Document Open Access Logo

On the Computational Power of Extensional ESO

Authors Manuel Bodirsky , Santiago Guzmán-Pro

PDF

File

Thumbnail PDF
PDF
LIPIcs.LICS.2026.20.pdf
  • Filesize: 0.98 MB
  • 26 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Finite Model Theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Existential second-order logic
  • constraint satisfaction problem
  • complexity classification
  • Hereditary first-order logic
  • NP-intermediate problems

Metrics

Abstract

Extensional ESO is a fragment of existential second-order logic (ESO) that captures the following family of problems. Given a fixed ESO sentence Ψ and an input structure A the task is to decide whether there is an extension B of A that satisfies the first-order part of Ψ, i.e., a structure B such that R^A ⊆ R^B for every existentially quantified predicate R of Ψ, and R^A = R^B for every non-quantified predicate R of Ψ. In particular, extensional ESO describes all pre-coloured finite-domain constraint satisfaction problems (CSPs).
In this paper we study the computational power of extensional ESO; we ask, for which problems in NP is there a polynomial-time equivalent problem in extensional ESO? One of our main results states that extensional ESO has the same computational power as hereditary first-order logic. We also characterize the computational power of the fragment of extensional ESO with monotone universal first-order part in terms of finitely bounded CSPs. These results suggest a rich computational power of this logic, and we conjecture that extensional ESO captures NP-intermediate problems. We further support this conjecture by showing that extensional ESO can express current candidate NP-intermediate problems such as Graph Isomorphism, and Monotone Dualization (up to polynomial-time equivalence). On the other hand, another main result proves that extensional ESO does not have the full computational power of NP: there are problems in NP that are not polynomial-time equivalent to a problem in extensional ESP (unless E=NE).

Cite As Get BibTex

Manuel Bodirsky and Santiago Guzmán-Pro. On the Computational Power of Extensional ESO. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 20:1-20:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.20

Author Details

Manuel Bodirsky
  • Institut für Algebra, TU Dresden, Germany
Santiago Guzmán-Pro
  • Institut für Algebra, TU Dresden, Germany

Funding

This project has received funding from the European Union (Project POCOCOP, ERC Synergy Grant 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.

Acknowledgements

The authors are thankful to Antoine Amarilli for a guided tour through Constrained Topological Sorting.

