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Automata Theory and Its Applications (Progress in Computer Science and Applied Logic

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Automata Theory and its Applications is a uniform treatment of the theory of finite state machines on finite and infinite strings and trees. Many books deal with automata on finite strings, but there are very few expositions that prove the fundamental results of automata on infinite strings and trees. These results have important applications to modeling parallel computation and concurrency, the specification and verification of sequential and concurrent programs, databases, operating systems, computational complexity, and decision methods in logic and algebra. Thus, this textbook  fills an important gap in the literature by exposing early fundamental results in automata theory and its applications.

Beginning with coverage of all standard fundamental results regarding finite automata, the book deals in great detail with B??chi and Rabin automata and their applications to various logical theories such as S1S and S2S, and describes game-theoretic models of concurrent operating and communication systems.

The book is self-contained with numerous examples, illustrations, exercises, and is suitable for a two-semester undergraduate course for computer science or mathematics majors, or for a one-semester graduate course/seminar. Since no advanced mathematical background is required, the text is also useful for self-study by computer science professionals who wish to understand the foundations of modern formal approaches to software development, validation, and verification.

Product Details

Table of Contents

1. Basic Notions.- 1.1 Sets.- 1.2 Sequences and Tuples.- 1.3 Functions, Relations, Operations.- 1.4 Equivalence Relations.- 1.5 Linearly Ordered Sets.- 1.6 Partially Ordered Sets.- 1.7 Graphs.- 1.8 Induction.- 1.9 Trees and Koenig's Lemma.- 1.10 Countable and Uncountable Sets.- 1.10.1 Countable Sets.- 1.10.2 Diagonalization and Uncountable Sets.- 1.11 Algorithms.- 2 Finite Automata.- 2.1 Two Examples.- 2.1.1 The Consumer-Producer Problem.- 2.1.2 A Monkey and Banana Problem.- 2.2 Finite Automata.- 2.2.1 Definition of Finite Automata and Languages.- 2.2.2 Runs (Computations) of Finite Automata.- 2.2.3 Accessibility and Recognizability.- 2.3 Closure Properties.- 2.3.1 Union and Intersection.- 2.3.2 Complementation and Nondeterminism.- 2.3.3 On the Exponential Blow-Up of Complementation.- 2.3.4 Some Other Operations.- 2.3.5 Projections of Languages.- 2.4 The Myhill-Nerode Theorem.- 2.5 The Kleene Theorem.- 2.5.1 Regular Languages.- 2.5.2 Regular Expressions.- 2.5.3 The Kleene Theorem.- 2.6 Generalized Finite Automata.- 2.7 The Pumping Lemma and Decidability.- 2.7.1 Basic Problems.- 2.7.2 The Pumping Lemma.- 2.7.3 Decidability.- 2.8 Relations and Finite Automata.- 2.9 Finite Automata with Equations.- 2.9.1 Preliminaries.- 2.9.2 Properties of E-Languages.- 2.10 Monadic Second Order Logic of Strings.- 2.10.1 Finite Chains.- 2.10.2 The Monadic Second Order Logic of Strings.- 2.10.3 Satisfiability.- 2.10.4 Discussion and Plan About SFS.- 2.10.5 From Automata to Formulas.- 2.10.6 From Formulas to Automata.- 3 Buchi Automata.- 3.1 Two Examples.- 3.1.1 The Dining Philosophers Problem.- 3.1.2 Consumer-Producer Problem Revisited.- 3.2 Buchi Automata.- 3.2.1 Basic Notions.- 3.2.2 Union, Intersection, and Projection.- 3.3 The Buchi Theorem.- 3.3.1 Auxiliary Results.- 3.3.2 Buchi's Characterization.- 3.4 Complementation for Buchi Automata.- 3.4.1 Basic Notations.- 3.4.2 Congruence ?.- 3.5 The Complementation Theorem.- 3.6 Determinism.- 3.7 Muller Automata.