Logics for Computer Science, 2nd Edition

431 Pages·2018·2.449 MB·English

by  Arindama Singh

Logics for Computer Science
Second Edition
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
Delhi-110092
2018

Contents
Preface ...............................................................................................................vii
Preface to the First Edition .................................................................................ix
1. Propositional Logic ....................................................................................1
1.1 Introduction .......................................................................................1
1.2 Syntax of PL ..................................................................................... 3
1.3 Is It a Proposition? ............................................................................ 6
1.4 Interpretations ..................................................................................11
1.5 Models ............................................................................................ 17
1.6 Equivalences and Consequences .................................................... 20
1.7 More About Consequence .............................................................. 25
1.8 Summary and Problems ................................................................. 27
2. A Propositional Calculus .........................................................................35
2.1 Axiomatic System PC .....................................................................35
2.2 Five Theorems about PC ................................................................ 41
2.3 Using the Metatheorems ................................................................. 45
2.4 Adequacy of PC to PL .....................................................................50
2.5 Compactness of PL ......................................................................... 55
2.6 Replacement Laws ......................................................................... 60
2.7 Quasi-proofs In PL ......................................................................... 64
2.8 Summary and Problems ................................................................. 66
3. Normal Forms and Resolution ................................................................70
3.1 Truth Functions ...............................................................................70
3.2 CNF and DNF .................................................................................72
3.3 Logic Gates .................................................................................... 77
3.4 Satisfiability Problem ......................................................................81
3.5 2SAT and Horn-SAT ..................................................................... 83
3.6 Resolution in PL ............................................................................. 86
3.7 Adequacy of resolution in PL ......................................................... 91
3.8 Resolution Strategies .......................................................................94
3.9 Summary and Problems ................................................................. 97
4. Other Proof Systems for PL ..................................................................103
4.1 Calculation ................................................................................... 103
4.2 Natural Deduction ........................................................................ 106
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4.3 Gentzen Sequent Calculus ............................................................ 110
4.4 Analytic Tableaux ........................................................................ 116
4.5 Adequacy of PT to PL ...................................................................122
4.6 Summary and Problems ............................................................... 127
5. First Order Logic ...................................................................................131
5.1 Syntax of FL ................................................................................. 131
5.2 Scope and Binding ....................................................................... 135
5.3 Substitutions ................................................................................. 139
5.4 Semantics of FL ............................................................................141
5.5 Translating into FL ........................................................................146
5.6 Satisfiability and Validity ............................................................. 149
5.7 Some Metatheorems ..................................................................... 152
5.8 Equality Sentences ....................................................................... 157
5.9 Summary and Problems ............................................................... 163
6. A First Order Calculus ..........................................................................168
6.1 Axiomatic System FC ...................................................................168
6.2 Six Theorems about FC ................................................................ 171
6.3 Adequacy of FC to FL ...................................................................177
6.4 Compactness of FL ....................................................................... 184
6.5 Laws in FL ................................................................................... 190
6.6 Quasi-proofs in FL ....................................................................... 195
6.7 Summary and Problems ............................................................... 198
7. Clausal Forms and Resolution ..............................................................201
7.1 Prenex Form ................................................................................. 201
7.2 Quantifier-free forms .....................................................................204
7.3 Clauses ......................................................................................... 211
7.4 Unification of Clauses ...................................................................213
7.5 Extending Resolution ....................................................................220
7.6 Factors and Pramodulants ............................................................ 223
7.7 Resolution for FL ......................................................................... 226
7.8 Horn Clauses in FL ...................................................................... 229
7.9 Summary and Problems ............................................................... 232
8. Other Proof Systems for FL ..................................................................236
8.1 Calculation ................................................................................... 236
