Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional calculus (i.e. Frege) proof is a proof starting from a set of axioms and deriving new Boolean formulas using a set of fixed sound derivation rules. Establishing any super-polynomial 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 by and large. Non-commutative arithmetic formulas, on the other hand, constitute a quite weak computational model, for which exponential-size lower bounds were shown already back in 1991 by Nisan [Nis91] who used a particularly transparent argument. In this work we show that Frege lower bounds in fact follow from correspond-ing size lower bounds on non-commutative formulas computing certain polynomi-als (and that such lower bounds on non-commutative formulas must exist, unless NP=coNP). More precisely, we demonstrate a natural association between tau-tologies T to non-commutative polynomials p, such that