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(linR)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm Res}({\rm lin}_R)$$\end{document}, 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(linR)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm Res}({\rm lin}_R)$$\end{document} 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.