Approximation and Small Depth Frege Proofs

Ajtai [Ajt] recently proved that if for some fixed d, every formula in a Frege proof of the propositional pigeonhole principle PHP, has depth at most d, then the proof size is not less than any polynomial in n. By introducing the notion of an “approximate proof” we demonstrate how to eliminate the non-standard model theory, including the non-constructive use of the compactness theorem, from Ajtai’s lower bound. An approximate proof is one in which each inference is sound on a subset of the possible truth assignments possibly a different subset for each inference. We also improve the lower bound, giving a specific superpolynomial function (nSl(’og’d+l’ ”)) bounding the proof size from below.