Homogeneity and Homogenizability: Hard Problems for the Logic SNP

Author Jakub Rydval



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.150.pdf
  • Filesize: 1.08 MB
  • 20 pages

Document Identifiers

Author Details

Jakub Rydval
  • Technische Universität Wien, Austria

Acknowledgements

The author thanks Manuel Bodirsky, Simon Knäuer, and Jakub Opršal for many inspiring discussions on the topic, and the anonymous reviewers for many helpful suggestions.

Cite AsGet BibTex

Jakub Rydval. Homogeneity and Homogenizability: Hard Problems for the Logic SNP. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 150:1-150:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.150

Abstract

The infinite-domain CSP dichotomy conjecture extends the finite-domain CSP dichotomy theorem to reducts of finitely bounded homogeneous structures. Every countable finitely bounded homogeneous structure is uniquely described by a universal first-order sentence up to isomorphism, and every reduct of such a structure by a sentence of the logic SNP. By Fraïssé’s Theorem, testing the existence of a finitely bounded homogeneous structure for a given universal first-order sentence is equivalent to testing the amalgamation property for the class of its finite models. The present paper motivates a complexity-theoretic view on the classification problem for finitely bounded homogeneous structures. We show that this meta-problem is EXPSPACE-hard or PSPACE-hard, depending on whether the input is specified by a universal sentence or a set of forbidden substructures. By relaxing the input to SNP sentences and the question to the existence of a structure with a finitely bounded homogeneous expansion, we obtain a different meta-problem, closely related to the question of homogenizability. We show that this second meta-problem is already undecidable, even if the input SNP sentence comes from the Datalog fragment and uses at most binary relation symbols. As a byproduct of our proof, we also get the undecidability of some other properties for Datalog programs, e.g., whether they can be rewritten in the logic MMSNP, whether they solve some finite-domain CSP, or whether they define a structure with a homogeneous Ramsey expansion in a finite relational signature.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Computational complexity and cryptography
Keywords
  • constraint satisfaction problems
  • finitely bounded
  • homogeneous
  • amalgamation property
  • universal
  • SNP
  • homogenizable

