From Proof Complexity to Circuit Complexity via Interactive Protocols

Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP P/poly, as recently observed by Pich and Santhanam (2023). We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Kraj\'icek (The Journal of Symbolic Logic, 2004), capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies \#P FP/poly (which would in turn imply, for example, PSPACE P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan (Journal of the ACM, 1992). This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.