Table Of ContentA lgorith m ic
LOGIC
Grazyna Mir Rows ka
Andrzej Salwicki
D. Reidel Publishing Company
PWN - Polish Scientific Publishers
G. MIRKOWSKA and A. SALWICKI
Institute of Mathematics Institute of Informatics
University of Warsaw University of Warsaw
ALGORITHMIC
LOGIC
D. REIDEL PUBLISHING COMPANY
A MEMBER OETHE KLUWER ACADEMIC PUBLISHERS GROUP
<f§? li
DORDRECHT / BOSTON / LANCASTER / TOKYO
PWN-POLISH SCIENTIFIC PUBLISHERS
WARSZAWA
CIF
Library of Congress Cataloging-in-Publication Data
Mirkowska-Salwicka, Grazyna.
Algorithmic logic.
Bibliography: p.
Includes index.
1. Formal languages. 2. Algorithms. 3. Logic,
Symbolic and mathematical. I. Salwicki, Andrzej.
II. Title.
QA267.3.M57 1986 511.3 85-2201
ISBN 90-277-1928-4
This edition published by PWN—Polish Scientific Publishers, Warszawa, Poland,
in co-publication with
D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland
Distributors for Albania, Bulgaria, Cuba, Czechoslovakia, German Democratic
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CONTENTS
PREFACE........................................................................................ IX
CHAPTER I. INTRODUCTION ............................................ 1
1. The motivations...................................................................... 1
2. An informal introduction to formalized languages............... 5
3. Assigning meanings to programs......................................... 11
4. Semantic properties of programs......................................... 16
5. Expressivity. An introduction to the language of algorithmic
logic............................................................................................ 18
6. On applications ...................................................................... 20
CHAPTER II. LOGIC OF DETERMINISTIC ITERATIVE
PROGRAMS ......................................................................... 23
1. Language.................................................................................... 24
2. Semantics.................................................................................... 30
3. Expressiveness......................................................................... 38
4. Properties of the semantic consequence operation............... 51
5. Axiomatization......................................................................... 56
6. Models and consistency ....................................................... 65
7. Useful tautologies and inference rules................................. 69
8. An example of a correctness proof..................................... 75
Bibliographic remarks ........................................................... 77
CHAPTER III. METAMATHEMATICAL INVESTIGATIONS
OF ALGORITHMIC LOGIC................................................ 79
1. Lindenbaum algebra............................................................... 79
2. The Completeness Theorem.................................................... 89
3. Two corollaries of the Completeness Theorem............... 95
4. The standard execution method is implicitly defined by the
axiomatization of algorithmic logic........................................ 97
VI CONTENTS
5. Gentzen type axiomatization................................................ 103
6. The normal form of programs............................................ 109
7. Equality.................................................................................... 115
8. Generalized term s.................................................................. 119
9. Partial functions...................................................................... 122
10. Many sorted structures........................................................... 127
11. Definability and programmability........................................ 131
12. Inessentiality of definitions................................................... 135
Bibliographic remarks........................................................... 137
CHAPTER IV. ALGORITHMIC PROPERTIES OF DATA
STRUCTURES...................................................................... 138
1. Data structures in programming........................................ 138
2. Dictionaries............................................................................. 141
3. Theory of dictionaries.......................................................... 142
4. Representation theorem for models of ATD ................... 149
5. On complexity of A T D ....................................................... 151
6. The theory of priority queues............................................ 154
7. The theory of natural numbers ........................................ 155
8. Stacks........................................................................................ 159
9. The theory of stacks.............................................................. 160
10. The representation theorem for stacks................................. 164
11. Implementation of arithmetic and dictionaries.................. 166
12. Theory of links and stacks—ATSL.................................... 167
13. Implementation of stacks in LOG LAN programming
language.................................................................................... 173
14. Queues.................................................................................... 176
35. Binary trees............................................................................. 179
16. Binary search trees.................................................................. 181
17. An interpretation of the theory of priority queues............... 184
18. An implementation of priority queues............................. 187
19. Arrays........................................................................................ 190
20. Hashtables................................................................................. 193
21. Rational numbers.................................................................. 194
22. Complex numbers.................................................................. 195
23. Real numbers.......................................................................... 200
24. Concluding remarks.............................................................. 202
Bibliographic remarks........................................................... 204
CONTENTS YU
CHAPTER V. PROPOSITIONAL ALGORITHMIC LOGIC 206
1. Syntax and semantics........................................................... 208
2. Semantic properties of program schemes.......................... 212
3. Properties of semantic structures........................................ 221
4. The semantic consequence operation is not compact . . . 228
5. The syntactic consequence operation................................. 229
6. Examples of propositional theories..................................... 233
7. Lindenbaum algebra.............................................................. 237
8. Deterministic and total interpretations of atomic programs 239
9. Partial functional interpretations........................................ 243
10. Bounded non-determinism: The Completeness Theorem . 248
11. Elimination of bounded non-deterministic program vari
ables ........................................................................................ 257
12. Yanov schemes...................................................................... 261
13. Application of PAL in microprogramming.......................... 263
Bibliographic remarks........................................................... 268
CHAPTER VI. NON-DETERMINISM IN ALGORITHMIC
LOGIC.................................................................................... 269
1. Non-deterministic while-programs and their semantics . . 270
2. Properties of non-deterministic programs.......................... 273
3. The Substitution Theorem................................................... 277
4. Non-deterministic algorithmic logic..................................... 282
5. Certain metamathematical results........................................ 286
6. On isomorphism of data structures..................................... 289
7. On the equivalence of non-deterministic programs............... 291
Bibliographic remarks........................................................... 297
CHAPTER VII. PROBLEMS AND THEORIES INSPIRED BY
THE LOGLAN PROJECT ................................................ 298
1. Concurrent programs ........................................................... 299
2. MAX semantics...................................................................... 300
3. Comparison with some other concepts of concurrency . . 303
4. A comparison of MAX and ARB semantics in the case
of Petri nets............................................................................. 311
5. Critical remarks concerning MAX semantics.................. 315
6. LIBERAL semantics.............................................................. 318
VIII CONTENTS
7. An algorithmic theory of references................................. 328
8. Representation theorem for ATR theory.......................... 332
9. Specification of univocal references..................................... 338
10. Virtual memory...................................................................... 339
11. Concatenable type declarations............................................ 341
12. An implementation of rational numbers.............................. 344
Bibliographic remarks........................................................... 346
Appendix A. Boolean algebras.................................................... 348
Appendix B. The proof of Lemma 2.2 from Chapter III . . 352
Bibliography..................................................................................... 356
Index................................................................................................... 369
PREFACE
The purpose of this book is manyfold. It is intended both to present
techniques useful in software engineering and to expose results of
research on properties of these techniques.
