Characterizing Propositional Proofs as Noncommutative Formulas

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 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].

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“…While the primary focus of this article is on lower bounds for restricted classes of the IPS proof system, we begin by discussing upper bounds to demonstrate that these restricted classes can prove the unsatisfiability of non-trivial systems of polynomials equations. In particular we go beyond existing work on upper bounds ( [32,66,65,35,50]) and place interesting refutations in IPS subsystems where we will also prove lower bounds, as such upper bounds demonstrate the non-triviality of our lower bounds.…”

“…Theorem 1.4 (Li, Tzameret and Wang [50]). Let ϕ = C 1 ∧• • •∧C m be an unsatisfiable CNF on n-variables, and let f 1 , .…”

“…Motivated by the fact that PIT of non-commutative formulas 2 can be solved deterministically ( [64]) and admit exponential-size lower bounds ( [52]), Li, Tzameret and Wang [50] have shown that IPS over non-commutative polynomials (along with additional commutator axioms) can simulate Frege (they also provided a quasipolynomial simulation of IPS over non-commutative formulas by Frege; see Li, Tzameret and Wang [50] for more details).…”

“…In a sequence of more recent papers, polynomial identity testing algorithms were devised for roABPs ([64, 25, 27, 24, 2], see also Section 3.1). In terms of proof complexity, Tzameret [83] studied a proof system with lines given by roABPs, and recently Li, Tzameret and Wang [50] (Theorem 1.4) showed that IPS over non-commutative formulas is essentially equivalent in power to the Frege proof system. Due to the close connections between non-commutative ABPs and roABPs, this last result suggests the importance of proving lower bounds for roABP-IPS as a way of attacking lower bounds for the Frege proof system (although we obtain roABP-IPS LIN lower bounds without obtaining non-commutative-IPS LIN lower bounds).…”

“…1-88 respectively. Li, Tzameret and Wang [50] (Theorem 1.4) have shown that even non-commutative-formula-IPS can simulate Frege. Grigoriev and Hirsch [32] have shown that proofs manipulating depth-3 algebraic formulas can refute hard axioms such as the pigeonhole principle, the subset-sum axiom, and Tseitin tautologies.…”