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Lower bounds to the size of constant-depth propositional proofs

Published online by Cambridge University Press:  12 March 2014

Jan Krajíček
Jan Krajíček*
Affiliation:
Mathematical Institute, Academy of Sciences, Žitná 25, Praha 1, 11567 Czech Republic, E-mail:[email protected]
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Abstract

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and ∧,∨ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set of depth d sequents of total size O(n3+d) which are refutable in LK by depth d + 1 proof of size exp(O(log2n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets express a weaker form of the pigeonhole principle.

Keywords

Lengths of proofspropositional calculusFrege systemspigeonhole principle
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 73 - 86
Copyright
Copyright © Association for Symbolic Logic 1994

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