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Serge Abiteboul, Richard Hull, and Victor Vianu. Foundations of databases, volume 8. Addison-Wesley, 1995. URL: http://webdam.inria.fr/Alice/.
  2. Ove Ahlman. Homogenizable structures and model completeness. Arch. Math. Log., 55(7-8):977-995, 2016. URL: https://doi.org/10.1007/s00153-016-0507-6.
  3. Reza Akhtar and Alistair H. Lachlan. On countable homogeneous 3-hypergraphs. Arch. Math. Log., 34(5):331-344, 1995. URL: https://doi.org/10.1007/BF01387512.
  4. Albert Atserias and Szymon Torunczyk. Non-homogenizable classes of finite structures. In Jean-Marc Talbot and Laurent Regnier, editors, 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France, volume 62 of LIPIcs, pages 16:1-16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CSL.2016.16.
  5. Franz Baader and Jakub Rydval. Using model theory to find decidable and tractable description logics with concrete domains. J. Autom. Reason., 66(3):357-407, 2022. URL: https://doi.org/10.1007/s10817-022-09626-2.
  6. Libor Barto, Michael Kompatscher, Miroslav Olsák, Trung Van Pham, and Michael Pinsker. The equivalence of two dichotomy conjectures for infinite domain constraint satisfaction problems. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-12. IEEE Computer Society, 2017. available at https://arxiv.org/abs/1612.07551. URL: https://doi.org/10.1109/LICS.2017.8005128.
  7. Meghyn Bienvenu, Balder ten Cate, Carsten Lutz, and Frank 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.
  8. Manuel Bodirsky. Complexity of Infinite-Domain Constraint Satisfaction. Cambridge University Press, 2021. URL: https://doi.org/10.1017/9781107337534.
  9. Manuel Bodirsky and Bertalan Bodor. Canonical polymorphisms of ramsey structures and the unique interpolation property. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021, pages 1-13. IEEE, IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470683.
  10. Manuel Bodirsky, David Bradley-Williams, Michael Pinsker, and András Pongrácz. The universal homogeneous binary tree. J. Log. Comput., 28(1):133-163, 2018. URL: https://doi.org/10.1093/logcom/exx043.
  11. Manuel Bodirsky and Víctor Dalmau. Datalog and constraint satisfaction with infinite templates. J. Comput. Syst. Sci., 79(1):79-100, 2013. A preliminary version appeared in the proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS'05). URL: https://doi.org/10.1016/j.jcss.2012.05.012.
  12. Manuel Bodirsky and Martin Grohe. Non-dichotomies in constraint satisfaction complexity. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part II - Track B: Logic, Semantics, and Theory of Programming & Track C: Security and Cryptography Foundations, volume 5126 of Lecture Notes in Computer Science, pages 184-196. Springer, July 2008. URL: https://doi.org/10.1007/978-3-540-70583-3_16.
  13. Manuel Bodirsky, Peter Jonsson, and Van Trung Pham. The complexity of phylogeny constraint satisfaction problems. ACM Trans. Comput. Log., 18(3):23:1-23:42, 2017. URL: https://doi.org/10.1145/3105907.
  14. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. J. ACM, 57(2):9:1-9:41, 2010. URL: https://doi.org/10.1145/1667053.1667058.
  15. Manuel Bodirsky, Simon Knäuer, and Florian Starke. ASNP: A tame fragment of existential second-order logic. In Marcella Anselmo, Gianluca Della Vedova, Florin Manea, and Arno Pauly, editors, Beyond the Horizon of Computability - 16th Conference on Computability in Europe, CiE 2020, Fisciano, Italy, June 29 - July 3, 2020, Proceedings, volume 12098 of Lecture Notes in Computer Science, pages 149-162. Springer, Springer, 2020. URL: https://doi.org/10.1007/978-3-030-51466-2_13.
  16. Manuel Bodirsky, Florent R. Madelaine, and Antoine Mottet. A proof of the algebraic tractability conjecture for monotone monadic SNP. SIAM J. Comput., 50(4):1359-1409, 2021. URL: https://doi.org/10.1137/19M128466X.
  17. Manuel Bodirsky and Antoine Mottet. Reducts of finitely bounded homogeneous structures, and lifting tractability from finite-domain constraint satisfaction. In Martin Grohe, Eric Koskinen, and Natarajan Shankar, editors, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016, pages 623-632. ACM, 2016. available at https://arxi.org/abs/1601.04520. URL: https://doi.org/10.1145/2933575.2934515.
  18. 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.
  19. Manuel Bodirsky and Jakub Rydval. On the descriptive complexity of temporal constraint satisfaction problems. J. ACM, 70(1):2:1-2:58, 2023. URL: https://doi.org/10.1145/3566051.
  20. Manuel Bodirsky, Jakub Rydval, and André Schrottenloher. Universal horn sentences and the joint embedding property. Discret. Math. Theor. Comput. Sci., 23(2), 2021. URL: https://doi.org/10.46298/dmtcs.7435.
  21. Mikolaj Bojanczyk, Luc Segoufin, and Szymon Torunczyk. Verification of database-driven systems via amalgamation. In Richard Hull and Wenfei Fan, editors, Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2013, New York, NY, USA - June 22-27, 2013, pages 63-74. ACM, 2013. URL: https://doi.org/10.1145/2463664.2465228.
  22. Samuel Braunfeld. The undecidability of joint embedding and joint homomorphism for hereditary graph classes. Discret. Math. Theor. Comput. Sci., 21(2), 2019. available at https://arxiv.org/abs/1903.11932. URL: https://doi.org/10.23638/DMTCS-21-2-9.
  23. 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, IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  24. Jakub Bulin, Dejan Delic, Marcel Jackson, and Todd Niven. A finer reduction of constraint problems to digraphs. Log. Methods Comput. Sci., 11(4), 2015. URL: https://doi.org/10.2168/LMCS-11(4:18)2015.
