A new proof of the weak pigeonhole principle

Abstract

The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [9] showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomial-size proofs. Their argument was further refined by Krajíček [5]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomial-size proofs where every formula consists of a single AND/OR. of polylog fan-in. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. © 2000 ACM.