) ; Sn-1 (s1 (stream-ref s 1)) ; Sn (s2 (stream-ref s 2))) ; Sn+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 :
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
Finally, we form a sequence by taking the first term in each row of the tableau:
The result is impressive. Taking eight terms of the sequence yields the correct value of to 14 decimal places. If we had used only the original sequence, we would need to compute on the order of 1013 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.
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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:
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 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
to compute three sequences of approximations to the natural logarithm of 2, in the same way we did above for . How rapidly do these sequences converge?
Infinite streams of pairs
In section 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 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 simply66
Our problem, then, is to produce the stream int-pairs. More generally, suppose we have two streams S = (Si) and T = (Tj), and imagine the infinite rectangular array
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
(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 (S0,T0), the rest of the pairs in the first row, and the remaining pairs:67
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
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 2.2.1:
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:68
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
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 (S0,T0) from the rest of the pairs in the first row, he proposes to work with the whole first row, as follows:
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 (Si,Tj,Uk) 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 i2 + j2 = k2.
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 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 (i1,j1) is less than (i2,j2) if W(i1,j1) < W(i2,j2). 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.69 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.70 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 i3 + j3 (see 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 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).
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Anonymous2021-03-16 9:45
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 = (xi), an initial value C, and a small increment dt, accumulates the sum
and returns the stream of values S = (Si). The following integral procedure is reminiscent of the ``implicit style'' definition of the stream of integers (section 3.5.2):
Figure 3.32: The integral procedure viewed as a signal-processing system.
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.
Exercise 3.73.
v = v0 + (1/C)0ti dt + R i
Figure 3.33: An RC circuit and the associated signal-flow diagram.
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 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 v0 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.
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
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:
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 exercise 3.50:
Complete the program by supplying the indicated <expression>.
Exercise 3.75. Unfortunately, Alyssa's zero-crossing detector in 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:
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 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.
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 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, 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).
Assuming we are given an initial value y0 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:
Now we can implement our solve procedure by delaying the evaluation of dy in the definition of y:71
(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 e 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:
Exercise 3.77. The integral procedure used above was analogous to the ``implicit'' definition of the infinite stream of integers in section 3.5.2. Alternatively, we can give a definition of integral that is more like integers-starting-from (also in section 3.5.2):
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.
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Anonymous2021-03-16 9:46
Exercise 3.78.
Figure 3.35: Signal-flow diagram for the solution to a second-order linear differential equation.
Consider the problem of designing a signal-processing system to study the homogeneous second-order linear differential equation
The output stream, modeling y, is generated by a network that contains a loop. This is because the value of d2y/dt2 depends upon the values of y and dy/dt and both of these are determined by integrating d2y/dt2. The diagram we would like to encode is shown in figure 3.35. Write a procedure solve-2nd that takes as arguments the constants a, b, and dt and the initial values y0 and dy0 for y and dy/dt and generates the stream of successive values of y.
Exercise 3.79. Generalize the solve-2nd procedure of exercise 3.78 so that it can be used to solve general second-order differential equations d2 y/dt2 = 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 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
and the circuit connections dictate the relations
Combining these equations shows that the state of the circuit (summarized by vC, the voltage across the capacitor, and iL, the current in the inductor) is described by the pair of differential equations
The signal-flow diagram representing this system of differential equations is shown in figure 3.37.
Figure 3.36: A series RLC circuit.
Figure 3.37: A signal-flow diagram for the solution to a series RLC circuit.
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 exercise 3.73, RLC should produce a procedure that takes the initial values of the state variables, vC0 and iL0, and produces a pair (using cons) of the streams of states vC and iL. 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 iL0 = 0 amps and vC0 = 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.72
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 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 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 exercises 3.51 and 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. 3.5.5 Modularity of Functional Programs and Modularity of Objects
As we saw in section 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 , which we examined in section 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 (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 . This version of the program doesn't need a parameter telling how many trials to perform. Better estimates of (from performing more experiments) are obtained by looking farther into the pi stream:
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. 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 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.