8.2 Natural Deduction ........................................................................ 240
8.3 Gentzen Sequent Calculus ............................................................ 245
8.4 Analytic Tableaux ........................................................................ 250
8.5 Adequacy of FT to FL ...................................................................255
8.6 Summary and Problems ............................................................... 260
CONTENTS v
9. Program Verification .............................................................................263
9.1 Debugging a Program ....................................................................263
9.2 Issue of Correctness ..................................................................... 265
9.3 The Core Language CL ................................................................ 268
9.4 Partial Correctness ........................................................................ 272
9.5 Axioms And Rules ....................................................................... 275
9.6 Hoare Proof .................................................................................. 279
9.7 Proof Summary .............................................................................282
9.8 Total Correctness .......................................................................... 288
9.9 A Predicate Transformer .............................................................. 292
9.10 Summary and Problems ............................................................... 300
10. First Order Theories ..............................................................................305
10.1 Structures and Axioms ................................................................. 305
10.2 Set Theory .................................................................................... 310
10.3 Arithmetic ..................................................................................... 313
10.4 Herbrand Interpretation ................................................................ 316
10.5 Herbrand Expansion ..................................................................... 318
10.6 Skolem-Löwenheim Theorems .................................................... 322
10.7 Decidability .................................................................................. 324
10.8 Expressibility ................................................................................ 328
10.9 Provability Predicate .....................................................................332
10.10 Summary and Problems ............................................................... 336
11. Modal Logic K ........................................................................................341
11.1 Introduction .................................................................................. 341
11.2 Syntax and Semantics of K ...........................................................343
11.3 Validity and Consequence in K .................................................... 350
11.4 Axiomatic System KC ...................................................................354
11.5 Adequacy of KC to K ....................................................................357
11.6 Natural Deduction in K ................................................................ 359
11.7 Analytic Tableau for K................................................................. 362
11.8 Other Modal Logics ..................................................................... 368
11.9 Various Modalities ....................................................................... 375
11.10 Computation Tree Logic .............................................................. 379
11.11 Summary and Problems ............................................................... 384
12. Some Other Logics .................................................................................387
12.1 Introduction .................................................................................. 387
12.2 Intuitionistic Logic ....................................................................... 388
12.3 Łukasiewicz Logics ...................................................................... 391
12.4 Probabilistic Logics ...................................................................... 395
12.5 Possibilistic and Fuzzy Logic ....................................................... 396
12.5.1 Crisp sentences and precise information .......................... 397
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12.5.2 Crisp sentences and imprecise information ..................... 397
12.5.3 Crisp sentences and fuzzy information .............................398
12.5.4 Vague sentences and fuzzy information .......................... 398
12.6 Default Logic ................................................................................ 398
12.7 Autoepistemic Logics ....................................................................402
12.8 Summary .......................................................................................404
References ......................................................................................................405
Index ................................................................................................................411
Preface
The first edition of this book was used by many academicians; and their comments
and suggestions had to be implemented sooner or later. For instance, Prof. Norman
Foo of University of New South Wales (UNSW) said:
“This is the best book available in the market that suits my course on logic
for CS masters students.”
However, one of his colleagues reported that he had to refer to other books for
applications of compactness. Now, it becomes somewhat obligatory on my part to
discuss applications of compactness in this edition.
In this revised version, the circularity in presenting logic via formal semantics
is brought to the fore in a very elementary manner. Instead of developing everything
from semantics, we now use an axiomatic system to model reasoning. Other proof
methods are introduced and worked out later as alternative models.
Elimination of the equality predicate via equality sentences is dealt with
semantically even before the axiomatic system for first order logic is presented.
The replacement laws and the quantifier laws are now explicitly discussed along
with the necessary motivation of using them in constructing proofs in mathematics.
Adequacy of the axiomatic system is now proved in detail. An elementary proof of
adequacy of Analytic Tableaux is now included.
Special attention is paid to the foundational questions such as decidability,
expressibility, and incompleteness. These important and difficult topics are dealt
with briefly and in an elementary manner.
