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

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 [RT08], through the work of [IS14] which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf.[Kra17, IS14, KO18, GK18]) 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 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 must pass through a fat clause containing an equation f=alpha for each alpha 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. (Abstract continued in the full paper.)