Characterizing Propositional Proofs as Noncommutative Formulas

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e., Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any superpolynomial-size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexity and among a handful of fundamental hardness questions in complexity theory. Noncommutative algebraic formulas, on the other hand, constitute a quite weak computational model, for which exponential-size lower bounds were shown back in 1991 by Nisan [Proceedings of STOC, 1991, pp. 410--418], using a particularly transparent argument. In this work we show that Frege lower bounds in fact follow from size lower bounds on noncommutative formulas computing certain families of polynomials (and that such lower bounds on noncommutative formulas must exist, unless (NP=coNP). More precisely, we demonstrate a natural associat...