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Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

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Theory and Applications of Satisfiability Testing – SAT 2021 (SAT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12831))

Abstract

Tseitin-formulas are systems of parity constraints whose structure is described by a graph. These formulas have been studied extensively in proof complexity as hard instances in many proof systems. In this paper, we prove that a class of unsatisfiable Tseitin-formulas of bounded degree has regular resolution refutations of polynomial length if and only if the treewidth of all underlying graphs G for that class is in \(O(\log |V(G)|)\). To do so, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length \(2^{\varOmega (tw(G))}/|V(G)|\), thus essentially matching the known \(2^{O(tw(G))}\text {poly}(|V(G)|)\) upper bound up. Our proof first connects the length of regular resolution refutations of unsatisfiable Tseitin-formulas to the size of representations of satisfiable Tseitin-formulas in decomposable negation normal form (DNNF). Then we prove that for every graph G of bounded degree, every DNNF-representation of every satisfiable Tseitin-formula with graph G must have size \(2^{\varOmega (tw(G))}\) which yields our lower bound for regular resolution.

This work has been partly supported by the PING/ACK project of the French National Agency for Research (ANR-18-CE40-0011).

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Notes

  1. 1.

    We remark that often branch decompositions are defined as unrooted trees. However, it is easy to see that our definition is equivalent, so we use it here since it is more convenient in our setting.

  2. 2.

    [20, Lemma 17] is for locally minimal 1-BP, which encompass minimal size 1-BP.

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Authors and Affiliations

  1. CNRS, UMR 8188, Centre de Recherche en Informatique de Lens (CRIL), 62300, Lens, France

    Alexis de Colnet & Stefan Mengel

  2. Univ. Artois, UMR 8188, 62300, Lens, France

    Alexis de Colnet & Stefan Mengel

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  1. Alexis de Colnet
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  2. Stefan Mengel
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Correspondence to Alexis de Colnet .

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Editors and Affiliations

  1. Laboratoire MIS, University of Picardie Jules Verne, Amiens, France

    Chu-Min Li

  2. IIIA-CSIC, Spanish National Research Council (CSIC), Bellaterra, Barcelona, Spain

    Felip Manyà

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de Colnet, A., Mengel, S. (2021). Characterizing Tseitin-Formulas with Short Regular Resolution Refutations. In: Li, CM., Manyà, F. (eds) Theory and Applications of Satisfiability Testing – SAT 2021. SAT 2021. Lecture Notes in Computer Science(), vol 12831. Springer, Cham. https://doi.org/10.1007/978-3-030-80223-3_9

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  • DOI: https://doi.org/10.1007/978-3-030-80223-3_9

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