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Resolution Over Linear Equations: Combinatorial Games for Tree-like Size and Space
Published 12 Apr 2024 in cs.CC | (2404.08370v2)
Abstract: We consider the proof system Res($\oplus$) introduced by Itsykson and Sokolov (Ann. Pure Appl. Log.'20), which is an extension of the resolution proof system and operates with disjunctions of linear equations over $\mathbb{F}_2$. We study characterizations of tree-like size and space of Res($\oplus$) refutations using combinatorial games. Namely, we introduce a class of extensible formulas and prove tree-like size lower bounds on it using Prover-Delayer games, as well as space lower bounds. This class is of particular interest since it contains many classical combinatorial principles, including the pigeonhole, ordering, and dense linear ordering principles. Furthermore, we present the width-space relation for Res($\oplus$) generalizing the results by Atserias and Dalmau (J. Comput. Syst. Sci.'08) and their variant of Spoiler-Duplicator games.
- Space Complexity in Propositional Calculus. SIAM J. Comput. 31, 4 (2002), 1184–1211. https://doi.org/10.1137/S0097539700366735
- Albert Atserias and VÃctor Dalmau. 2008. A combinatorial characterization of resolution width. J. Comput. Syst. Sci. 74, 3 (2008), 323–334. https://doi.org/10.1016/J.JCSS.2007.06.025
- Using the Groebner Basis Algorithm to Find Proofs of Unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, Gary L. Miller (Ed.). ACM, Philadelphia, Pennsylvania, USA, 174–183. https://doi.org/10.1145/237814.237860
- A machine program for theorem-proving. Commun. ACM 5, 7 (1962), 394–397. https://doi.org/10.1145/368273.368557
- Martin Davis and Hilary Putnam. 1960. A Computing Procedure for Quantification Theory. J. ACM 7, 3 (1960), 201–215. https://doi.org/10.1145/321033.321034
- Juan Luis Esteban and Jacobo Torán. 2001. Space Bounds for Resolution. Inf. Comput. 171, 1 (2001), 84–97. https://doi.org/10.1006/inco.2001.2921
- Nicola Galesi and Massimo Lauria. 2010. Optimality of size-degree tradeoffs for polynomial calculus. ACM Trans. Comput. Log. 12, 1 (2010), 4:1–4:22. https://doi.org/10.1145/1838552.1838556
- Michal GarlÃk and Leszek Aleksander Kolodziejczyk. 2018. Some Subsystems of Constant-Depth Frege with Parity. ACM Trans. Comput. Log. 19, 4 (2018), 29:1–29:34. https://doi.org/10.1145/3243126
- Mika Göös and Toniann Pitassi. 2018. Communication Lower Bounds via Critical Block Sensitivity. SIAM J. Comput. 47, 5 (2018), 1778–1806. https://doi.org/10.1137/16M1082007
- Svyatoslav Gryaznov. 2019. Notes on Resolution over Linear Equations. In Computer Science - Theory and Applications - 14th International Computer Science Symposium in Russia, CSR 2019, Novosibirsk, Russia, July 1-5, 2019, Proceedings (Lecture Notes in Computer Science, Vol. 11532), René van Bevern and Gregory Kucherov (Eds.). Springer, Novosibirsk, Russia, 168–179. https://doi.org/10.1007/978-3-030-19955-5_15
- Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43, 4 (2006), 439–561. https://doi.org/10.1090/S0273-0979-06-01126-8
- Dmitry Itsykson and Alexander Knop. 2017. Hard Satisfiable Formulas for Splittings by Linear Combinations. In Theory and Applications of Satisfiability Testing - SAT 2017 - 20th International Conference, Melbourne, VIC, Australia, August 28 - September 1, 2017, Proceedings (Lecture Notes in Computer Science, Vol. 10491), Serge Gaspers and Toby Walsh (Eds.). Springer, Melbourne, VIC, Australia, 53–61. https://doi.org/10.1007/978-3-319-66263-3_4
- Dmitry Itsykson and Dmitry Sokolov. 2020. Resolution over linear equations modulo two. Ann. Pure Appl. Log. 171, 1 (2020), 31Â pages. https://doi.org/10.1016/j.apal.2019.102722
- Mladen Miksa and Jakob Nordström. 2015. A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds. In 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA (LIPIcs, Vol. 33), David Zuckerman (Ed.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Portland, Oregon, USA, 467–487. https://doi.org/10.4230/LIPIcs.CCC.2015.467
- Fedor Part and Iddo Tzameret. 2021. Resolution with Counting: Dag-Like Lower Bounds and Different Moduli. Comput. Complex. 30, 1 (2021), 2. https://doi.org/10.1007/s00037-020-00202-x
- Pavel Pudlák and Russell Impagliazzo. 2000. A lower bound for DLL algorithms for k-SAT (preliminary version). In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA, David B. Shmoys (Ed.). ACM/SIAM, San Francisco, CA, USA, 128–136. http://dl.acm.org/citation.cfm?id=338219.338244
- Ran Raz and Iddo Tzameret. 2008. Resolution over linear equations and multilinear proofs. Ann. Pure Appl. Log. 155, 3 (2008), 194–224. https://doi.org/10.1016/j.apal.2008.04.001
- Søren Riis. 2001. A complexity gap for tree resolution. Comput. Complex. 10, 3 (2001), 179–209. https://doi.org/10.1007/S00037-001-8194-Y
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