Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Abstract

Resolution over linear equations (introduced in [RT08]) emerged recently as an important object of study. This refutation system, denoted Res(lin R ), operates with disjunction of linear equations over a ring R. 1 On the one hand, the system captures a natural "minimal" extension of resolution in which efficient counting can be achieved; while on the other hand, as observed by, e.g., Krajíček [Kra17] (cf. [IS14, KO18, GK17]), when considered over prime fields, and specifically F 2 , super-polynomial lower bounds on (dag-like) Res(lin F 2 ) is a first step towards the long-standing open problem of establishing constant-depth Frege with counting gates (AC 0 [2]-Frege) lower bounds.In this work we develop new lower bound techniques for resolution over linear equations and extend existing ones to work over different rings. We obtain a host of new lower bounds, separations and upper bounds, while calibrating the relative strength of different sub-systems. We first establish, over fields of characteristic zero, exponential-size dag-like lower bounds against resolution over linear equations refutations of instances with large coefficients. Specifically, we demonstrate that the subset sum principle α 1 x 1 + . . . + α n x n = β, for β not in the image of the linear form, requires refutations proportional to the size of the image. Moreover, for instances with small coefficients, we separate the tree and dag-like versions of Res(lin F ), when F is of characteristic zero, by employing the notion of immunity from Alekhnovich-Razborov [AR01], among other techniques.We then study resolution over linear equations over different finite fields, extending the work of Itsykson and Sokolov [IS14] who developed tree-like Res(lin F 2 ) lower bounds techniques. We obtain new lower bounds and separations as follows: (i) exponential-size lower bounds for tree-like Res(lin Fp ) Tseitin mod q formulas, for every pair of distinct primes p, q. As a corollary we obtain an exponential-size separation between tree-like Res(lin Fp ) and tree-like Res(lin Fq ); (ii) exponential-size lower bounds for tree-like Res(lin Fp ) refutations of random k-CNF formulas, 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.