Extended Frege proofs, circuits and rewriting

2026-06-11

AI Summary

This work investigates the connection between circuit logical equivalence and the Extended Frege proof system, introducing a binary relation ≈ decidable in polynomial time. This relation not only implies logical equivalence but also guarantees that any two circuits related by ≈ can be transformed into one another by deleting some gates and adding at most seven new gates. The main contribution is the first polynomial correspondence between the length of equivalence proofs in Extended Frege and the number of local rewriting steps between circuits: if two circuits admit a proof of size $s$, then there exists a chain of intermediate circuits connecting them of length $s^{O(1)}$. This result reveals a deep link between proof complexity and structural circuit transformations and defines a notion of approximate circuit equivalence that simultaneously ensures decidability and structural guarantees.

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 $E_i$ $$ C = E_0 \approx \dots \approx E_t = D $$ with $t \le s^{O(1)}$.

Problem

• Extended Frege
• circuit equivalence
• proof complexity
• rewriting
• logical equivalence

Innovation

• Extended Frege
• circuit equivalence
• rewriting
• proof complexity
• polynomial-time relation