 
Summary: Plausibly Hard Combinatorial Tautologies
Jeremy Avigad
Abstract. We present a simple propositional proof system which con
sists of a single axiom schema and a single rule, and use this system to
construct a sequence of combinatorial tautologies that, when added to
any Frege system, psimulates extendedFrege systems.
1. Introduction
As was pointed out in [6], the conjecture that NP = coNP can be con
strued as the assertion that there is no proof system (broadly interpreted)
in which there are short (polynomiallength) proofs of every propositional
tautology. Though showing NP = coNP seems to be difficult, the above
formulation suggests an obvious restriction, namely the assertion that spe
cific proof systems are inefficient. One of the nicest results of this form to
date is the fact that there are no short proofs of tautologies representing the
pigeonhole principle in a fixeddepth Frege system (see, for example, [1, 2]).
This approach to demonstrating a proof system's inefficiency seems natu
ral: choose a suitable sequence of propositional formulas that express some
true combinatorial assertion, and then show that these tautologies can't be
proven efficiently by the system in question.
Unfortunately, in the case of Frege systems, there is a shortage of good
