Exponential Lower Bounds for AC⁰-Frege Imply Superpolynomial Frege Lower Bounds

We give a general transformation which turns polynomial-size Frege proofs to subexponential-size AC⁰-Frege proofs. This indicates that proving exponential lower bounds for AC⁰-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.