Table Of ContentAn Introduction to Programming
with Specifications
This is volume 35 in the A.P.I.C. Series
General Editors: M.J.R. Shave and I.C. Wand
A complete list of titles in this series appears at the end of
this volume
The A.P.I.C. Series
No 35
AN INTRODUCTION TO PROGRAMMING
WITH SPECIFICATIONS
A MATHEMATICAL APPROACH
Ryszard Kubiak
Roman Rudzinski
Stefan Sokolowski
Polish Academy of Sciences
Institute of Computer Science
Gdansk Division of the PAS
Gdansk, POLAND
ACADEMIC PRESS
Harcourt Brace Jovanovich, Publishers
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Copyright ©1991 by
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British Library Cataloguing in Publication Data
Kubiak, Ryszard
An Introduction to programming with specifications
1. Computer systems. Programming I, Title II. Rudzinski, Roman
III. Sokolowski, Stephen
ISBN 0-12-427620-2
Printed and bound in Great Britain by The University Press, Cambridge
Contents
0 INTRODUCTION 1
0.1 Evolution in programming styles 1
0.2 Specified programming 3
0.3 On rigour and accuracy 5
0.4 Computer mathematics 5
0.5 About this book 7
1 BASIC MATHEMATICAL VOCABULARY 9
1.1 Types and their elements 9
1.2 Expression evaluation 11
1.3 Partial functions 14
1.4 Erroneous expressions 16
1.5 Exercises 17
2 ON DEFINITIONS AND SPECIFICATIONS 19
2.1 What is and what is not a definition? 19
2.2 Collective definitions or specifications 22
2.3 Function specifications and realizations 23
2.4 What are the definitions for? 26
2.5 Exercises 27
3 CONDITIONAL EXPRESSIONS 30
3.1 Definition and typing requirements 30
3.2 Basic laws 32
3.3 Application to drawing graphs 35
3.4 Exercises 37
4 LOCAL DEFINITIONS 40
4.1 Introduction to local definitions 40
4.2 Operators that bind variables 42
4.3 Reduction laws 44
4.4 Exercises 47
ν
vi CONTENTS
5 RECURSIVE FUNCTION DEFINITIONS 50
5.1 Recursive definitions and related evaluations 50
5.2 Getting familiar with recursive definitions 54
5.3 Generalizing vs instantiating 56
5.4 On definitions that computers can handle 58
5.5 Exercises 60
6 DESIGNING RECURSIVE DEFINITIONS 61
6.1 Two ways of evaluating powers 61
6.2 Series \ + \ + ... + \ 62
6.3 Integer division and remainder hit with one stone 63
6.4 Two ways of finding square roots 65
6.5 How to design recursive definitions 70
6.6 Exercises 71
7 MORE COMPLEX RECURSION 75
7.1 Multiple recursive calls: Fibonacci and binomial 75
7.2 Mutual recursion: a strange definition of factorial 78
7.3 Ackermann's function 80
7.4 Exercises 81
8 HIGH-ORDER DATA: SEQUENCES 84
8.1 Linear and binary search in a sequence 84
8.2 Updating a sequence 86
8.3 Sorting 88
8.4 Exercises 91
9 OUR MAIN TOOL: INDUCTION 94
9.1 Mathematical induction revisited 94
9.2 Generalized induction 95
9.3 How not to misuse the rule 98
9.4 Exercises 100
10 PROVING FACTS ABOUT RECURSIVELY DEFINED
FUNCTIONS 102
10.1 Correctness of divmod 102
10.2 Correctness of two mutually recursive functions 104
10.3 Finding the quantity over which to run induction 106
10.4 Equinumerosity of Nat and Nat χ Nat 109
10.5 Exercises Ill
11 TAIL RECURSION 114
11.1 Factorial revisited 114
11.2 Discussion of tail recursion 116
11.3 Making definitions tail-recursive 118
CONTENTS vii
11.4 Maximum selection 121
11.5 Fibonacci numbers 123
11.6 Exercises 125
12 ITERATIVE ALGORITHMS 127
12.1 Variables, values and valuations 127
12.2 Commands 129
12.3 The imperative point of view 134
12.4 Exercises 137
13 FROM RECURSION TO ITERATION AND BACK 140
13.1 From tail recursion to iteration (greatest common divisor) . 140
13.2 From iteration to tail recursion (Fibonacci) 141
13.3 The correspondence theorem 143
13.4 Construction of iterative algorithms 148
13.5 Exercises 149
14 PARTIAL AND TOTAL CORRECTNESS 152
14.1 Assertions 152
14.2 Correctness properties and their interdependence 153
14.3 Invariants 154
14.4 Loop counters 158
14.5 Exercises 160
15 HOW TO DESIGN ITERATIVE ALGORITHMS 162
15.1 Square of a natural number 162
15.2 Checking a prime 167
15.3 Nested loops 172
15.4 Exercises 174
16 COMPUTER PROGRAMMING 177
16.1 What is programming about? 177
16.2 Functional language: Standard ML 179
16.3 Imperative language: Pascal 182
A SOLUTIONS OF EXERCISES 188
A.l Solutions to Chapter 1 188
A.2 Solutions to Chapter 2 191
A.3 Solutions to Chapter 3 195
A.4 Solutions to Chapter 4 203
A.5 Solutions to Chapter 5 207
A.6 Solutions to Chapter 6 208
A.7 Solutions to Chapter 7 210
A.8 Solutions to Chapter 8 213
A.9 Solutions to Chapter 9 218
viii CONTENTS
A. 10 Solutions to Chapter 10 225
A. 11 Solutions to Chapter 11 234
A. 12 Solutions to Chapter 12 237
A. 13 Solutions to Chapter 13 241
A. 14 Solutions to Chapter 14 245
A.15 Solutions to Chapter 15 247
Β RECOMMENDATIONS FOR FURTHER READING 254
C INDEX OF TECHNICAL TERMS 259
Preface
In the good old days a man who wanted a new house would harness a horse,
and a cart, and take an axe and go into the nearest forest; then he would
come back home with a load of wooden logs and nail them together as best
he could. Some of those carpenters were really able men: they could even
work such wonders as four-storey buildings.
We are more sophisticated nowadays. Science and teamwork were nec
essary to create something as grand and pompous as the Palace of Culture
in Warsaw. Just try to imagine building it using the horse, axe and log
approach!
In computing, however, we are still back in the Dark Ages. We are
often told about the magic power that some schoolchildren have over their
computers. We are made to believe that what really matters is bold hearts
and goodwill rather than a thorough study. What is more, sometimes we
do believe the best way to master programming is by practice.
This is not the view of the authors of this book. Today computer
programming is a craft on the verge of becoming a technology with firm
scientific roots. Appropriate theories are ready and waiting to be applied
and, like it or not, boring professionals with mathematical backgrounds are
going to take over from the wonder kids as they have done so many times
before.
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