Lower Bounds to the size of constant-depth propositional proofs

Journal of Symbolic Logic 59 (1):73-86 (1994)
LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O(n3 + d) which are refutable in LK by depth d + 1 proof of size exp(O(log2 n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets Td n express a weaker form of the pigeonhole principle
Keywords Lengths of proofs   propositional calculus   Frege systems   pigeonhole principle
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    Samuel R. Buss (2012). Towards–Via Proof Complexity and Search. Annals of Pure and Applied Logic 163 (7):906-917.
    Emil Jeřábek (2012). Proofs with Monotone Cuts. Mathematical Logic Quarterly 58 (3):177-187.
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