The Classical Decision Problem

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This book is addressed to all those - logicians, computer scientists, mathe maticians, philosophers of science as well as the students in all these disci plines - who may be interested in the development and current status of one of the major themes of mathematical logic in the twentieth century, namely the classical decision problem known also as Hilbert's Entscheidungsproblem. The text provides a comprehensive modern treatment of the subject, includ ing complexity theoretic analysis. We have made an effort to combine the features of a research monograph and a textbook. Only the basic knowledge of the language of first-order logic is required for understanding of the main parts of the book, and we use standard terminology. The chapters are written in such a way that various combinations of them can be used for introductory or advanced courses on undecidability, decidability and complexity of logical decision problems. This explains a few intended redundancies and repetitions in some of the chapters. The annotated bibliography, the historical remarks at the end of the chap ters and the index allow the reader to use the text also for quick reference purposes.

This book, useful to students and researchers alike, is an up-to-date account of a topic of central importance in mathematical logic and theoretical computer science

Includes supplementary material:

Egon Börger ist Professor für Informatik an der Universität Pisa (Italien) und Alexander-von-Humboldt-Forschungspreisträger.

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 Surányi 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-Schönfinkel-Ramsey Class.- 6.2.3 The Gödel-Kalmár-Schütte 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.

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The Classical Decision Problem
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