A lower bound for polynomial calculus with extension rule

Abstract

A major proof complexity problem is to prove a superpolynomial lower bound on the length of Frege proofs of arbitrary depth. A more general question is to prove an Extended Frege lower bound. Surprisingly, proving such bounds turns out to be much easier in the algebraic setting. In this paper, we study a proof system that can simulate Extended Frege: an extension of the Polynomial Calculus proof system where we can take a square root and introduce new variables that are equivalent to arbitrary depth algebraic circuits. We prove that an instance of the subset-sum principle, the binary value principle 1 + x1 + 2x2 + ... + 2n−1xn = 0 (BVPn), requires refutations of exponential bit size over Q in this system. Part and Tzameret [18] proved an exponential lower bound on the size of Res-Lin (Resolution over linear equations [22]) refutations of BVPn. We show that our system p-simulates Res-Lin and thus we get an alternative exponential lower bound for the size of Res-Lin refutations of BVPn.