Algebraic Proofs over Noncommutative Formulas

TL;DR

This paper explores algebraic proof systems using noncommutative formulas, introducing a new system called PC over ordered formulas, which is stronger than existing systems and can efficiently refute certain combinatorial principles.

Contribution

It defines and analyzes PC over ordered formulas, demonstrating its strength over traditional proof systems and proposing methods for establishing lower bounds based on noncommutative formula properties.

Findings

PC over ordered formulas is stronger than resolution, PCR, and polynomial calculus.

It admits polynomial-size refutations for pigeonhole principle and Tseitin's formulas.

The paper proposes a framework for lower bounds using noncommutative formula properties.

Abstract

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege---yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in [BIKPRS96,GH03]. We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over ordered formulas, for short): an ordered polynomial is a noncommutative polynomial in which the order of products in every monomial respects a fixed linear order on variables; an algebraic formula is ordered if the polynomial computed by each of its subformulas is ordered. We show that PC over ordered formulas is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR) and admits polynomial-size refutations for the pigeonhole principle and the Tseitin's formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas. The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in algebraic circuit complexity) for establishing lower bounds on propositional proofs.