The major goal of the book is to help the reader in elaboration of his
own views on foundations of computing. The present authors believe
that semantics of programs will always be the necessary foundation
for every student of computing. On this foundation one can construct
subsequent layers of skill and knowledge in computer science. Later
one discovers more questions of a different nature, e.g. on cost and
optimality of algorithms. This book shall be mainly concerned with
semantics.
Secondly, the book aims to supply a new set of logical axioms and
inference rules appropriate for reasoning about the properties of algo
rithms. Such tools are useful for formalizing the verification and analy
sis of algorithms. The tools should be of quality—they should be
consistent and complete. These and similar requirements lead us toward
metamathematical questions concerning the structure of algorithmic
logic.
Algorithmic properties are expressed by algorithmic formulas in a
straigthforward way. Therefore the analysis of algorithms, i.e. their verifi
cation and evaluation of their effectiveness can be based on algorithmic
logic. Our third aim is to expose the possible applications of algorithmic
logic in the description of structures and systems, especially those
appearing in computer science.
Finally, we wish to stress strong connections between the formal
methods described in this book and the methodologies supported
by modern programming languages. This phenomenon has two aspects
commercial and scientific. Scientific—since modern tools of programm
ing inspire many problems. Commercial—since formal methods
X PREFACE
of AL can be used in software engineering for creating industrial means
of production of software.
We are aware that algorithmics, i.e. the creation of new more efficient
algorithms and the discovery of new data structures, differs from study
of rules of reasoning about algorithms. The book may be useful for
those who wish to learn about formal, logical methods of computer
science, but we cannot assure, however, that the reader will learn how
to conduct a research in computer science. The topics presented in this
book belong to the mathematical foundations of computer science.
The main questions considered are: analysis of algorithms and the
analysis of the process of analysing algorithms. The formal counterparts
of these notions are the notions of proof of a semantical property
of a program and metamathematical properties of the system of algo
rithmic logic. The formal tools developed by algorithmic logic have
many applications in specification of abstract data types, in verification
of algorithms and the implementation of data structures, and in de
fining the semantics of programming languages.
This book can serve as a textbook for a course on the theory of pro
grams or logic of programs or as a textbook of logic for computer scien
tists. It does not assume any special mathematical background from
the reader, but skill in programming and experience with mathematical
reasoning are desirable.
The book can also serve as an auxiliary textbook for courses on
programming languages and on methods of programming. Indicating
the elements of the logic of programs may be helpful in courses for
beginners.
This book arose from lectures that both authors have given on algo
rithmic logic at the University of Warsaw, Christian-Albrechts Uni-
versitat in Kiel, Universite Paris 6 and in IAC Roma.
During one semester couise we skip Chapters V, VI and final parts
of the Chapters III and IV. For a two semester course it is advisable
to add material on the logic of recursive procedures. An introductory
course of computer science or a course on methods of programming
can use the material contained in Chapters II (methods of verification)
and IV (specification of data structures and related topics). In these
lectures we stress the relationship between ideas of hierarchical and
modular programming and the ideas contained in the book.
The defects of this book are caused by the authors. One such defect
PREFACE XI
is the omission of recursion and procedures. The authors presented
elsewhere the results concerned to algorithmic logic of programs with
block structures and recursive procedures and also their own approach
toward semantics of functional procedures. We do not include these
results here since, so far, they have found little application in the practice
of verification. We hope that future research will bring answer to our
doubts.
We are sure that new branches of algorithmic logic will appear in con
nection with new methods and tools of programming, especially a
logic of concurrent programs. One can foresee a broader, commercial
application of AL in specification of data types and their implementation
leading toward production of software modules in programming lan
guages which allow extension of modules by their concatenation.
We would like to express our gratitude to dr L. Banachowski and
prof. Z. Pawlak for their critical remarks which helped us to improve
several parts of the manuscript. We have also profited from the com
ments of many colleagues and students, we thank to all of them.
The book would never appear without the sympathetical help and
patience of the Polish publishers. We thank Mrs K. Regulska and
others for the help in preparation of the manuscript and Mr J. Roguski
for the help in proof-reading.
