Bounded arithmetic, propositional logic, and complexity theory
著者
書誌事項
Bounded arithmetic, propositional logic, and complexity theory
（Encyclopedia of mathematics and its applications / edited by G.-C. Rota, v. 60）
Cambridge University Press, 1995
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注記
Includes bibliographical references (p. 327-334) and indexes
内容説明・目次
内容説明
This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area.
目次
- 1. Introduction
- 2. Preliminaries
- 3. Basic complexity theory
- 4. Basic propositional logic
- 5. Basic bounded arithmetic
- 6. Definability of computations
- 7. Witnessing theorems
- 8. Definability and witnessing in second order theories
- 9. Translations of arithmetic formulas
- 10. Finite axiomatizability problem
- 11. Direct independence proofs
- 12. Bounds for constant-depth Frege systems
- 13. Bounds for Frege and extended Frege systems
- 14. Hard tautologies and optimal proof systems
- 15. Strength of bounded arithmetic
- References
- Index.
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