Automata Theory and Its Applications

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. Beginning with coverage of all standard fundamental results regarding finite automata, the book deals in great detail with Büchi 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. Self-contained with numerous examples, illustrations, exercises. Suitable for a two-semester undergraduate course for computer science or math majors, or for a one-semester graduate course/seminar. No advanced mathematical background is required, thus 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.

"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

Inhalt

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 König's Lemma.- 1.10 Countable and Uncountable Sets.- 1.11 Algorithms.- 2 Finite Automata.- 2.1 Two Examples.- 2.2 Finite Automata.- 2.3 Closure Properties.- 2.4 The MyhillNerode Theorem.- 2.5 The Kleene Theorem.- 2.6 Generalized Finite Automata.- 2.7 The Pumping Lemma and Decidability.- 2.8 Relations and Finite Automata.- 2.9 Finite Automata with Equations.- 2.10 Monadic Second Order Logic of Strings.- 3 Büchi Automata.- 3.1 Two Examples.- 3.2 Büchi Automata.- 3.3 The Büchi Theorem.- 3.4 Complementation for Büchi Automata.- 3.5 The Complementation Theorem.- 3.6 Determinism.- 3.7 Müller Automata.- 3.8 The McNaughton Theorem.- 3.9 Decidability.- 3.10 Büchi Automata and the Successor Function.- 3.11 An Application of the McNaughton Theorem.- 4 Games Played on Finite Graphs.- 4.1 Introduction.- 4.2 Finite Games.- 4.3 Infinite Games.- 4.4 Update Games and Update Networks.- 4.5 Solving Games.- 5 Rabin Automata.- 5.1 Rabin Automata.- 5.2 Special Automata.- 5.3 Game Automata.- 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?.- 5.11 The Determinacy Theorem.- 5.12 Complementation and 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.- 6.6 The Structure with ? Successors.- 6.7 Applications to LinearlyOrdered Sets.- 6.8 Application to Unary Algebras.- 6.9 Applications to Cantor's Discontinuum.- 6.10 Application to Boolean Algebras.