Automata Theory and its Applications

'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 Co