The Classical Decision Problem

Transcription

1 Egon Borger Erich Gradel Yuri Gurevich The Classical Decision Problem Springer

2 Table of Contents Preface VII 1. Introduction: The Classical Decision Problem The Original Problem The Transformation of the Classical Decision Problem What Is and What Isn't in this Book 8 Part I. Undecidable Classes 2. Reductions Undecidability and Conservative Reduction The Church-Turing Theorem and Reduction Classes Trakhtenbrot's Theorem and Conservative Reductions Inseparability and Model Complexity Logic and Complexity Prepositional Satisfiability The Spectrum Problem and Fagin's Theorem Capturing Complexity Classes A Decidable Prefix-Vocabulary Class The Classifiability Problem The Problem Well Partially Ordered Sets The Well Quasi Ordering of Prefix Sets The Well Quasi Ordering of Arity Sequences The Classifiability of Prefix-Vocabulary Sets Historical Remarks Undecidable Standard Classes for Pure Predicate Logic The Kahr Class Domino Problems Formalization of Domino Problems by [V3V, (0, u)]- Formulae Graph Interpretation of [V3V, (0, w)]-formulae 98

3 X Table of Contents The Remaining Cases Without 3* Existential Interpretation for [V 3 3*, (0,1)] The Gurevich Class The Proof Strategy Reduction to Diagonal-Freeness Reduction to Shift-Reduced Form Reduction to Fi-Elimination Form Elimination of Monadic Fi The Kostyrko-Genenz and Suranyi Classes Historical Remarks Undecidable Standard Classes with Functions or Equality Classes with Functions and Equality Classes with Functions but Without Equality Classes with Equality but Without Functions: the Goldfarb Classes Formalization of Natural Numbers in [V 2 3*, (u>,u), (0)]= Using Only One Existential Quantifier Encoding the Non-Auxiliary Binary Predicates Encoding the Auxiliary Binary Predicates of NUM* Historical Remarks Other Undecidable Cases Krom and Horn Formulae Krom Prefix Classes Without Functions or Equality Krom Prefix Classes with Functions or Equality Few Atomic Subformulae Few Function and Equality Free Atoms Few Equalities and Inequalities Horn Clause Programs With One Krom Rule Undecidable Logics with Two Variables First-Order Logic with the Choice Operator Two-Variable Logic with Cardinality Comparison Conjunctions of Prefix-Vocabulary Classes Reduction to the Case of Conjunctions Another Classifiability Theorem Some Results and Open Problems Historical Remarks 233

4 Table of Contents XI Part II. Decidable Classes and Their Complexity 6. Standard Classes with the Finite Model Property Techniques for Proving Complexity Results Domino Problems Revisited Succinct Descriptions of Inputs The Classical Solvable Cases Monadic Formulae The Bernays-Schonfinkel-Ramsey Class The Godel-Kalmar-Schiitte Class: a Probabilistic Proof Formulae with One V A Satisfiability Test for [3*V3*, all, all] The Ackermann Class The Ackermann Class with Equality Standard Classes of Modest Complexity The Relational Classes in P, NP and Co-NP Fragments of the Theory of One Unary Function Other Functional Classes Finite Model Property vs. Infinity Axioms Historical Remarks Monadic Theories and Decidable Standard Classes with Infinity Axioms Automata, Games and Decidability of Monadic Theories Monadic Theories Automata on Infinite Words and the Monadic Theory of One Successor Tree Automata, Rabin's Theorem and Forgetful Determinacy The Forgetful Determinacy Theorem for Graph Games The Monadic Second-Order Theory of One Unary Function Decidability Results for One Unary Function The Theory of One Unary Function is not Elementary Recursive The Shelah Class Algebras with One Unary Operation Canonic Sentences Terminology and Notation Satisfiability Satisfiability Refinements Villages Contraction Towns 369

5 XII Table of Contents The Final Reduction Historical Remarks Other Decidable Cases First-Order Logic with Two Variables Unification and Applications to the Decision Problem Unification Herbrand Formulae Positive First-Order Logic Decidable Classes of Krom Formulae The Chain Criterion The Aanderaa-Lewis Class The Maslov Class Historical Remarks 404 A. Appendix: Tiling Problems 407 A.I Introduction 407 A.2 The Origin Constrained Domino Problem 408 A.3 Robinson's Aperiodic Tile Set 410 A.4 The Unconstrained Domino Problem 414 A.5 The Periodic Problem and the Inseparability Result 419 Annotated Bibliography 421 Index 477