The Strength of Multilinear Proofs

Raz, Ran; Tzameret, Iddo

doi:10.1007/s00037-008-0246-0

Cited by 13 publications

(44 citation statements)

References 33 publications

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“…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].…”

Section: Summary Of Resultsmentioning

confidence: 96%

“…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.…”

Section: Comparison To Earlier Workmentioning

confidence: 96%

“…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.…”

Section: Introductionmentioning

confidence: 97%

“…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.…”

Section: Introductionmentioning

confidence: 97%

“…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).…”

Section: Introductionmentioning

confidence: 98%

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Resolution over linear equations and multilinear proofs

Raz

Tzameret

2008

Annals of Pure and Applied Logic

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a b s t r a c tWe develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using (monotone) interpolation we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment.We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [Ran Raz, Iddo Tzameret, The strength of multilinear proofs. Comput. Complexity (in press)]. Specifically, we show the following:• Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations.• Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the functional pigeonhole principle. The latter are different from previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0.We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.

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Section: Summary Of Resultsmentioning

confidence: 96%

Section: Comparison To Earlier Workmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

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Resolution over linear equations and multilinear proofs

Raz

Tzameret

2008

Annals of Pure and Applied Logic

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Algebraic Proofs over Noncommutative Formulas

Tzameret

2010

Lecture Notes in Computer Science

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Algebraic proofs over noncommutative formulas

Tzameret

2011

Information and Computation

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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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The Strength of Multilinear Proofs

Cited by 13 publications

References 33 publications

Resolution over linear equations and multilinear proofs

Resolution over linear equations and multilinear proofs

Algebraic Proofs over Noncommutative Formulas

Algebraic proofs over noncommutative formulas

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