Mathematical Logic for Computer Science

310 Pages·2001·10.04 MB·English

by  Mordechai Ben-Ari PhD (auth.)

Mathematical Logic for Computer Science
Springer-Verlag London Ltd.
Mordechai Ben -Ari
Mathematical Logic
for Computer Science
(Second edition)
, Springer
Mordechai Ben-Ari, PhD
Department of Science Teaching, Weizmann Institute of Science,
Rehovot 76100, Israel
ISBN 978-1-85233-319-5 ISBN 978-1-4471-0335-6 (eBook)
DOI 10.1007/978-1-4471-0335-6
British Library Cataloguing in Publication Data
Ben-Ari.M .• 1948-
Mathematicallogic for computer science. -2nd rev. ed.
I.Logic, Symbolic and mathematical
I.Titie
511.3
ISBN 978-1-85233-319-55
Library of Congress Cataloging-in-Publication Data
Ben-Ari., 1948-
Mathematicallogic for computer science I Mordechai Ben-Ari.-·2nd rev. ed.
p.cm.
Indudes bibliographical references and indexes.
ISBN 1-85233-319-7 (acid-free paper)
1. Logic, Symbolic and mathematical. I. Title
QA9 .B3955 2001
511.3--dc21 00-066113
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© Springer-Verlag London 2001
Originally published by Springer-Verlag London Berlin Heidelberg in 2001
2nd printing with corrections 2003
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4th printing 2004
First published 1993
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Preface
Students of science and engineering are required to study mathematics during their
first years at a university. Traditionally, they concentrate on calculus, linear algebra
and differential equations, but in computer science and engineering, logic, combina
torics and discrete mathematics are more appropriate. Logic is particularly important
because it is the mathematical basis of software: it is used to formalize the seman
tics of programming languages and the specification of programs, and to verify the
correctness of programs.
Mathematical Logic for Computer Science is a mathematics textbook, just as a first
year calculus text is a mathematics textbook. A scientist or engineer needs more than
just a facility for manipulating formulas and a firm foundation in mathematics is an
excellent defense against technological obsolescence. Tempering this requirement for
mathematical competence is the realization that applications use only a fraction of the
theoretical results. Just as the theory of calculus can be taught to students of engineer
ing without the full generality of measure theory, students of computer science need
not be taught the full generality of uncountable structures. Fortunately (as shown by
Raymond M. Smullyan), tableaux provide an elegant way to teach mathematical logic
that is both theoretically sound and yet sufficiently elementary for the undergraduate.
Audience
The book is intended for undergraduate computer science students. No specific mathe
matical knowledge is assumed aside from informal set theory which is summarized in
an appendix, but elementary knowledge of concepts from computer science (graphs,
languages, programs) are used. Prolog implementations of many of the algorithms are
given; for a computer science student, the study of a concrete program can reinforce
the study of an abstract algorithm. An ideal course in logic would supplement the
theory in this book with practical study of logic programming.
My approach can be characterized as broad, elementary and rigorous: I want to cover
many topics rigorously, but I have chosen elementary material, generally the minimal
subset of a logic that can be profitably studied. Examples are: the predicate calculus
without equality, a future fragment of temporal logic and partial rather than total cor
rectness. The student who has learned the results and techniques in these elementary
cases will be well-prepared to study the more general systems in advanced courses.
viii Preface
Organization
Chapter 1 is introductory and surveys the topics in the book. Appendix A summarizes
the elementary set theory that the reader should know, and Appendix B contains a
guide to the literature. The rest of the book covers five main topics:
• Propositional calculus (Chapters 2,3,4).
• Predicate calculus (Chapters 5, 6).
• Resolution and logic programming (Chapters 7, 8).
• Program specification and verification (Chapters 9, 10).
• Temporal logic (Chapters 11, 12).
The first two topics form the core of classical mathematical logic, though I have aug
mented them with algorithms and programs, and material of interest to computer sci
entists. The other three topics are chosen for their specific relevance in modern com
puter science.
The general progression in each main topic is: syntax and semantics of formulas,
semantic tableaux, deductive systems and algorithms. While many modern textbooks
