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

We give a general transformation that turns polynomial-size Frege proofs into subexponential-size AC0-Frege proofs. This indicates that proving truly exponential lower bounds for AC0-Frege is hard, as it is a long-standing open problem to prove superpolynomial lower bounds for Frege. Our construction is optimal for proofs of formulas of unbounded depth. As a consequence of our main result, we are able to shed some light on the question of automatizability for bounded-depth Frege systems. First, we present a simpler proof of the results of Bonet et al. showing that under cryptographic assumptions, bounded-depth Frege proofs are not automatizable. Second, we show that because our proof is more general, under the right cryptographic assumptions, it could resolve the automatizability question for lower-depth Frege systems.

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