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

## Abstract

Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted $${\rm Res}({\rm lin}_R)$$, 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 $${\rm Res}({\rm lin}_R)$$ 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 $$1+\sum\nolimits_{i=1}^{n}2^i x_i = 0$$ requires refutations of exponential size over $$\mathbb{Q}$$. 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 $$f=\alpha$$ for every $$\alpha$$ 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 $$\mathbb{F}_2$$ 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 $${\rm Res}({\rm lin}{\mathbb{F}_p})$$ that require exponential-size tree-like $${\rm Res}({\rm lin}{\mathbb{F}_q})$$ refutations; (ii) random k-CNF formulas require exponential-size tree-like $${\rm Res}({\rm lin}{\mathbb{F}_p})$$ refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like $${\rm Res}({\rm lin}{\mathbb{F}})$$ refutations of the pigeonhole principle, for every field $$\mathbb{F}$$.

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## Acknowledgements

We wish to thank Dima Itsykson and Dima Sokolov for very helpful comments concerning this work, and telling us about the lower bound on random k-CNF formulas for tree-like Res$$({\rm lin}_{\mathbb{F}_{2}})$$ that can be achieved using the results of Garlik and Kołodziejczyk. We thank Edward Hirsch for spotting a gap in the initial proof of the dag-like lower bound concerning the use of the contraction rule, and Fedor Petrov for very useful discussions. Lastly, we are grateful to the anonymous reviewers of the extended abstract as well as the journal version of this work, who contributed to improving the exposition. An extended abstract of this paper has appeared as Part & Tzameret (2020).

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