Lower Bounds to the Size of Constant-depth Propositional Proofs

1 LK is a natural modiication of Gentzen sequent calculus for pro-positional 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 (1)). 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 tau-tology 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 rst order and reasonably axiomatized, e.g. by schemes) having the property that if a statement does not have a short proof in T then there is a short veriication of it in S. 2 This problem is, with a general notion of a proof system, equivalent to a principal question of complexity theory, namely whether the class of predicates acceptable in non-deterministic polynomial time is closed under complementa-tion, 8]. One can show that if there exists such an optimal proof system it can be formed by augmenting the usual textbook calculus (called a Frege system in the terminology of 8]) by the extension rule (allowing to abbreviate formulas by new propositional variables) and by an additional, polynomial time recognizable , set of tautologies as extra axioms. Equivalently, for such an optimal proof system one could take a fragment T of true arithmetic ((nitely axiomatizable and 0 1 , in fact) where for a proof of the tautology ' is a taken a T-proof of the arithmetization Taut(d'e) of "' is a tautology", see 14] for more details. principle. 2 This problem was mentioned in a similar form in K. GG odel's letter to J. von Neumann in 1956. One can restrict to T =predicate calculus.