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Journal of Logic and Computation 2005 15(4):433-446; doi:10.1093/logcom/exi035
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Vol. 15 No. 4, © The Author, 2005. Published by Oxford University Press. All rights reserved.

Original Articles

Uniform Proof Complexity

Arnold Beckmann

University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK. Email: A.Beckmann{at}swansea.ac.uk

We define the notion of the uniform reduct of a propositional proof system as the set of those bounded formulas in the language of Peano Arithmetic which have polynomial size proofs under the Paris-Wilkie-translation. With respect to the arithmetic complexity of uniform reducts, we show that uniform reducts are {Pi}10-hard and obviously in {Sigma}20. We also show under certain regularity conditions that each uniform reduct is closed under bounded generalisation; that in the case the language includes a symbol for exponentiation, a uniform reduct is closed under modus ponens if and only if it already contains all true bounded formulas; and that each uniform reduct contains all true {Pi}1b({alpha})-formulas.

Keywords: Length of proofs, propositional calculus, translations, bounded arithmetic

Received May 2005.


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