Extended Frege proofs, circuits and rewriting

Abstract

Inspired by a statement about Extended Frege proof systems by Jain and Jin (FOCS 2022) we prove that: - there is a p-time binary relation $\approx$ between circuits that implies their logical equivalence, - the relation $\approx$ implies that each of the two circuits can be rewritten into the other one by possibly deleting some gates and adding at most seven new gates, - if the equivalence $C\equiv D$ has a size $s$ proof in an Extended Frege or a Circuit Frege proof system then there is a chain of circuits $E_i$ $$C=E_0\approx \cdots \approx E_t=D$$ with $t\le s^{O(1)}$.

10 pp