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The Polynomial Hierarchy and ω-Categorical CSPs

Authors Santiago Guzmán-Pro , Jakub Rydval

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LIPIcs.MFCS.2026.96.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Computational complexity and cryptography
Keywords
  • monadic second-order logic
  • constraint satisfaction
  • homomorphism-closed
  • ω-categoricity
  • polynomial hierarchy
  • conjunctive query
  • primitive positive formula
  • quantifiers

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Abstract

In 2008, Bodirsky and Grohe showed that for every Π_n^P-level of the Polynomial Hierarchy (PH) there are ω-categorical Constraint Satisfaction Problems (CSPs) complete for this level. We show that, in fact, there are ω-categorical CSPs complete for any level of the PH. To this end, we use a recent result of Bodirsky, Knäuer, and Rudolph for constructing ω-categorical CSPs from sentences of Monadic Second-Order logic (MSO) with certain preservation properties. As a secondary contribution, we develop a new tool for producing MSO sentences satisfying said preservation properties.

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Santiago Guzmán-Pro and Jakub Rydval. The Polynomial Hierarchy and ω-Categorical CSPs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 96:1-96:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.96

Author Details

Santiago Guzmán-Pro
  • Technische Universität Dresden, Germany
Jakub Rydval
  • Technische Universität Wien, Austria

Funding

  • Guzmán-Pro, Santiago: This research is funded by the European Union (ERC, POCOCOP, 101071674). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
  • Rydval, Jakub: This research was funded in whole or in part by the Austrian Science Fund (FWF) [ESP 1571225]. For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

References

  1. Libor Barto, Bertalan Bodor, Marcin Kozik, Antoine Mottet, and Michael Pinsker. Symmetries of Graphs and Structures that Fail to Interpret a Finite Thing. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-13, 2023. URL: https://doi.org/10.1109/LICS56636.2023.10175732.
  2. Libor Barto, Michael Kompatscher, Miroslav Olšák, Van Pham Trung, and Michael Pinsker. Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures. J. Math. Log., 19(02):Article 1950010, 2019. URL: https://doi.org/10.1142/S0219061319500107.
  3. Libor Barto, Jakub Opršal, and Michael Pinsker. The wonderland of reflections. Isr. J. Math., 223(1):363-398, 2018. URL: https://doi.org/10.1007/s11856-017-1621-9.
  4. Libor Barto and Michael Pinsker. Topology is irrelevant (in a dichotomy conjecture for infinite domain constraint satisfaction problems). SIAM J. Comput., 49(2):365-393, 2020. URL: https://doi.org/10.1137/18M1216213.
  5. Manuel Bodirsky. Complexity of infinite-domain constraint satisfaction, volume 52. Cambridge University Press, 2021. https://wwwpub.zih.tu-dresden.de/~bodirsky/Book.pdf. URL: https://doi.org/10.1017/9781107337534.
  6. Manuel Bodirsky, Thomas Feller, Simon Knäuer, and Sebastian Rudolph. On logics and homomorphism closure. In 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-13. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470511.
  7. Manuel Bodirsky and Martin Grohe. Non-dichotomies in constraint satisfaction complexity. In Proceedings of the 35th International Colloquium on Automata, Languages, and Programming - ICALP'08, pages 184-196. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-70583-3_16.
  8. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. J. ACM., 57(2):9:1-9:41, 2010. A conference version appeared in the proceedings of the 40th ACM Symposium on Theory of Computing - STOC'08. URL: https://doi.org/10.1145/1667053.1667058.
  9. Manuel Bodirsky, Simon Knäuer, and Sebastian Rudolph. Datalog-Expressibility for Monadic and Guarded Second-Order Logic. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming - ICALP'21, volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 120:1-120:17, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.120.
  10. Manuel Bodirsky and Jaroslav Nešetřil. Constraint satisfaction with countable homogeneous templates. J. Log. Comput, 16(3):359-373, 2006. A conference version appeared in the proceedings of Computer Science Logic - CSL'03. URL: https://doi.org/10.1093/logcom/exi083.
  11. Manuel Bodirsky, Michael Pinsker, and András Pongrácz. Projective clone homomorphisms. J. Symb. Log., 86(1):148-161, 2021. URL: https://doi.org/10.1017/jsl.2019.23.
  12. Johanna Brunar, Marcin Kozik, Tomáš Nagy, and Michael Pinsker. The sorrows of a smooth digraph: the first hardness criterion for infinite directed graph-colouring problems. In 2025 40th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 403-416, 2025. URL: https://doi.org/10.1109/LICS65433.2025.00037.
  13. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science - FOCS'17, pages 319-330. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  14. A. K. Chandra and P. M. Merlin. Optimal implementation of conjunctive queries in relational data bases. In Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, STOC '77, pages 77-90, New York, NY, USA, 1977. Association for Computing Machinery. URL: https://doi.org/10.1145/800105.803397.
  15. Hubie Chen. Existentially restricted quantified constraint satisfaction. Inf. Comput., 207(3):369-388, 2009. URL: https://doi.org/10.1016/j.ic.2008.11.001.
  16. Grädel E. Back and Forth Between Logic and Games, pages 99-145. Cambridge University Press, 2011. Google Scholar
  17. 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. URL: https://doi.org/10.1137/S0097539794266766.
  18. Pierre Gillibert, Julius Jonusas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker. When symmetries are not enough: A hierarchy of hard constraint satisfaction problems. SIAM J. Comput., 51(2):175-213, 2022. URL: https://doi.org/10.1137/20M1383471.
  19. Wilfrid Hodges. A Shorter Model Theory. Cambridge University Press, 1997. Google Scholar
  20. Barnaby Martin, Manuel Bodirsky, and Martin Hils. On the scope of the universal-algebraic approach to constraint satisfaction. Logical Methods in Computer Science, 8, 2012. URL: https://doi.org/10.2168/LMCS-8(3:13)2012.
  21. Antoine Mottet and Michael Pinsker. Smooth approximations: An algebraic approach to CSPs over finitely bounded homogeneous structures. J. ACM., 71(5):36:1-36:47, 2024. URL: https://doi.org/10.1145/3689207.
  22. Mark H. Siggers. A strong Mal'cev condition for locally finite varieties omitting the unary type. Algebra Univers., 64:15-20, 2010. URL: https://doi.org/10.1007/s00012-010-0082-3.
  23. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science - FOCS'17, pages 331-342. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
  24. Dmitriy Zhuk. A proof of the CSP dichotomy conjecture. J. ACM., 67(5):30:1-30:78, 2020. URL: https://doi.org/10.1145/3402029.
  25. Dmitriy Zhuk and Barnaby Martin. QCSP monsters and the demise of the Chen Conjecture. J. ACM, 69(5):35:1-35:44, 2022. URL: https://doi.org/10.1145/3563820.
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