New Lower Bounds for Polynomial Calculus over Non-Boolean Bases

Abstract

In this paper, we obtain new size lower bounds for proofs in the Polynomial Calculus (PC) proof system, in two different settings. When the Boolean variables are encoded using ±1 (as opposed to 0, 1): We establish a lifting theorem using an asymmetric gadget G, showing that for an unsatisfiable formula F, the lifted formula F ◦ G requires PC size 2Ω(d), where d is the degree required to refute F. Our lower bound does not depend on the number of variables n, and holds over every field. The only previously known size lower bounds in this setting were established quite recently in [Sokolov, STOC 2020] using lifting with another (symmetric) gadget. The size lower bound there is 2Ω((d−d0)2/n) (where d0 is the degree of the initial equations arising from the formula), and is shown to hold only over the reals. When the PC refutation proceeds over a finite field Fp and is allowed to use extension variables: We show that there is an unsatisfiable AC0[p] formula with N variables for which any PC refutation using N1+ϵ(1−δ) extension variables, each of arity at most N1−ϵ and size at most Nc, must have size exp(Ω(Nϵδ/poly log N)). Our proof achieves these bounds by an XOR-ification of the generalised PHPm,rn formulas from [Razborov, CC 1998]. The only previously known lower bounds for PC in this setting are those obtained in [Impagliazzo-Mouli-Pitassi, CCC 2023]

in those bounds the number of extension variables is required to be sub-quadratic, and their arity is restricted to logarithmic in the number of original variables. Our result generalises these, and demonstrates a tradeoff between the number and the arity of extension variables. Since our tautology is represented by a small AC0[p] formula, our results imply lower bounds for a reasonably strong fragment of AC0[p]-Frege. © Yogesh Dahiya, Meena Mahajan, and Sasank Mouli.