Characterizing Propositional Proofs as Non-Commutative Formulas

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 superpolynomial 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 noncommutative 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 [Tza11] that considered adding the commutator as an axiom in algebraic propositional proof systems. This also 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]. An extended abstract of this work entitled “Non-commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs” appeared in Proceedings of the 30th Annual Computational Complexity Conference (CCC): June 17-19, 2015. The University of Texas at Austin, Department of Computer Science. Email: [email protected] Royal Holloway, University of London, Department of Computer Science. Supported in part by NSFC grant 61373002. Email: [email protected] Harvard University, Department of Computer Science. Email: [email protected]