Resolution over linear equations and multilinear proofs

Abstract

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.

Algebraic proofs over noncommutative formulas

“…(2) For the second proof system, OFPC, we show that, despite its apparent weakness, it is stronger than Polynomial Calculus with Resolution (PCR; and hence it is also stronger than both PC and resolution), and also can polynomially simulate a proof system operating with restricted forms of disjunctions of linear equalities called R 0 (lin) (introduced in [RT08a]). The latter implies polynomial-size refutations for the pigeonhole principle and the Tseitin graph formulas, due to corresponding upper bounds demonstrated in [RT08a].…”

“…Pudlák et al [Pud99,AGP02] studied proofs based on monotone circuits-motivated by known exponential lower bounds on monotone circuits. Raz and the author [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).…”

Lower Bounds for Splittings by Linear Combinations

“…We also show that treelike versions of Res-Lin and Sem-Lin are equivalent to linear splitting trees; the latter implies that our lower bounds hold for tree-like Res-Lin and Sem-Lin. Raz and Tzameret studied a system R(lin) which operates with disjunctions of linear equalities with integer coefficients [12]. It is possible to p-simulate Res-Lin in R(lin) but the existence of the simulation in the other direction is an open problem.…”

Sparser Random 3-SAT Refutation Algorithms and the Interpolation Problem

“…This way we obtain the system R(lin), that is similar to R(quad) but with linear instead of quadratic equations. It was shown in [28] that even when we allow disjunctions of only a constant number of generalized 5 linear equations in each proof-line, R(lin) has short refutations of the Tseitin formulas; this shows that using (fairly restricted) disjunctions of linear equations allows to improve the ability of cutting planes with small coefficients to refute contradictions that involve counting.…”

“…Proof. We can reason in a case-by-case manner as follows (see [28] on how to carry out informal case-analysis reasoning inside R(lin)): assume that j∈J y j = a, for a ∈ {0, 1, . .…”