Resolution with Counting: Dag-Like Lower Bounds and Different Moduli - Part, Fedor; Tzameret, Iddo (Computational complexity, 2021)
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 ( lin R ) , 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 Raz & Tzameret (2008), through the work of Itsykson & Sokolov (2020) which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. Garlik & Kołodziejczyk 2018; Itsykson & Sokolov 2020; Krajícek 2017; Krajícek & Oliveira 2018) made it evident that establishing lower bounds against general Res ( lin R ) 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 against general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular, we prove that the subset-sum principle 1 + ∑ i = 1 n 2 i x i = 0 requires refutations of exponential size over Q . We use a novel lower bound technique: We show that under certain conditions every refutation of a subset-sum instance f = 0 must pass through a fat clause consisting of the equation f = α for every α in the image of f under Boolean assignments, or can be efficiently reduced to a proof containing such a clause. We then modify this 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 & Sokolov (2020), where tree-like lower bounds over F 2 were obtained, 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 F p ) that require exponential-size tree-like Res ( lin F q ) refutations; ( ii ) random k -CNF formulas require exponential-size tree-like Res ( lin F p ) 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 .