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 𝔽 2 . 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 ( PC d ,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 n k for any fixed k ) and also exponential (2 nϵ for an ϵ > 0) lower bounds for tree-like Res ( PC d ,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.