Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

Abstract

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e. Frege) proof  is any  proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of 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 [STOC 1991], using  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.

The argument is a characterization of Frege proofs as non-commutative formulas: we show that the Frege  system is (quasi-)polynomially equivalent to a non-commutative Ideal Proof System (IPS), following the recent work of Grochow and Pitassi [FOCS 2014] that introduced a propositional proof system in which proofs are arithmetic circuits, and the work in [Tzameret 2011] that considered adding the commutator as an axiom in algebraic propositional proof systems. This gives a characterization of propositional Frege proofs in terms of (non-commutative) arithmetic formulas that is tighter than (the formula version of IPS) in Grochow and Pitassi [FOCS 2014], in the following sense:

(i) The non-commutative IPS is polynomial-time checkable - whereas the original IPS was checkable in probabilistic polynomial-time; and
(ii) Frege proofs unconditionally quasi-polynomially simulate the non-commutative IPS - whereas Frege was shown to  efficiently simulate IPS only assuming that the decidability of PIT for (commutative) arithmetic formulas by polynomial-size circuits is efficiently provable in Frege.