References

  1. J.D. Alvarado, Simone Dantas, and Dietrich Rautenbach. Sandwiches missing two ingredients of order four. Annals of Operations Research, 280:47-63, 2019. URL: https://doi.org/10.1007/s10479-019-03174-6.
  2. Antoine Amarilli and Charles Paperman. Topological Sorting with Regular Constraints. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 115:1-115:14, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.115.
  3. Klaus Ambos-Spies and Jürgen Kämper. On disjunctive self-reducibility. In Egon Börger, Hans Kleine Büning, and Michael M. Richter, editors, CSL '88, 2nd Workshop on Computer Science Logic, Duisburg, Germany, October 3-7, 1988, Proceedings, volume 385 of Lecture Notes in Computer Science, pages 1-13. Springer, 1988. URL: https://doi.org/10.1007/BFB0026292.
  4. Lázló Babai. Graph isomorphism in quasipolynomial time. arXiv:1512.03547, 2015. Google Scholar
  5. Richard N. Ball, Jaroslav Nešetřil, and Aleš Pultr. Dualities in full homomorphisms. European Journal of Combinatorics, 31:106-199, 2010. Google Scholar
  6. J. Bang-Jensen, J. Huang, and X. Zhu. Completing orientations of partially oriented graphs. Journal of Graph Theory, 87(3):285-304, 2017. URL: https://doi.org/10.1002/jgt.22157.
  7. Max Bannach, Florian Chudigiewitsch, and Till Tantau. On the Descriptive Complexity of Vertex Deletion Problems. In Rastislav Královič and Antonín Kučera, editors, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), volume 306 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1-17:14, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2024.17.
  8. Alexey Barsukov and Florent R. Madelaine. On guarded extensions of MMSNP. In Gianluca Della Vedova, Besik Dundua, Steffen Lempp, and Florin Manea, editors, Unity of Logic and Computation - 19th Conference on Computability in Europe, CiE 2023, Batumi, Georgia, July 24-28, 2023, Proceedings, volume 13967 of Lecture Notes in Computer Science, pages 202-213. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-36978-0_17.
  9. Libor Barto, Jakub Bulín, Andrei Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. J. ACM, 68(4), July 2021. URL: https://doi.org/10.1145/3457606.
  10. Mihir Bellare and Shafi Goldwasser. The complexity of decision versus search. SIAM J. Comput., 23(1):97-119, 1994. URL: https://doi.org/10.1137/S0097539792228289.
  11. M. Bienvenu, B. ten Cate, C. Lutz, and F. Wolter. Ontology-Based Data Access: A Study through Disjunctive Datalog, CSP, and MMSNP. ACM Trans. Database Syst., 39(4):33:1-33:44, 2014. URL: https://doi.org/10.1145/2661643.
  12. M. Bodirsky, S. Knäuer, and F. Starke. ASNP: A tame fragment of existential second-order logic. In CiE 2020, Proceedings, volume 12098 of Lecture Notes in Computer Science, pages 149-162. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-51466-2_13.
  13. Manuel Bodirsky. Complexity of Infinite-Domain Constraint Satisfaction. Lecture Notes in Logic (52). Cambridge University Press, Cambridge, United Kingdom; New York, NY, 2021. URL: https://doi.org/10.1017/9781107337534.
  14. Manuel Bodirsky and Martin Grohe. Non-dichotomies in constraint satisfaction complexity. In Luca Aceto, Ivan Damgard, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2008), pages 184-196. Springer Verlag, July 2008. URL: https://doi.org/10.1007/978-3-540-70583-3_16.
  15. Manuel Bodirsky and Santiago Guzmán-Pro. Hereditary first-order logic: the tractable quantifier prefixes. arXiv preprint, 2025. URL: https://arxiv.org/abs/2411.10860.
  16. Manuel Bodirsky and Santiago Guzmán-Pro. On the computational power of eso. arXiv preprint, 2025. URL: https://arxiv.org/abs/2511.08515.
  17. Manuel Bodirsky and Santiago Guzmán-Pro. The generic circular triangle-free graph. Journal of Graph Theory, 109(4):426-445, 2025. URL: https://doi.org/10.1002/jgt.23235.
  18. John Adrian Bondy and Uppaluri Siva Ramachandra Murty. Graph Theory. Springer, Berlin, 2008. Google Scholar
  19. Egon Börger, Erich Grädel, and Yuri Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997. Google Scholar
  20. Joshua Brakensiek and Venkatesan Guruswami. Promise constraint satisfaction: Algebraic structure and a symmetric boolean dichotomy. SIAM J. Comput., 50(6):1663-1700, 2021. URL: https://doi.org/10.1137/19M128212X.
  21. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, pages 319-330, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  22. Maria Chudnovsky, Neil Roberston, Paul Seymour, and Robin Thomas. The strong perfect graph theorem. Annals of Mathematics, 164:51-229, 2006. Google Scholar
  23. Simone Dantas, Celina M.H. de Figueiredo, Murilo V.G. da Silva, and Rafael B. Teixeira. On the forbidden induced subgraph sandwich problem. Discrete Applied Mathematics, 159:1717-1725, 2015. Google Scholar
