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Notes on Resolution over Linear Equations

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Computer Science – Theory and Applications (CSR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11532))

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Abstract

We consider the proof system \(\mathrm {Res}(\oplus )\) introduced by Itsykson and Sokolov [8] which is an extension of Resolution proof system and operates with disjunctions of linear equations over \(\mathbb {F}_2\). In this paper we prove exponential lower bounds on tree-like \(\mathrm {Res}(\oplus )\) refutations for Ordering and Dense Linear Ordering principles by Prover-Delayer games.

We also consider the following problem: given two disjunctions of linear equations over ring R decide whether all Boolean satisfying assignments of one of them satisfy another. Part and Tzameret conjectured that for rings \(R \ne \mathbb {F}_2\) this problem is \(\mathrm {coNP}\)-hard, but proved it only for rings with \({{\,\mathrm{char}\,}}(R)=0\) and \({{\,\mathrm{char}\,}}(R) \ge 5\) [10]. We completely prove the conjecture.

S. Gryaznov: The research is supported by Russian Science Foundation (project 18-71-10042).

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Acknowledgements

The author is grateful to Dmitry Itsykson, Dmitry Sokolov, and Anastasia Sofronova for fruitful discussions. The author also thanks Dmitry Itsykson for the statement of the problem and Dmitry Sokolov for telling about Dense Linear Ordering principle and related problems.

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Authors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, 27 Fontanka, St. Petersburg, 191023, Russia

    Svyatoslav Gryaznov

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  1. Svyatoslav Gryaznov
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Correspondence to Svyatoslav Gryaznov .

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Editors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russia

    René van Bevern

  2. Laboratoire d'Informatique Gaspard-Monge, Marne-la-Vallée, France

    Gregory Kucherov

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Gryaznov, S. (2019). Notes on Resolution over Linear Equations. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_15

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  • DOI: https://doi.org/10.1007/978-3-030-19955-5_15

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