  25. Gregory L. Cherlin. Forbidden substructures and combinatorial dichotomies: WQO and universality. Discret. Math., 311(15):1543-1584, 2011. URL: https://doi.org/10.1016/j.disc.2011.03.014.
  26. Gregory L. Cherlin. Homogeneous ordered graphs, metrically homogeneous graphs, and beyond, volume 1. Cambridge University Press, 2022. URL: https://doi.org/10.1017/9781009229661.
  27. Gregory L. Cherlin, Saharon Shelah, and Niandong Shi. Universal graphs with forbidden subgraphs and algebraic closure. Advances in Applied Mathematics, 22(4):454-491, 1999. URL: https://doi.org/10.1006/aama.1998.0641.
  28. Jacinta Covington. Homogenizable relational structures. Illinois Journal of Mathematics, 34(4):731-743, 1990. URL: https://doi.org/10.1215/ijm/1255988065.
  29. Péter L. Erdös, Claude Tardif, and Gábor Tardos. Caterpillar dualities and regular languages. SIAM J. Discret. Math., 27(3):1287-1294, 2013. URL: https://doi.org/10.1137/120879270.
  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 J. Comput., 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  31. Haim Gaifman, Harry G. Mairson, Yehoshua Sagiv, and Moshe Y. Vardi. Undecidable optimization problems for database logic programs. J. ACM, 40(3):683-713, 1993. URL: https://doi.org/10.1145/174130.174142.
  32. Nicola Gigante, Andrea Micheli, Angelo Montanari, and Enrico Scala. Decidability and complexity of action-based temporal planning over dense time. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, New York, NY, USA, February 7-12, 2020, pages 9859-9866. AAAI Press, 2020. URL: https://doi.org/10.1609/aaai.v34i06.6539.
  33. Wilfrid Hodges. A shorter model theory. Cambridge University Press, Cambridge, 1997. Google Scholar
  34. Jan Hubička and Jaroslav Nešetřil. Homomorphism and embedding universal structures for restricted classes. J. Multiple Valued Log. Soft Comput., 27(2-3):229-253, 2016. available at https://arxiv.org/abs/0909.4939. URL: http://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-27-number-2-3-2016/mvlsc-27-2-3-p-229-253/.
  35. Jan Hubička and Jaroslav Nešetřil. All those ramsey classes (ramsey classes with closures and forbidden homomorphisms). Advances in Mathematics, 356:1-89, 2019. URL: https://doi.org/10.1016/j.aim.2019.106791.
  36. Julia F. Knight and Alistair H. Lachlan. Shrinking, stretching, and codes for homogeneous structures. In Classification Theory: Proceedings of the US-Israel Workshop on Model Theory in Mathematical Logic held in Chicago, Dec. 15-19, 1985, volume 1292, pages 192-229. Springer, Springer, Berlin, Heidelberg, 2006. URL: https://doi.org/10.1007/BFb0082239.
  37. Gábor Kun. Constraints, MMSNP and expander relational structures. Comb., 33(3):335-347, 2013. URL: https://doi.org/10.1007/s00493-013-2405-4.
  38. Alistair H. Lachlan. Homogeneous structures. In Proceedings of the International Congress of Mathematicians, volume 1, pages 314-321, Berkeley, 1986. AMS. Google Scholar
  39. Carsten Lutz and Maja Milicic. A tableau algorithm for description logics with concrete domains and general tboxes. J. Autom. Reason., 38(1-3):227-259, 2007. URL: https://doi.org/10.1007/s10817-006-9049-7.
  40. Dugald Macpherson. A survey of homogeneous structures. Discret. Math., 311(15):1599-1634, 2011. URL: https://doi.org/10.1016/j.disc.2011.01.024.
  41. Florent R. Madelaine. Universal structures and the logic of forbidden patterns. Log. Methods Comput. Sci., 5(2), 2009. URL: https://doi.org/10.2168/LMCS-5(2:13)2009.
  42. Florent R. Madelaine and Iain A. Stewart. Some problems not definable using structure homomorphisms. Ars Comb., 67:153-160, 2003. Google Scholar
  43. Antoine Mottet. Promise and infinite-domain constraint satisfaction. In Aniello Murano and Alexandra Silva, editors, 32nd EACSL Annual Conference on Computer Science Logic, CSL 2024, February 19-23, 2024, Naples, Italy, volume 288 of LIPIcs, pages 41:1-41:19. Schloss-Dagstuhl-Leibniz Zentrum für Informatik, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CSL.2024.41.
  44. Antoine Mottet and Michael Pinsker. Smooth approximations and csps over finitely bounded homogeneous structures. In Christel Baier and Dana Fisman, editors, LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2-5, 2022, pages 36:1-36:13. ACM, 2022. available at https://arxiv.org/abs/2011.03978. URL: https://doi.org/10.1145/3531130.3533353.
  45. Antoine Mottet, Michael Pinsker, and Tomáš Nagy. An order out of nowhere: a new algorithm for infinite-domain CSPs. In 51st EATCS International Colloquium on Automata, Languages and Programming (ICALP 2024). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. to appear, available at https://arxiv.org/abs/2301.12977. URL: https://doi.org/10.48550/arXiv.2301.12977.
  46. Shan-Hwei Nienhuys-Cheng and Ronald de Wolf. Foundations of Inductive Logic Programming, volume 1228 of Lecture Notes in Computer Science. Springer, 1997. URL: https://doi.org/10.1007/3-540-62927-0.
  47. Jakub Rydval. Homogeneity and homogenizability: Hard problems for the logic snp. CoRR, abs/2108.00452, 2021. URL: https://doi.org/10.48550/arXiv.2108.00452.
  48. François Schwarzentruber. The complexity of tiling problems. CoRR, abs/1907.00102, 2019. URL: https://doi.org/10.48550/arXiv.1907.00102.
  49. Jeffrey O. Shallit. A second course in formal languages and automata theory. Cambridge University Press, 2008. URL: http://www.cambridge.org/gb/knowledge/isbn/item1173872/?site_locale=en_GB.
  50. Peter van Emde Boas. The convenience of tilings. In Complexity, Logic, and Recursion Theory, pages 331-363. CRC Press, 2019. URL: https://doi.org/10.1201/9780429187490.
  51. Dmitriy 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, 2017, pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
  52. 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.