The Classical Decision Problem

This book is addressed to all those — logicians, computer scientists, mathematicians, philosophers of science as well as the students in all these disciplines — 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, including 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 (over 50 pages), the historical remarks at the end of the chapters and the index allow the reader to use the text also for quick reference purposes. 1. Introduction: The Classical Decision Problem Part I. Undecidable Classes 2. Reductions 3. Undecidable Standard Classes for Pure Predicate Logic 4. Undecidable Standard Classes with Functions or Equality 5. Other Undecidable Cases Part II. Decidable Classes and Their Complexity 6. Standard Classes with the Finite Model Property 7. Monadic Theories and Decidable Standard Classes with Infinity Axioms 8. Other Decidable Cases Appendix: Tiling Problems Annotated Bibliography Index This treatment of the classical decision problem of mathematical logic and its role in modern computer science provides an analysis of the natural order of decidable and undecidable cases. Of particular interest is the complete classification of the standard cases of the classical decision problem.