Resolution with Counting: Lower Bounds for Proofs of Membership in the Complement of a Linear Map Image of the Boolean Cube

Abstract

We propose a new approach to proving lower bounds for sizes of dag-like proofs in the proof system Res(lin Fp ), where F p is a finite field of prime order p ≥ 5. An exponential lower bound on sizes of arbitrary dag-like refutations in proof systems Res(lin F ) has previously been proven in [13] in case F is a field of characteristic 0 for an instance, which is not CNF: for the binary value principle [13] substantially uses peculiarities of characteristic 0 regime and does not give a clue on how to prove lower bounds neither for finite fields nor for CNFs. Aiming at constructing a bridge between lower bounds in [13] and CNF lower bounds we initiate the development of methods for proving dag-like Res(lin Fp ) lower bounds for tautologies of the form b / ∈ A({0, 1} n ), where A is a linear map. The negation of such a tautology can be represented in the language of Res(lin Fp ) as a system of linear equations A • x = b unsatisfiable over the boolean assignments. Instances of this form are in some ways simpler than CNFs, this makes analysis of their Res(lin Fp ) refutations more approachable and might be aided by tools from linear algebra and additive combinatorics.We identify hardness criterions for instances of the form A • x = b using notions of an error correcting code and what we call (s, r)-robustness, a combinatorial, algebraic property of linear systems A • x = b, which we introduce. We prove two lower bounds for fragments of Res(lin Fp ) that capture two complementary aspects of Res(lin Fp ) refutations and constitute a combinatorial toolbox for approaching general dag-like Res(lin Fp