Algebraic proofs over noncommutative formulas

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 [BIK + 97, 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.

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“…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).…”

“…In Appendix A we describe various other algebraic proof systems and their relations to IPS, such as the dynamic Polynomial Calculus of Clegg, Edmonds, and Impagliazzo [13], the ordered formula proofs of Tzameret [83], and the multilinear proofs of Raz and Tzameret [66]. In Appendix B we give an explicit description of a multilinear polynomial occurring in our IPS upper bounds.…”

“…However, the above measures of degree and sparsity are rather rough measures of a complexity of a proof. Accordingly, Grochow and Pitassi [35] have recently advocated measuring the complexity of such proofs by their algebraic circuit size and shown that the resulting proof system can polynomially simulate strong proof systems such as the Frege system (see also prior work on measuring algebraic proofs by their algebraic circuit size [59,32,66,65,83] as well as the subsequent survey [60]). This naturally leads to the question of establishing lower bounds for this stronger proof system, even for restricted classes of algebraic circuits.…”

Characterizing Propositional Proofs as Noncommutative Formulas

“…Raz and Tzameret [RT08b,RT08a,Tza08] investigated algebraic proof systems operating with multilinear formulas, motivated by lower bounds on multilinear formulas for the determinant, permanent and other explicit polynomials [Raz09,Raz06]. Atserias et al [AKV04], Krajíček [Kra08] and Segerlind [Seg07] have considered proofs operating with ordered binary decision diagrams (OBDDs), and the second author [Tza11] initiated the study of proofs operating with non-commutative formulas (see Sec. 1.4 for a comparison with the current work).…”

“…Let us also mention the work in [Tza11] that dealt with propositional proof systems over noncommutative formulas. In [Tza11] the choice was made to define all proof systems as polynomial calculus-style systems in which proof-lines are written as non-commutative formulas (as well as the more restricted class of ordered-formulas). This meant that the characterization of a proof system in terms of a single non-commutative polynomial is lacking from that work (as well as the consequences we obtained in the current work).…”