Polynomial Calculus Sizes Over the Boolean and Fourier Bases are Incomparable

Abstract

For every n > 0, we show the existence of a CNF tautology over O(n2) variables of width O(log n) such that it has a Polynomial Calculus Resolution refutation over 0,1 variables of size O(n3 polylog} (n)) but any Polynomial Calculus refutation over {+1, -1} variables requires size 2Ω(n). This shows that Polynomial Calculus sizes over the {0, 1} and +1, -1 bases are incomparable (since Tseitin tautologies show a separation in the other direction) and answers an open problem posed by Sokolov [1] and Razborov [2]. © 2024 IEEE.