### Lower Bounds to the Size of Constant-Depth Propositional Proofs

Jan Krajicek
Source: J. Symbolic Logic Volume 59, Issue 1 (1994), 73-86.

#### Abstract

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives $\neg$ and $\bigwedge, \bigvee$ (both of bounded arity). Then for every $d \geq 0$ and $n \geq 2$, there is a set $T^d_n$ of depth $d$ sequents of total size $O(n^{3 + d})$ which are refutable in LK by depth $d + 1$ proof of size $\exp(O(\log^2 n))$ but such that every depth $d$ refutation must have the size at least $\exp(n^{\Omega(1)})$. The sets $T^d_n$ express a weaker form of the pigeonhole principle.

Primary Subjects: 03F20
Secondary Subjects: 03F07, 68Q15, 68R99
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