## Lower Bounds to the Size of Constant-Depth Propositional Proofs (1994)

Citations: | 54 - 6 self |

### BibTeX

@MISC{Krajícek94lowerbounds,

author = {Jan Krajícek},

title = {Lower Bounds to the Size of Constant-Depth Propositional Proofs},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 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\Gamma21 ). The sets T d n express a weaker form of the pigeonhole principle. It is a fundamental problem of mathematical logic and complexity theory whether there exists a proof system for propositional logic in which every tautology has a short proof, where the length (equivalently the size) of a proof is measured essentially by the total number of symbols in it and short means polynomial in the length of the tautology. Equivalently one can ask whether for every theory T there is another theory S (both first order and reasonably axiomatized, e.g. by schemes) having the property that if a statement...