Bounded Arithmetic, Propositional Logic and Complexity Theory

ISBN-13: 9780521452052 / Angielski / Twarda / 1995 / 360 str.

Jan Krajicek; G. -C Rota; B. Doran

Opis

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 is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. Then more advanced topics are 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, simple independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the approximation method and the method of Boolean valuations, 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 his comprehensive treatment an excellent guide to this expanding interdisciplinary area.

Recenzje

'It can be strongly recommended especially to mathematicians and computer scientists working in the field and to graduate students.' European Mathematical Society Newsletter

Spis treści

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.