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The Expressive Power of CSP-Quantifiers

Author Lauri Hella



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Lauri Hella
  • Faculty of Information Technology and Communication Sciences, Tampere University, Finland

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Lauri Hella. The Expressive Power of CSP-Quantifiers. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 25:1-25:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CSL.2023.25

Abstract

A generalized quantifier Q_𝒦 is called a CSP-quantifier if its defining class 𝒦 consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n ≥ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic L^k_{∞ω}(CSP^+_n), where CSP^+_n is the union of the class Q₁ of all unary quantifiers and the class CSP_n of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n ≥ 2 there exists a CSP-quantifier with template of size n+1 which is not definable in L^ω_{∞ω}(CSP^+_n). The proof of this result is based on a new variation of the well-known Cai-Fürer-Immerman construction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • generalized quantifiers
  • constraint satisfaction problems
  • pebble games
  • finite variable logics
  • descriptive complexity theory

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References

  1. Albert Atserias, Andrei A. Bulatov, and Anuj Dawar. Affine systems of equations and counting infinitary logic. Theor. Comput. Sci., 410(18):1666-1683, 2009. URL: https://doi.org/10.1016/j.tcs.2008.12.049.
  2. Andrei A. Bulatov. A dichotomy theorem for nonuniform csps. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  3. Jin-yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identifications. Comb., 12(4):389-410, 1992. URL: https://doi.org/10.1007/BF01305232.
  4. Anuj Dawar, Erich Grädel, and Wied Pakusa. Approximations of isomorphism and logics with linear-algebraic operators. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 112:1-112:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.112.
  5. Anuj Dawar, Martin Grohe, Bjarki Holm, and Bastian Laubner. Logics with rank operators. In Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, 11-14 August 2009, Los Angeles, CA, USA, pages 113-122. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/LICS.2009.24.
  6. Anuj Dawar and Bjarki Holm. Pebble games with algebraic rules. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part II, volume 7392 of Lecture Notes in Computer Science, pages 251-262. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31585-5_25.
  7. Heinz-Dieter Ebbinghaus and Jörg Flum. Finite model theory. Perspectives in Mathematical Logic. Springer, 1995. Google Scholar
  8. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput., 28(1):57-104, 1998. Google Scholar
  9. Erich Grädel and Wied Pakusa. Rank logic is dead, long live rank logic! In Stephan Kreutzer, editor, 24th EACSL Annual Conference on Computer Science Logic, CSL 2015, September 7-10, 2015, Berlin, Germany, volume 41 of LIPIcs, pages 390-404. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.CSL.2015.390.
  10. Lauri Hella. Logical hierarchies in PTIME. Inf. Comput., 129(1):1-19, 1996. URL: https://doi.org/10.1006/inco.1996.0070.
  11. Neil Immerman and Eric Lander. Describing graphs: A first-order approach to graph canonization. In Alan L. Selman, editor, Complexity Theory Retrospective: In Honor of Juris Hartmanis on the Occasion of His Sixtieth Birthday, July 5, 1988, pages 59-81. Springer New York, New York, NY, 1990. URL: https://doi.org/10.1007/978-1-4612-4478-3_5.
  12. Phokion G. Kolaitis and Moshe Y. Vardi. A logical approach to constraint satisfaction. In Nadia Creignou, Phokion G. Kolaitis, and Heribert Vollmer, editors, Complexity of Constraints - An Overview of Current Research Themes [Result of a Dagstuhl Seminar], volume 5250 of Lecture Notes in Computer Science, pages 125-155. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-92800-3_6.
  13. Leonid Libkin. Elements of Finite Model Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2004. URL: https://doi.org/10.1007/978-3-662-07003-1.
  14. Moritz Lichter. Separating rank logic from polynomial time. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021, pages 1-13. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470598.
  15. Per Lindström. First order predicate logic with generalized quantifiers. Theoria, 32:186-195, 1966. URL: https://doi.org/10.1111/j.1755-2567.1966.tb00600.x.
  16. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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