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Revision #1 to TR10-097 | 2nd August 2010 13:24

#### Algebraic Proofs over Noncommutative Formulas

Revision #1
Authors: Iddo Tzameret
Accepted on: 2nd August 2010 13:24
Keywords:

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.

Changes to previous version:

Changed definition of ordered formulas (to, essentially, "syntactic" ordered formulas. This allows to verify in polytime whether a formula is an ordered one. Upper bounds for PC over ordered formulas stay the same); added some missing proofs of certain claims, and missing definitions of known concepts; improved introduction; other local/cosmetic changes.

### Paper:

TR10-097 | 16th June 2010 17:04

#### Algebraic Proofs over Noncommutative Formulas

TR10-097
Authors: Iddo Tzameret
Publication: 17th June 2010 09:48