Exponential Lower Bounds for AC-Frege Imply Superpolynomial Frege Lower Bounds

We give a general transformation which turns polynomialsize Frege proofs to subexponential-size AC-Frege proofs. This indicates that proving exponential lower bounds for AC-Frege is hard, since it is a longstanding open problem to prove super-polynomial lower bounds for Frege. Our construction is optimal for tree-like proofs. As a consequence of our main result, we are able to shed some light on the question of weak automatizability for bounded-depth Frege systems. First, we present a simpler proof of the results of Bonet et al. [5] showing that under cryptographic assumptions, bounded-depth Frege proofs are not weakly automatizable. Secondly, we show that because our proof is more general, under the right cryptographic assumptions, it could resolve the weak automatizability question for lower depth Frege systems. ? Supported by NSERC