The Classical Decision Problem

1. Introduction: The Classical Decision Problem. - 1.1 The Original Problem. - 1.2 The Transformation of the Classical Decision Problem. - 1.3 What Is and What Isn't in this Book. - I. Undecidable Classes. - 2. Reductions. - 2.1 Undecidability and Conservative Reduction. - 2.1.1 The Church-Turing Theorem and Reduction Classes. - 2.1.2 Trakhtenbrot's Theorem and Conservative Reductions. - 2.1.3 Inseparability and Model Complexity. - 2.2 Logic and Complexity. - 2.2.1 Propositional Satisfiability. - 2.2.2 The Spectrum Problem and Fagin's Theorem. - 2.2.3 Capturing Complexity Classes. - 2.2.4 A Decidable Prefix-Vocabulary Class. - 2.3 The Classifiability Problem. - 2.3.1 The Problem. - 2.3.2 Well Partially Ordered Sets. - 2.3.3 The Well Quasi Ordering of Prefix Sets. - 2.3.4 The Well Quasi Ordering of Arity Sequences. - 2.3. 5 The Classifiability of Prefix-Vocabulary Sets. - 2.4 Historical Remarks. - 3. Undecidable Standard Classes for Pure Predicate Logic. - 3.1 The Kahr Class. - 3.1.1 Domino Problems. - 3.1.2 Formalization of Domino Problems by $$[\forall \exists \forall , (0,\omega )]$$-Formulae. - 3.1.3 Graph Interpretation of $$[\forall \exists \forall , (0,\omega )]$$-Formulae. - 3.1. 4 The Remaining Cases Without $$\exists *$$. - 3.2 Existential Interpretation for $$[{{\forall }^{3}}\exists *, (0,1)]$$. - 3.3 The Gurevich Class. - 3.3.1 The Proof Strategy. - 3.3.2 Reduction to Diagonal-Freeness. - 3.3.3 Reduction to Shift-Reduced Form. - 3.3.4 Reduction toFi-Elimination Form. - 3.3. 5 Elimination of MonadicFi. - 3.3. 6 The Kostyrko-Genenz and Suranyi Classes. - 3.4 Historical Remarks. - 4. Undecidable Standard Classes with Functions or Equality. - 4.1 Classes with Functions and Equality. - 4.2 Classes with Functions but Without Equality. - 4.3 Classes with Equality but Without Functions: the Goldfarb Classes 161 4.3.1 Formalization of Natural Numbers in $$[{{\forall }^{3}}\exists *, (\omega ,\omega ),(0)]$$=. - 4.3.2 Using Only One Existential Quantifiers. - 4.3.3 Encoding the Non-Auxiliary Binary Predicates. - 4.3.4 Encoding the Auxiliary Binary Predicates of NUM*. - 4.4 Historical Remarks. - 5. Other Undecidable Cases. - 5.1 Krom and Horn Formulae. - 5.1.1 Krom Prefix Classes Without Functions or Equality. - 5.1.2 Krom Prefix Classes with Functions or Equality. - 5.2 Few Atomic Subformulae. - 5.2.1 Few Function and Equality Free Atoms. - 5.2.2 Few Equalities and Inequalities. - 5.2.3 Horn Clause Programs With One Krom Rule. - 5.3 Undecidable Logics with Two Variables. - 5.3.1 First-Order Logic with the Choice Operator. - 5.3.2 Two-Variable Logic with Cardinality Comparison. - 5.4 Conjunctions of Prefix-Vocabulary Classes. - 5.4.1 Reduction to the Case of Conjunctions. - 5.4.2 Another Classifiability Theorem. - 5.4.3 Some Results and Open Problems. - 5.5 Historical Remarks. - II. Decidable Classes and Their Complexity. - 6. Standard Classes with the Finite Model Property. - 6.1 Techniques for Proving Complexity Results. - 6.1.1 Domino Problems Revisited. - 6.1.2 Succinct Descriptions of Inputs. - 6.2 The Classical Solvable Cases. - 6.2.1 Monadic Formulae. - 6.2.2 The Bernays-Schonfinkel-Ramsey Class. - 6.2.3 The Godel-Kalmar-Schutte Class: a Probabilistic Proof. - 6.3 Formulae with One ? - 6.3.1 A Satisfiability Test for [? *? *, all, all]. - 6.3.2 The Ackermann Class. - 6.3.3 The Ackermann Class with Equality. - 6.4 Standard Classes of Modest Complexity. - 6.4.1 The Relational Classes in P, NP and Co-NP. - 6.4.2 Fragments of the Theory of One Unary Function. - 6.4.3 Other Functional Classes. - 6.5 Finite Model Property vs. Infinity Axioms. - 6.6 Historical Remarks. - 7. Monadic Theories and Decidable Standard Classes with Infinity Axioms. - 7.1 Automata, Games and Decidability of Monadic Theories. - 7.1.1 Monadic Theories. - 7.1.2 Automata on Infinite Words and the Monadic Theory of One Successor. - 7.1.3 Tree Automata, Rabin's Theorem and Forgetful De terminacy. - 7.1. 4 The Forgetful Determinacy Theorem for Graph Games. - 7.2 The Monadic Second-Order Theory of One Unary Function. - 7.2.1 Decidability Results for One Unary Function. - 7.2.2 The Theory of One Unary Function is not Elementary Recursive. - 7.3 The Shelah Class. - 7.3.1 Algebras with One Unary Operation. - 7.3.2 Canonic Sentences. - 7.3.3 Terminology and Notation. - 7.3.4 1-Satisfiability. - 7.3. 5 2-Satisfiability. - 7.3. 6 Refinements. - 7.3. 7 Villages. - 7.3. 8 Contraction. - 7.3. 9 Towns. - 7.3.10 The Final Reduction. - 7.4 Historical Remarks. - 8. Other Decidable Cases. - 8.1 First-Order Logic with Two Variables. - 8.2 Unification and Applications to the Decision Problem. - 8.2.1 Unification. - 8.2.2 Herbrand Formulae. - 8.2.3 Positive First-Order Logic. - 8.3 Decidable Classes of Krom Formulae. - 8.3.1 The Chain Criterion. - 8.3.2 The Aanderaa-Lewis Class. - 8.3.3 The Maslov Class. - 8.4 Historical Remarks. - A. Appendix: Tiling Problems. - A.1 Introduction. - A.2 The Origin Constrained Domino Problem. - A.3 Robinson's Aperiodic Tile Set. - A.4 The Unconstrained Domino Problem. - A.5 The Periodic Problem and the Inseparability Result. - Annotated Bibliography.