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.
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.
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