Algebraic Proofs over Noncommutative Formulas

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 CookReckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in [BIK+ 97, 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). This is an algebraic propositional proof system that operates with noncommutative polynomials in which the order of products in all monomials respects a fixed linear order on the variables, and where proof-lines are written as noncommutative formulas. We show that the latter proof system is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR) and admits polynomial-size refutations for the pigeonhole principle and t...

• Type: Conference
• Year: 2010
• Where: TAMC
• Authors: Iddo Tzameret
• Keywords: Algebraic Propositional Proof, Noncommutative Formulas, Polynomial Calculus, TAMC 2010