Characterizing Propositional Proofs as Non-Commutative Formulas

TL;DR

This paper connects propositional proof complexity to non-commutative algebra, showing that lower bounds in Frege proofs can be derived from lower bounds on non-commutative formulas computing certain polynomials, linking proof complexity to algebraic complexity.

Contribution

It establishes a novel correspondence between propositional proofs and non-commutative formulas, enabling transfer of lower bounds from algebraic models to proof complexity.

Findings

Frege proof size lower bounds relate to non-commutative formula lower bounds.

Polynomial-size Frege proofs imply polynomial-size non-commutative formulas for associated polynomials.

For DNF tautologies, non-commutative formulas over GF(2) imply quasi-polynomial Frege proofs.

Abstract

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional calculus (i.e. Frege) proof is a proof starting from a set of axioms and deriving new Boolean formulas using a set of fixed sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexity, and among a handful of fundamental hardness questions in complexity theory by and large. Non-commutative arithmetic formulas, on the other hand, constitute a quite weak computational model, for which exponential-size lower bounds were shown already back in 1991 by Nisan [Nis91] who used a particularly transparent argument. In this work we show that Frege lower bounds in fact follow from corresponding size lower bounds on non-commutative formulas computing certain polynomials (and that such lower bounds on non-commutative formulas must exist, unless NP=coNP). More precisely, we demonstrate a natural association between tautologies $T$ to non-commutative polynomials $p$, such that: if $T$ has a polynomial-size Frege proof then $p$ has a polynomial-size non-commutative arithmetic formula; and conversely, when $T$ is a DNF, if $p$ has a polynomial-size non-commutative arithmetic formula over $GF(2)$ then $T$ has a Frege proof of quasi-polynomial size.