The Strength of Multilinear Proofs

Abstract

We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show the following:1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) proofs;2. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for the functional pigeonhole principle;3. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for Tseitin's graph tautologies.By known lower bounds, this demonstrates that algebraic proof systems manipulating depth 3 multilinear formulas are strictly stronger than Resolution, PC and PCR, and have an exponential gap over boundeddepth Frege for both the functional pigeonhole principle and Tseitin's graph tautologies. We also illustrate a connection between lower bounds on multilinear proofs and lower bounds on multilinear circuits. In particular, we show that (an explicit) super-polynomial size separation between proofs manipulating general arithmetic circuits and proofs manipulating multilinear circuits implies a super-polynomial size lower bound on multilinear circuits for an explicit family of polynomials.

Resolution over linear equations and multilinear proofs

“…The upper bounds for the pigeonhole principle and Tseitin mod p formulas in fragments of resolution over linear equations are new. By generalizing the machinery developed in [26], these upper bounds yield new and improved results concerning multilinear proofs. The lower bound for the clique-coloring formulas in a fragment of resolution over linear equations employs the standard monotone interpolation by a communication game technique, and specifically utilizes the theorem of Bonet, Pitassi and Raz from [5].…”

“…Whereas previous results concerned primarily with extending the cutting planes proof system, our foremost motivation is to extend and improve previous results on algebraic proof systems operating with multilinear formulas obtained in [26]. In that paper the concept of multilinear proofs was introduced and several basic results concerning multilinear proofs were proved.…”

“…In this paper, we are particularly interested in algebraic proof systems that operate with multilinear polynomials represented as multilinear arithmetic formulas, called by the generic name multilinear proofs (a polynomial is multilinear if the power of each variable in its monomials is at most one). The investigation into such proof systems was initiated in [26], and here we continue this line of research. This research is motivated on the one hand by the apparent considerable strength of such systems; and on the other hand, by the known super-polynomial size lower bounds on multilinear formulas computing certain important functions [24,25], combined with the general working assumption that establishing lower bounds on the size of objects a proof system manipulates (in this case, multilinear formulas) is close to establishing lower bounds on the size of the proofs themselves.…”

“…The basic idea connecting resolution operating with disjunctions of linear equations and multilinear proofs is this: whenever a disjunction of linear equations is simple enough -and specifically, when it is close to a symmetric function, in a manner made precise -then it can be represented by a small size and small depth multilinear arithmetic formula over fields of characteristic 0. This idea was already used (somewhat implicitly) in [26] to obtain polynomial-size multilinear proofs operating with depth-3 multilinear formulas of the functional pigeonhole principle (this principle is weaker than the pigeonhole principle). In the current paper we generalize previous results on multilinear proofs by fully using this idea: We show how to polynomially simulate with multilinear proofs, operating with small depth multilinear formulas, certain short proofs carried inside resolution over linear equations.…”

“…In the current paper we generalize previous results on multilinear proofs by fully using this idea: We show how to polynomially simulate with multilinear proofs, operating with small depth multilinear formulas, certain short proofs carried inside resolution over linear equations. This enables us to provide new polynomial-size multilinear proofs for certain hard tautologies, improving results from [26]. More specifically, we introduce a certain fragment of R(lin), which can be polynomially simulated by depth-3 multilinear proofs (that is, multilinear proofs operating with depth-3 multilinear formulas).…”