The complexity of the Pigeonhole Principle

The Pigeonhole Principle forn is the statement that there is no one-to-one function between a set of sizen and a set of sizen–1. This statement can be formulated as an unlimited fan-in constant depth polynomial size Boolean formulaPHP _n inn(n–1) variables. We may think that the truth-value of the variablex _i,j will be true iff the function maps thei-th element of the first set to thej-th element of the second (see Cook and Rechkow 5]).PHP _n can be proved in the propositional calculus. That is, a sequence of Boolean formulae can be given so that each one is either an axiom of the propositional calculus or a consequence of some of the previous ones according to an inference rule of the propositional calculus, and the last one isPHP _n . Our main result is that the Pigeonhole Principle cannot be proved this way, if the size of the proof (the total number or symbols of the formulae in the sequence) is polynomial inn and each formula is constant depth (unlimited fan-in), polynomial size and contains only the variables ofPHP _n .