are heavily weighted in favor of algorithmic semantic methods, I have tried to give
equal time to syntactic deduction. Deduction is still the language of mathematical
reasoning and is the only fall-back when semantic methods fail.
The propositional and predicate calculi are the central topics in logic and should be
taught in any course. The other three topics are independent of each other and the
instructor can select or rearrange the topics as desired. Sections marked * contain ad
vanced material that may be skipped in lower-level undergraduate classes, as well as
interesting results presented without proof that are beyond the scope of this book. Sec
tions marked P contain the Prolog programs. The printed programs are only fragments;
the full source code of the programs (including routines for input-output and testing)
is available online (http://www.springer.co.uk/com_pubs/cLmics.htm). The web
site will also include answers to exercises. The programs have been run on the free im
plementation SWI Prolog (http://www.swi.psy.uva.nl/projects/SWI-Prolog/). but
should run on any implementation of the Prolog standard.
Second edition
The second edition has been totally rewritten. Major additions are sections on binary
decision diagrams, constraint logic programming and the completeness of Hoare logic.
One curiosity: I had used Fermat's Last Theorem as an example of a formula whose
truth is not known. Since then, Andrew Wiles has proved the theorem (Singh 1997)!
I have chosen to replace it by Goldbach's conjecture.
Preface ix
Notation
'If and only if' is abbreviated iff. Definitions by convention use iff to emphasize that
the definition is restrictive. Consider the definition: a natural number is even iff it can
be expressed as 2k for some natural number k. In this context, iff means that numbers
expressible as 2k are even and these are the only even numbers.
Definitions, theorems and examples are consecutively numbered within each chapter
to make them easy to locate. The end of a proof is denoted by I and the end of an
example or definition is denoted by 0 .
Acknowledgements
I am very grateful to Mark Ryan and Anna LaBella for their merciless comments on
the manuscript. Amir Pnueli helped bring me up to date on temporal logic, as did
Ehud Shapiro on logic programming. I appreciate receiving corrections to the first
edition from Arnon Avron, Anna LaBella and Rudiger Reischuk.
I would like to thank Beverley Ford and her team at Springer-Verlag London for their
enthusiasm and support during the past two years. Rebecca Mowat deserves a special
citation for expertly fielding dozens of email messages.
Finally, the students in my logic class at the Weizmann Institute of Science (in partic
ular Dana Fisman) were extremely helpful in debugging a preliminary draft.
M. Ben-Ari
Rehovot, 2000
Contents
Preface . . . . . ................................ vii
1 Introduction . ................ 1
1.1 The origins of mathematical logic 1
1.2 Propositional calculus . 2
1.3 Predicate calculus . . . 3
1.4 Theorem proving and logic programming 5
1.5 Systems of logic . 6
1.6 Exercise ..... 7
2 Propositional Calculus: Formulas, Models, Tableaux 9
2.1 Boolean operators ... 9
2.2 Propositional formulas 12
2.3 Interpretations . . . . . 17
2.4 Logical equivalence and substitution 19
2.5 Satisfiability, validity and consequence . 24
2.6 Semantic tableaux . . . . . . 29
2.7 Soundness and completeness 33
2.8 ImplementationP 38
2.9 Exercises .... 40
3 Propositional Calculus: Deductive Systems . 43
3.1 Deductive proofs ... 43
3.2 The Gentzen system C . 45
3.3 The Hilbert system 11. . 48
3.4 Soundness and completeness of 11. 56
3.5 A proof checker ......... 59
3.6 Variant forms of the deductive systems· 60
3.7 Exercises ................ 64
xii Contents
4 Propositional Calculus: Resolution and BDDs 67
4.1 Resolution . . . . . . . . . . . . . 67
4.2 Binary decision diagrams (BDDs) 81
4.3 Algorithms on BDDs 88
4.4 Complexity* 95
4.5 Exercises 99
5 Predicate Calculus: Formulas, Models, Tableaux 101
5.1 Relations and predicates. 101
5.2 Predicate formulas . 102
5.3 Interpretations . . . 105
5.4 Logical equivalence and substitution 107
5.5 Semantic tableaux . 109
5.6 ImplementationP 118
5.7 Finite and infinite models* 120
5.8 Decidability* 121
5.9 Exercises .. 125
6 Predicate Calculus: Deductive Systems . 127
6.1 The Gentzen system C . 127
6.2 The Hilbert system 11 129
6.3 ImplementationP .. 134
6.4 Complete and decidable theories* . 135
6.5 Exercises ....... . . . . . . 138
7 Predicate Calculus: Resolution 139
7.1 Functions and terms . 139
7.2 Clausal form . . . 142
7.3 Herbrand models 148
7.4 Herbrand's Theorem* 150
7.5 Ground resolution 152
7.6 Substitution 153
7.7 Unification. 155
7.8 General resolution . 164