Rudiments of Calculus (eBook)

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2001 | 1. Auflage
298 Seiten
Elsevier Science (Verlag)
978-0-08-051645-5 (ISBN)

This book presents what in our opinion constitutes the basis of the theory of the mu-calculus, considered as an algebraic system rather than a logic. We have wished to present the subject in a unified way, and in a form as general as possible. Therefore, our emphasis is on the generality of the fixed-point notation, and on the connections between mu-calculus, games, and automata, which we also explain in an algebraic way.
This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially for researchers and students interested in logic in computer science, comuter aided verification, and general aspects of automata theory. We have aimed at gathering in a single place the fundamental results of the theory, that are currently very scattered in the literature, and often hardly accessible for interested readers.
The presentation is self-contained, except for the proof of the Mc-Naughton's Determinization Theorem (see, e.g., [97]. However, we suppose that the reader is already familiar with some basic automata theory and universal algebra. The references, credits, and suggestions for further reading are given at the end of each chapter.

This book presents what in our opinion constitutes the basis of the theory of the mu-calculus, considered as an algebraic system rather than a logic. We have wished to present the subject in a unified way, and in a form as general as possible. Therefore, our emphasis is on the generality of the fixed-point notation, and on the connections between mu-calculus, games, and automata, which we also explain in an algebraic way. This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially for researchers and students interested in logic in computer science, comuter aided verification, and general aspects of automata theory. We have aimed at gathering in a single place the fundamental results of the theory, that are currently very scattered in the literature, and often hardly accessible for interested readers. The presentation is self-contained, except for the proof of the Mc-Naughton's Determinization Theorem (see, e.g., [97]. However, we suppose that the reader is already familiar with some basic automata theory and universal algebra. The references, credits, and suggestions for further reading are given at the end of each chapter.

Cover 1
Copyright 5
Dedication 6
Preface 8
Table of Contents 14
Chapter 1. Complete lattices and fixed-point theorems 20
1.1 Complete lattices 20
1.2 Fixed-point theorems 27
1.3 Some properties of fixed points 38
1.4 Fixed points on product lattices 43
1.5 Conway identities 58
1.6 Bibliographical notes and sources 58
Chapter 2. The µ-calculi: Syntax and semantics 60
2.1 µ-calculi 61
2.2 Functional µ-calculi 63
2.3 Fixed-point terms 65
2.4 Quotient µ-calculi 69
2.5 Powerset interpretations 72
2.6 Alternation-depth hierarchy 74
2.7 Vectorial µ-calculi 77
2.8 Bibliographic notes and sources 88
Chapter 3. The Boolean µ-calculus 90
3.1 Monotone Boolean functions 90
3.2 Powerset interpretations and the Boolean µ-calculus 91
3.3 The selection property 94
3.4 Bibliographic notes and sources 102
Chapter 4. Parity games 104
4.1 Games and strategies 105
4.2 Positional strategies 106
4.3 The µ-calculus of games 107
4.4 Games for the µ-calculus 114
4.5 Weak parity games 118
4.6 Bibliographic notes and sources 120
Chapter 5. The µ-calculus on words 124
5.1 Rational languages 124
5.2 Nondeterministic automata 134
5.3 Terms with intersection 145
5.4 Bibliographic notes and sources 157
Chapter 6. The µ-calculus over powerset algebras 160
6.1 Powerset algebras 160
6.2 Modal µ-calculus 164
6.3 Homomorphisms, µ-homomorphisms, and bisimulations 167
6.4 Bibliographic notes and sources 172
Chapter 7. The µ-calculus vs. automata 174
7.1 Automata over semi-algebras 174
7.2 Automata in the µ-calculus perspective 190
7.3 Equivalences 199
7.4 Bibliographic notes and sources 206
Chapter 8. Hierarchy problems 210
8.1 Introduction 210
8.2 The hierarchy of alternating parity tree automata 211
8.3 Weak alternating automata 220
8.4 Bibliographic notes and sources 222
Chapter 9. Distributivity and normal form results 224
9.1 The propositional µ-calculus 224
9.2 Guarded terms 227
9.3 Intersection-free terms 230
9.4 The powerset construction 235
9.5 The vµ case 236
9.6 The simulation theorem 240
9.7 Bibliographic notes and sources 248
Chapter 10. Decision problems 252
10.1 Disjunctive mappings 253
10.2 Decidability of emptiness for disjunctive µ-terms 254
10.3 The regularity theorem 259
10.4 The satisfiability over powerset algebras 261
10.5 Bibliographic notes and sources 269
Chapter 11. Algorithms 272
11.1 Evaluation of vectorial fixed-point terms 272
11.2 Winning positions and winning strategies 284
11.3 Bibliographic notes and sources 286
Bibliography 288
Index 294

Erscheint lt. Verlag	7.2.2001
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Technik
ISBN-10	0-08-051645-9 / 0080516459
ISBN-13	978-0-08-051645-5 / 9780080516455

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