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Birkhäuser - Birkhäuser Mathematics | Automata Theory and its Applications

Automata Theory and its Applications

Khoussainov, Bakhadyr, Nerode, Anil

2001, XIV, 432 p.

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The theory of finite automata on finite stings, infinite strings, and trees has had a dis­ tinguished history. First, automata were introduced to represent idealized switching circuits augmented by unit delays. This was the period of Shannon, McCullouch and Pitts, and Howard Aiken, ending about 1950. Then in the 1950s there was the work of Kleene on representable events, of Myhill and Nerode on finite coset congruence relations on strings, of Rabin and Scott on power set automata. In the 1960s, there was the work of Btichi on automata on infinite strings and the second order theory of one successor, then Rabin's 1968 result on automata on infinite trees and the second order theory of two successors. The latter was a mystery until the introduction of forgetful determinacy games by Gurevich and Harrington in 1982. Each of these developments has successful and prospective applications in computer science. They should all be part of every computer scientist's toolbox. Suppose that we take a computer scientist's point of view. One can think of finite automata as the mathematical representation of programs that run us­ ing fixed finite resources. Then Btichi's SIS can be thought of as a theory of programs which run forever (like operating systems or banking systems) and are deterministic. Finally, Rabin's S2S is a theory of programs which run forever and are nondeterministic. Indeed many questions of verification can be decided in the decidable theories of these automata.

Content Level » Research

Keywords » Automat - automata - complexity - finite automata - logic - modeling - verification

Related subjects » Birkhäuser Computer Science - Birkhäuser Mathematics

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 König’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 Büchi Automata.- 3.1 Two Examples.- 3.1.1 The Dining Philosophers Problem.- 3.1.2 Consumer—Producer Problem Revisited.- 3.2 Büchi Automata.- 3.2.1 Basic Notions.- 3.2.2 Union, Intersection, and Projection.- 3.3 The Büchi Theorem.- 3.3.1 Auxiliary Results.- 3.3.2 Büchi’s Characterization.- 3.4 Complementation for Büchi Automata.- 3.4.1 Basic Notations.- 3.4.2 Congruence ?.- 3.5 The Complementation Theorem.- 3.6 Determinism.- 3.7 Müller Automata.- 3.7.1 Motivation and Definition.- 3.7.2 Properties of Müller 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 Büchi 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 Büchi Automata to Formulas.- 3.10.5 From Formulas to Büchi 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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