On Proof Complexity of Resolution over Polynomial Calculus

Abstract

The proof system Res (PC_d,R) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree d multivariate polynomials over a ring R with Boolean variables. Proving super-polynomial lower bounds for the size of Res(PC_1,R)-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for Res(PC_1,𝔽) when 𝔽 is a finite field, such as 𝔽₂. In this article, we investigate Res(PC_d,R) and tree-like Res(PC_d,R) and prove size-width relations for them when R is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows:

(1)	We prove almost quadratic lower bounds for Res(PCd,𝔽)-refutations for every fixed d. The new lower bounds are for the following CNFs: (a) Mod q Tseitin formulas (char(𝔽)≠ q) and Flow formulas, (b) Random k-CNFs with linearly many clauses.
(2)	We also prove super-polynomial (more than nk for any fixed k) and also exponential (2nϵ for an ϵ > 0) lower bounds for tree-like Res(PCd,𝔽)-refutations based on how big d is with respect to n for the following CNFs: (a) Mod q Tseitin formulas (char(𝔽)≠ q) and Flow formulas, (b) Random k-CNFs of suitable densities, (c) Pigeonhole principle and Counting mod q principle.

The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case d=1. The lower bounds for the tree-like systems were known for the case d=1 (except for the Counting mod q principle, in which lower bounds for the case d> 1 were known too). Our lower bounds extend those results to the case where d> 1 and also give new proofs for the case d=1.