- 3.7.1 Motivation and Definition.- 3.7.2 Properties of Muller Automata.- 3.7.3 Sequential Rabin Automata.- 3.8 The McNaughton Theorem.- 3.8.1 Flag Points.- 3.8.2 The Theorem.- 3.9 Decidability.- 3.10 Buchi Automata and the Successor Function.- 3.10.1 ?-Strings as Structures.- 3.10.2 Monadic Second Order Formalism.- 3.10.3 Satisfiability.- 3.10.4 From Buchi Automata to Formulas.- 3.10.5 From Formulas to Buchi Automata.- 3.10.6 Decidability and Definability in S1S.- 3.11 An Application of the McNaughton Theorem.- 4 Games Played on Finite Graphs.- 4.1 Introduction.- 4.2 Finite Games.- 4.2.1 Informal Description.- 4.2.2 Definition of Finite Games and Examples.- 4.2.3 Finding The Winners.- 4.3 Infinite Games.- 4.3.1 Informal Description and an Example.- 4.3.2 Formal Definition of Games.- 4.3.3 Strategies.- 4.4 Update Games and Update Networks.- 4.4.1 Update Games and Examples.- 4.4.2 Deciding Update Networks.- 4.5 Solving Games.- 4.5.1 Forgetful Strategies.- 4.5.2 Constructing Forgetful Strategies.- 4.5.3 No-Memory Forcing Strategies.- 4.5.4 Finding Winning Forgetful Strategies.- 5 Rabin Automata.- 5.1 Rabin Automata.- 5.1.1 Union, Intersection, and Projection.- 5.2 Special Automata.- 5.2.1 Basic Properties of Special Automata.- 5.2.2 A Counterexample to Complementation.- 5.3 Game Automata.- 5.3.1 What Is a Game?.- 5.3.2 Game Automata.- 5.3.3 Strategies.- 5.4 Equivalence of Rabin and Game Automata.- 5.5 Terminology: Arenas, Games, and Strategies.- 5.6 The Notion of Rank.- 5.7 Open Games.- 5.8 Congruence Relations.- 5.9 Sewing Theorem.- 5.10 Can Mr. (?) Visit C Infinitely Often?.- 5.10.1 Determinacy Theorem for Games (?, [C], (?).- 5.10.2 An Example of More Complex Games.- 5.11 The Determinacy Theorem.- 5.11.1 GH-Games and Last Visitation Record.- 5.11.2 The Restricted Memory Determinacy Theorem.- 5.12 Complementation and Decidability.- 5.12.1 Forgetful Determinacy Theorem.- 5.12.2 Solution of the Complementation Problem.- 5.12.3 Decidability.- 6 Applications of Rabin Automata.- 6.1 Structures and Types.- 6.2 The Monadic Second Order Language.- 6.3 Satisfiability and Theories.- 6.4 Isomorphisms.- 6.5 Definability in T and Decidability of S2S.- 6.5.1 ?-Valued Trees as Structures.- 6.5.2 Definable Relations.- 6.5.3 From Rabin Automata to Formulas.- 6.5.4 From Formulas to Rabin Automata.- 6.5.5 Definability and Decidability.- 6.6 The Structure with ? Successors.- 6.7 Applications to Linearly Ordered Sets.- 6.7.1 Two Algebraic Facts.- 6.7.2 Decidability.- 6.8 Application to Unary Algebras.- 6.8.1 Unary Structures.- 6.8.2 Enveloping Algebras.- 6.8.3 Decidability.- 6.9 Applications to Cantor's Discontinuum.- 6.9.1 A Brief Excursion to Cantor's Discontinuum.- 6.9.2 Cantor's Discontinuum as a Topological Space.- 6.9.3 Expressing Subsets of CD in S2S.- 6.9.4 Decidability Results.- 6.10 Application to Boolean Algebras.- 6.10.1 A Brief Excursion into Boolean Algebras.- 6.10.2 Ideals, Factors, and Subalgebras of Boolean Algebras.- 6.10.3 Maximal Ideals of Boolean Algebras.- 6.10.4 The Stone Representation Theorem.- 6.10.5 Homomorphisms of Boolean Algebras.- 6.10.6 Decidability Results.

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"The aim of this book is to present a theory of several types of automata and applications of these facts in logic, concurrency and algebra. ...The book contains suitable material for a two-semester course for students of computer science or mathematics. It is completely self-contained and one can really enjoy reading it. It is strongly recommended for researchers and postgraduate students interested in logic, automata and/or concurrency." --EMS

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