The material on Program Verification, Modal Logics, and Other Logics in
Chapters 9, 11 and 12 have undergone minimal change. Attempt has been made to
correct all typographical errors pointed out by the readers. However, rearrangement
of the old material and the additional topics might have brought in new errors.
Numerous relevant results, examples, exercises and problems have been added.
The correspondence of topics to chapters and sections have changed considerably,
compared to the first edition. A glance through the table of contents will give you a
comprehensive idea.
The book now contains enough material to keep students busy for two semesters.
However, by carefully choosing the topics, the essential portion of the book can be
covered in a single semester. The core topics are discussed in Chapters 1, 2, 5, 6, 10,
and Sections 3.1–3.5 and 7.1–7.2. As an alternative to the axiomatic system, one may
replace Chapters 2 and 6 with Sections 4.4–4.5, 8.4–8.5, 2.5 and 6.4. Topics from
other chapters may be discussed depending on the requirement of the students.
I cheerfully thank all those whose suggestions were the driving force in
bringing out this edition. I thank all those students who wanted me to relate to
them the rigorous foundation of mathematics. I acknowledge the trouble taken by
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viii PREFACE
Norman Foo of UNSW, S. H. Kulkarni, Sounaka Mishra, Kalpana Mahalingam of
IIT Madras, Balasubramaniam Jayaram of IIT Hyderabad, and my long-time friend
Biswa R. Patnaik of Toronto, in carefully reading the earlier drafts, finding mistakes,
and suggesting improvements.
Most part of this edition was drafted during my three months stay at Fayetteville,
Arkansas with my family. The pleasure of working on the book was doubled due to
frequent get together arranged by Brajendra Panda of University of Arkansas, and
his wife Rashmi; I thank them profusely. I thank the administrators of IIT Madras
for granting me sabbatical so that I could devote full time on the book. I also thank
the publisher, PHI Learning, Delhi and their production team for timely help in
typesetting.
Arindama Singh
Preface to the First Edition
Each chapter in this book has an introduction, the first section. But there is no
introduction to the book itself. So, let me use the mask of the Preface for this purpose.
Of course, a plain introduction to the book is just to place the book in your hand; but
while you are reading it, you yourself have already done that. Well done!
This is the way I will be talking to you throughout. I will ask you to do exercises
on the spot, often waiting for you up to some point, and then give you a hint to
proceed. It is something like the following commercial for the book:
I would like to ask you three questions, would you answer them with a
plain ‘Yes’ or ‘No’?
Good, you have answered ‘Yes’, whatever be the reason. But see, that
was my first question. Would you answer the same to the second as to
the third?
Very good, you have answered ‘Yes’ again; of course, it does not
matter. If you have not bought a copy of this book, are you going to
buy it soon?
Look, if you have answered ‘Yes’ to the second question, you are
also answering ‘Yes’ to the third, and if you have answered ‘No’ to
the second, you are not answering ‘No’ to the third, i.e., your answer
to the third is undoubtedly, ‘Yes’. Excellent. That is the end of the
commercial.
You have participated well in the commercial; I hope you will be with me
throughout. It is easy to learn logic and easier to teach it; that is the spirit of this
book. I would not reveal what is logic; you will discover it eventually. My aim is to
equip you with the logical methods so that when you take up computer science as
your profession, you will feel like a fish in water. A warning: though the book is ideal
for self-learning, it would not replace a teacher. At least on one count: you can ask a
question to your teacher and, hopefully, he1 will give you an answer, or you will be
inspired by him for finding your own answer; the book cannot do that always.
If you are a teacher, you may not need to learn all the topics, for you had probably
learnt them. But you can really teach logic better by helping your students in the
exercises. You can supplement the book with more computer science applications
1The masculine gender is used throughout the book not because of some chauvinistic
reasons, as it is easy to do that. It is easy to write ‘he’ rather than ‘she’ or ‘one’ or ‘a person’
or ‘a human being’ etc. You need not be offended by it.
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