  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
corresponding size lower bounds on non-commutative formulas computing certain
polynomials (and that such lower bounds on non-commutative formulas must exist,
unless NP=coNP). More precisely, we demonstrate a natural association between
tautologies $T$ to non-commutative polynomials $p$, such that: if $T$ has a
polynomial-size Frege proof then $p$ has a polynomial-size non-commutative
arithmetic formula; and conversely, when $T$ is a DNF, if $p$ has a
polynomial-size non-commutative arithmetic formula over $GF(2)$ then $T$ has a
Frege proof of quasi-polynomial size.