  24. Simone Dantas, Celina M.H. de Figueiredo, Priscila Petito, and Rafael B. Teixeira. A general method for forbidden induced subgraph sandwich problem np-completeness. Electronic Notes in Theoretical Computer Science, 346:393-400, 2019. URL: https://doi.org/10.1016/j.entcs.2019.08.035.
  25. Celina M.H. de Figueiredo and Sophie Spirkl. Sandwich and probe problems for excluding paths. Discrete Applied Mathematics, 251:146-154, 2018. URL: https://doi.org/10.1016/j.dam.2018.05.054.
  26. Thomas Eiter, Georg Gottlob, and Kazuhisa Makino. New results on monotone dualization and generating hypergraph transversals. SIAM Journal on Computing, 32(2):514-537, 2003. URL: https://doi.org/10.1137/S009753970240639X.
  27. Thomas Eiter, Kazuhisa Makino, and Georg Gottlob. Computational aspects of monotone dualization: A brief survey. Discrete Applied Mathematics, 156(11):2035-2049, 2008. In Memory of Leonid Khachiyan (1952 - 2005 ). URL: https://doi.org/10.1016/j.dam.2007.04.017.
  28. Shimon Even, Alan L. Selman, and Yacov Yacobi. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61(2):159-173, 1984. URL: https://doi.org/10.1016/S0019-9958(84)80056-X.
  29. R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. Complexity of Computation, 7:43-73, 1974. Google Scholar
  30. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1999. URL: https://doi.org/10.1137/S0097539794266766.
  31. F.V. Fomin, P.A. Golovach, and D.M Thilikos. On the parameterized complexity of graph modification to first-order logic properties. Theory of Computing Systems, 64:251-271, 2020. URL: https://doi.org/10.1007/s00224-019-09938-8.
  32. Michael L. Fredman and Leonid Khachiyan. On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms, 21(3):618-628, 1996. URL: https://doi.org/10.1006/jagm.1996.0062.
  33. M.C. Golumbic, H. Kaplan, and R. Shamir. Graph Sandwich Problems. Journal of Algorithms, 19:449-473, 1995. URL: https://doi.org/10.1006/jagm.1995.1047.
  34. Martin Grohe and Pascal Schweitzer. The graph isomorphism problem. Commun. ACM, 63(11):128-134, 2020. URL: https://doi.org/10.1145/3372123.
  35. Edith Hemaspaandra, Ashish V. Naik, Mitsunori Ogihara, and Alan L. Selman. P-selective sets and reducing search to decision vs self-reducibility. Journal of Computer and System Sciences, 5:194-209, 1996. URL: https://doi.org/10.1006/JCSS.1996.0061.
  36. Lane A. Hemaspaandra. The Power of Self-Reducibility: Selectivity, Information, and Approximation, pages 19-47. Springer International Publishing, Cham, 2020. URL: https://doi.org/10.1007/978-3-030-41672-0_3.
  37. Wilfrid Hodges. A shorter model theory. Cambridge University Press, Cambridge, 1997. Google Scholar
  38. N. Immerman. Descriptive Complexity. Graduate Texts in Computer Science, Springer, New York, 1998. Google Scholar
  39. Ker-I Ko. On self-reducibility and weak p-selectivity. Journal of Computer and System Sciences, 26:209-221, 1983. URL: https://doi.org/10.1016/0022-0000(83)90013-2.
  40. Gábor Kun and Jaroslav Nešetřil. Dichotomy for orderings?, 2025. URL: https://doi.org/10.48550/arXiv.2504.13268.
  41. Alistair H. Lachlan and Robert E. Woodrow. Countable ultrahomogeneous undirected graphs. Transactions of the AMS, 262(1):51-94, 1980. Google Scholar
  42. Richard E. Ladner. On the structure of polynomial time reducibility. J. ACM, 22(1):155-171, January 1975. URL: https://doi.org/10.1145/321864.321877.
  43. Dugald Macpherson. A survey of homogeneous structures. Discrete Mathematics, 311(15):1599-1634, 2011. Infinite Graphs: Introductions, Connections, Surveys. URL: https://doi.org/10.1016/j.disc.2011.01.024.
  44. Khaled Maghout. Sur la determination des nombres de stabilite et du nombre chromatique dun graphe. Comptes rendus hebdomadaires des séances de l'Académie des sciences, 248(25):3522-3523, 1959. Google Scholar
  45. A. Meyer and M. Patersons. With what frequency are apparently intractable problems difficult?, 1979. Tech. Report MIT/LCS/TM-126, Massachusetts Institute of Technology, Cambridge, MA. Google Scholar
  46. Claus-Peter Schnorr. Optimal Algorithms for Self-Reducible Problems. In Proceedings of the Third International Colloquium on Automata, Languages and Programming, pages 322-337, 1976. Google Scholar
  47. Alan L. Selman. Promise problems for complexity classes. Information and Computation, 78:87-98, 1988. Google Scholar
  48. Howard Straubing. First-order logic and aperiodic languages: a revisionist history. ACM SIGLOG News, 5(3):4-20, 2018. URL: https://doi.org/10.1145/3242953.3242956.
  49. Dmitriy N. Zhuk. A proof of CSP dichotomy conjecture. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, pages 331-342, 2017. URL: https://arxiv.org/abs/1704.01914.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact