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Smooth Approximations and Relational Width Collapses

Authors Antoine Mottet , Tomáš Nagy , Michael Pinsker , Michał Wrona



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Author Details

Antoine Mottet
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Tomáš Nagy
  • Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Michael Pinsker
  • Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Michał Wrona
  • Theoretical Computer Science Department, Jagiellonian University, Kraków, Poland

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Antoine Mottet, Tomáš Nagy, Michael Pinsker, and Michał Wrona. Smooth Approximations and Relational Width Collapses. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 138:1-138:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.138

Abstract

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ≥ 3) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al.. In particular, the bounded width hierarchy collapses in those cases as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • local consistency
  • bounded width
  • constraint satisfaction problems
  • polymorphisms
  • smooth approximations

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References

  1. Samson Abramsky, Anuj Dawar, and Pengming Wang. The pebbling comonad in finite model theory. 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. URL: https://doi.org/10.1109/LICS.2017.8005129.
  2. Albert Atserias, Andrei A. Bulatov, and Víctor Dalmau. On the power of k -consistency. In Lars Arge, Christian Cachin, Tomasz Jurdzinski, and Andrzej Tarlecki, editors, Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Proceedings, volume 4596 of Lecture Notes in Computer Science, pages 279-290. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-73420-8_26.
  3. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 684-697. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897542.
  4. Libor Barto. The collapse of the bounded width hierarchy. J. Log. Comput., 26(3):923-943, 2016. URL: https://doi.org/10.1093/logcom/exu070.
  5. Libor Barto and Marcin Kozik. Congruence distributivity implies bounded width. SIAM J. Comput., 39(4):1531-1542, 2009. URL: https://doi.org/10.1137/080743238.
  6. Libor Barto and Marcin Kozik. Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Log. Methods Comput. Sci., 8(1), 2012. URL: https://doi.org/10.2168/LMCS-8(1:7)2012.
  7. Libor Barto and Marcin Kozik. Constraint satisfaction problems solvable by local consistency methods. J. ACM, 61(1):3:1-3:19, 2014. URL: https://doi.org/10.1145/2556646.
  8. Libor Barto, Jakub Opršal, and Michael Pinsker. The wonderland of reflections. Israel Journal of Mathematics, 223(1):363-398, 2018. Google Scholar
  9. Libor Barto and Michael Pinsker. The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems. In Proceedings of the 31th Annual IEEE Symposium on Logic in Computer Science - LICS'16, pages 615-622, 2016. Preprint URL: https://arxiv.org/abs/1602.04353.
  10. Libor Barto and Michael Pinsker. Topology is irrelevant. SIAM Journal on Computing, 49(2):365-393, 2020. Google Scholar
  11. 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.
  12. Manuel Bodirsky. Ramsey classes: Examples and constructions. In Surveys in Combinatorics. London Mathematical Society Lecture Note Series 424. Cambridge University Press, 2015. Invited survey article for the British Combinatorial Conference; URL: https://arxiv.org(abs/1502.05146.
  13. Manuel Bodirsky and Víctor Dalmau. Datalog and constraint satisfaction with infinite templates. J. Comput. Syst. Sci., 79(1):79-100, 2013. URL: https://doi.org/10.1016/j.jcss.2012.05.012.
  14. Manuel Bodirsky and Jan Kára. The complexity of equality constraint languages. Theory of Computing Systems, 3(2):136-158, 2008. A conference version appeared in the proceedings of Computer Science Russia (CSR'06). Google Scholar
  15. Manuel Bodirsky, Florent R. Madelaine, and Antoine Mottet. A universal-algebraic proof of the complexity dichotomy for monotone monadic SNP. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 105-114. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209156.
  16. Manuel Bodirsky, Barnaby Martin, and Antoine Mottet. Discrete temporal constraint satisfaction problems. J. ACM, 65(2):9:1-9:41, 2018. URL: https://doi.org/10.1145/3154832.
  17. Manuel Bodirsky, Barnaby Martin, Michael Pinsker, and András Pongrácz. Constraint satisfaction problems for reducts of homogeneous graphs. SIAM J. Comput., 48(4):1224-1264, 2019. A conference version appeared in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, pages 119:1-119:14. URL: https://doi.org/10.1137/16M1082974.
  18. Manuel Bodirsky and Antoine Mottet. A dichotomy for first-order reducts of unary structures. Log. Methods Comput. Sci., 14(2), 2018. URL: https://doi.org/10.23638/LMCS-14(2:13)2018.
  19. Manuel Bodirsky, Antoine Mottet, Miroslav Olsák, Jakub Oprsal, Michael Pinsker, and Ross Willard. Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems). In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1-12. IEEE, 2019. URL: https://doi.org/10.1109/LICS.2019.8785883.
  20. Manuel Bodirsky, Wied Pakusa, and Jakub Rydval. Temporal constraint satisfaction problems in fixed-point logic. In Holger Hermanns, Lijun Zhang, Naoki Kobayashi, and Dale Miller, editors, LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8-11, 2020, pages 237-251. ACM, 2020. URL: https://doi.org/10.1145/3373718.3394750.
  21. Manuel Bodirsky and Michael Pinsker. Canonical Functions: a Proof via Topological Dynamics. To appear. Preprint URL: https://arxiv.org/abs/1610.09660.
  22. Manuel Bodirsky and Michael Pinsker. Schaefer’s theorem for graphs. J. ACM, 62(3):19:1-19:52, 2015. URL: https://doi.org/10.1145/2764899.
  23. Manuel Bodirsky and Michael Pinsker. Topological Birkhoff. Transactions of the American Mathematical Society, 367:2527-2549, 2015. Google Scholar
  24. Manuel Bodirsky, Michael Pinsker, and András Pongrácz. Projective clone homomorphisms. Journal of Symbolic Logic, 2019. URL: https://doi.org/10.1017/jsl.2019.23.
  25. Manuel Bodirsky, Michael Pinsker, and Todor Tsankov. Decidability of definability. Journal of Symbolic Logic, 78(4):1036-1054, 2013. A conference version appeared in the Proceedings of the Twenty-Sixth Annual IEEE Symposium on Logic in Computer Science (LICS) 2011, pages 321-328. Google Scholar
  26. Manuel Bodirsky and MichałWrona. Equivalence constraint satisfaction problems. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL, CSL 2012, September 3-6, 2012, Fontainebleau, France, pages 122-136, 2012. URL: https://doi.org/10.4230/LIPIcs.CSL.2012.122.
  27. Pierre Bourhis and Carsten Lutz. Containment in monadic disjunctive datalog, mmsnp, and expressive description logics. In Chitta Baral, James P. Delgrande, and Frank Wolter, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifteenth International Conference, KR 2016, Cape Town, South Africa, April 25-29, 2016, pages 207-216. AAAI Press, 2016. URL: http://www.aaai.org/ocs/index.php/KR/KR16/paper/view/12847.
  28. Hans-Jürgen Bückert and Bernhard Nebel. Reasoning about temporal relations: a max-imal tractable subclass of allen’s interval algebra. J. ACM, 42(1):43-66, 1995. Google Scholar
  29. Andrei A. Bulatov. Bounded relational width, 2009. URL: https://www2.cs.sfu.ca/~abulatov/papers/relwidth.pdf.
  30. 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.
  31. Hubie Chen and Benoît Larose. Asking the metaquestions in constraint tractability. ACM Trans. Comput. Theory, 9(3):11:1-11:27, 2017. URL: https://doi.org/10.1145/3134757.
  32. Víctor Dalmau and Justin Pearson. Closure functions and width 1 problems. In Proceedings of the International Conference on Principles and Practice of Constraint Programming (CP), pages 159-173, 1999. Google Scholar
  33. 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.
  34. Cristina Feier, Antti Kuusisto, and Carsten Lutz. Rewritability in monadic disjunctive datalog, mmsnp, and expressive description logics. Log. Methods Comput. Sci., 15(2), 2019. URL: https://doi.org/10.23638/LMCS-15(2:15)2019.
  35. Pierre Gillibert, Julius Jonusas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker. Hrushovski’s encoding and ω-categorical CSP monsters. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 131:1-131:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.131.
  36. Martin Grohe. Fixed-point definability and polynomial time on graphs with excluded minors. J. ACM, 59(5):27:1-27:64, 2012. URL: https://doi.org/10.1145/2371656.2371662.
  37. Alexander Kechris, Vladimir Pestov, and Stevo Todorčević. Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geometric and Functional Analysis, 15(1):106-189, 2005. Google Scholar
  38. Phokion G. Kolaitis and Moshe Y. Vardi. Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci., 61(2):302-332, 2000. URL: https://doi.org/10.1006/jcss.2000.1713.
  39. Michael Kompatscher and Trung Van Pham. A complexity dichotomy for poset constraint satisfaction. FLAP, 5(8):1663-1696, 2018. URL: https://www.collegepublications.co.uk/downloads/ifcolog00028.pdf.
  40. Marcin Kozik, Andrei Krokhin, Matt Valeriote, and Ross Willard. Characterizations of several Maltsev conditions. Algebra universalis, 73(3-4):205-224, 2015. URL: https://doi.org/10.1007/s00012-015-0327-2.
  41. Benoit Larose and László Zádori. Bounded width problems and algebras. Algebra Universalis, 56(3-4):439-466, 2007. Google Scholar
  42. Florent R. Madelaine and Iain A. Stewart. Constraint satisfaction, logic and forbidden patterns. SIAM J. Comput., 37(1):132-163, 2007. URL: https://doi.org/10.1137/050634840.
  43. Miklós Maróti and Ralph McKenzie. Existence theorems for weakly symmetric operations. Algebra Universalis, 59(3), 2008. Google Scholar
  44. Antoine Mottet, Tomáš Nagy, Michael Pinsker, and Michał Wrona. Smooth approximations and relational width collapses, 2021. URL: http://arxiv.org/abs/2102.07531.
  45. Antoine Mottet and Michael Pinsker. Smooth approximations and CSPs over finitely bounded homogeneous structures. CoRR, abs/2011.03978, 2020. URL: http://arxiv.org/abs/2011.03978.
  46. Michael Pinsker, Pierre Gillibert, and Julius Jonušas. Pseudo-loop conditions. Bulletin of the London Mathematical Society, 51(5):917-936, 2019. Google Scholar
  47. Matthew A. Valeriote. A subalgebra intersection property for congruence distributive varieties. Canadian Journal of Mathematics, 61(2):451–464, 2009. URL: https://doi.org/10.4153/CJM-2009-023-2.
  48. MichałWrona. On the relational width of first-order expansions of finitely bounded homogeneous binary cores with bounded strict width. In Holger Hermanns, Lijun Zhang, Naoki Kobayashi, and Dale Miller, editors, LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8-11, 2020, pages 958-971. ACM, 2020. URL: https://doi.org/10.1145/3373718.3394781.
  49. MichałWrona. Relational width of first-order expansions of homogeneous graphs with bounded strict width. In Christophe Paul and Markus Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, March 10-13, 2020, Montpellier, France, volume 154 of LIPIcs, pages 39:1-39:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.39.
  50. 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.
  51. 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.
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