Resolution over Linear Equations and Multilinear Proofs

TL;DR

This paper introduces and analyzes proof systems extending resolution with linear equations, demonstrating their efficiency on certain tautologies and establishing bounds, while connecting them to multilinear proofs and cutting planes.

Contribution

It develops resolution over linear equations, proves polynomial-size refutations for key tautologies, and links these systems to multilinear proofs and cutting planes.

Findings

Polynomial-size refutations for pigeonhole principle and Tseitin tautologies.

Exponential lower bounds for certain fragments of resolution over linear equations.

Polynomial upper bounds for non-monotone interpolants in these fragments.

Abstract

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using the (monotone) interpolation by a communication game technique we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [RazTzameret06]. Specifically, we show the following: 1. Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations. 2. Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the \emph{functional} pigeonhole principle. The latter are different than previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.