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Bounded Arithmetic and Propositional Proof Complexity

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Logic of Computation

Part of the book series: NATO ASI Series ((NATO ASI F,volume 157))

Abstract

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories and of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses.

We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the Razborov-Rudich notion of natural proofs of P ≠ NP and discuss Razborov’s theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given.

Supported in part by NSF grants DMS-9503247 and DMS-9205181.

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Authors and Affiliations

  1. Departments of Mathematics and Computer Science, University of California, San Deigo La Jolla, CA, 92093-0112, USA

    Samuel R. Buss

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  1. Samuel R. Buss
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Editors and Affiliations

  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, D-80333, München, Germany

    Helmut Schwichtenberg

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© 1997 Springer-Verlag Berlin Heidelberg

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Buss, S.R. (1997). Bounded Arithmetic and Propositional Proof Complexity. In: Schwichtenberg, H. (eds) Logic of Computation. NATO ASI Series, vol 157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59048-1_3

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  • DOI: https://doi.org/10.1007/978-3-642-59048-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63832-9

  • Online ISBN: 978-3-642-59048-1

  • eBook Packages: Springer Book Archive

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