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.

We study arithmetic proof systems P c (F) and P f (F) operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field F. We establish a series of structural theorems about these proof systems, the main one stating that P c (F) proofs can be balanced: if a polynomial identity of syntactic degree d and depth k has a P c (F) proof of size s, then it also has a P c (F) proof of size poly(s, d) in which every circuit has depth O(k + log 2 d + log d • log s). As a corollary, we obtain a quasipolynomial simulation of P c (F) by P f (F).Using these results we obtain the following: consider the identitieswhere X, Y and Z are n×n square matrices and Z is a triangular matrix with z 11 , . . . , z nn on the diagonal (and det is the determinant polynomial). Then we can construct a polynomial-size arithmetic circuit det such that the above identities have P c (F) proofs of polynomial-size using circuits of O(log 2 n) depth. Moreover, there exists an arithmetic formula det of size n O(log n) such that the above identities have P f (F) proofs of sizeThis yields a solution to a basic open problem in propositional proof complexity, namely, whether there are polynomial-size NC 2 -Frege proofs for the determinant identities and the hard matrix identities, as considered, e.g. in Soltys and Cook [SC04] (cf., Beame and Pitassi [BP98]). We show that matrix identities like AB = I → BA = I (for matrices over the two element field) as well as basic properties of the determinant have polynomial-size NC 2 -Frege proofs, and quasipolynomial-size Frege proofs.

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.

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Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted Res(linR), this refutation system operates with disjunctions of linear equations with boolean variables over a ring R, to refute unsatisfiable sets of such disjunctions. Beginning in the work of [26], through the work of [17] which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. [18, 17, 19, 13]) made it evident that establishing lower bounds against general Res(linR) refutations is a challenging and interesting task since the system captures a "minimal" extension of resolution with counting gates for which no super-polynomial lower bounds are known to date.We provide the first super-polynomial size lower bounds on general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular we prove that the subset-sum principle 1 + x1 + • • • + 2 n xn = 0 requires refutations of exponential-size over Q. Our proof technique is nontrivial and novel: roughly speaking, we show that under certain conditions every refutation of a subset-sum instance f = 0, where f is a linear polynomial over Q, must pass through a fat clause containing an equation f = α for each α in the image of f under boolean assignments. We develop a somewhat different approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on n variables, hence also separating tree-like from dag-like refutations over the rationals.We then turn to the finite fields regime, showing that the work of Itsykson and Sokolov [17] who obtained tree-like lower bounds over F2 can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (i) for every pair of distinct primes p, q, there exist CNF formulas with short tree-like refutations in Res(lin Fp ) that require exponential-size treelike Res(lin Fq ) refutations; (ii) random k-CNF formulas require exponential-size tree-like Res(lin Fp ) refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like Res(lin F ) refutations of the pigeonhole principle, for every field F.