Bounded-Depth Frege Lower Bounds for Weaker Pigeonhole Principles

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHPm n where m 1 1 polylog n n. This lower bound qualitatively matches the known quasi-polynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for boundeddepth Frege proofs, is novel in that the tautology to which this switching lemma is applied remains random throughout the argument. Research supported by NSF grant CCR-0098066 †Research supported by US-Israel BSF grant 98-00349 ‡Research supported by US-Israel BSF grant 98-00349 1 Electronic Colloquium on Computational Complexity, Report No. 23 (2002)