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Link to original content: https://doi.org/10.4230/LIPIcs.MFCS.2024.47
Preservation Theorems on Sparse Classes Revisited

Preservation Theorems on Sparse Classes Revisited

Authors Anuj Dawar , Ioannis Eleftheriadis



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Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Ioannis Eleftheriadis
  • Department of Computer Science and Technology, University of Cambridge, UK

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Anuj Dawar and Ioannis Eleftheriadis. Preservation Theorems on Sparse Classes Revisited. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.47

Abstract

We revisit the work studying homomorphism preservation for first-order logic in sparse classes of structures initiated in [Atserias et al., JACM 2006] and [Dawar, JCSS 2010]. These established that first-order logic has the homomorphism preservation property in any sparse class that is monotone and addable. It turns out that the assumption of addability is not strong enough for the proofs given. We demonstrate this by constructing classes of graphs of bounded treewidth which are monotone and addable but fail to have homomorphism preservation. We also show that homomorphism preservation fails on the class of planar graphs. On the other hand, the proofs of homomorphism preservation can be recovered by replacing addability by a stronger condition of amalgamation over bottlenecks. This is analogous to a similar condition formulated for extension preservation in [Atserias et al., SiCOMP 2008].

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • Homomorphism preservation
  • sparsity
  • finite model theory
  • planar graphs

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References

  1. Miklos Ajtai and Yuri Gurevich. Datalog vs first-order logic. Journal of Computer and System Sciences, 49(3):562-588, 1994. 30th IEEE Conference on Foundations of Computer Science. URL: https://doi.org/10.1016/S0022-0000(05)80071-6.
  2. Albert Atserias, Anuj Dawar, and Martin Grohe. Preservation under extensions on well-behaved finite structures. SIAM Journal on Computing, 38(4):1364-1381, 2008. URL: https://doi.org/10.1137/060658709.
  3. Albert Atserias, Anuj Dawar, and Phokion G Kolaitis. On preservation under homomorphisms and unions of conjunctive queries. Journal of the ACM (JACM), 53(2):208-237, 2006. Google Scholar
  4. Gary Chartrand and Frank Harary. Planar Permutation Graphs. Annales de l'institut Henri Poincaré. Section B. Calcul des probabilités et statistiques, 3(4):433-438, 1967. URL: http://www.numdam.org/item/AIHPB_1967__3_4_433_0/.
  5. Peter Damaschke. Induced subgraphs and well-quasi-ordering. Journal of Graph Theory, 14(4):427-435, 1990. Google Scholar
  6. Anuj Dawar. Finite model theory on tame classes of structures. In International Symposium on Mathematical Foundations of Computer Science, pages 2-12. Springer, 2007. Google Scholar
  7. Anuj Dawar. Homomorphism preservation on quasi-wide classes. Journal of Computer and System Sciences, 76(5):324-332, 2010. Google Scholar
  8. Jan Dreier, Ioannis Eleftheriadis, Nikolas Mählmann, Rose McCarty, Michał Pilipczuk, and Szymon Toruńczyk. First-order model checking on monadically stable graph classes. arXiv preprint, 2023. Accepted to FOCS 2024. URL: https://arxiv.org/abs/2311.18740.
  9. Jan Dreier, Nikolas Mählmann, and Szymon Toruńczyk. Flip-breakability: A combinatorial dichotomy for monadically dependent graph classes. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1550-1560, 2024. Google Scholar
  10. H-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 2nd edition, 1999. Google Scholar
  11. Robert Ganian, Petr Hliněnỳ, Jaroslav Nešetřil, Jan Obdržálek, and Patrice Ossona De Mendez. Shrub-depth: Capturing height of dense graphs. Logical Methods in Computer Science, 15, 2019. Google Scholar
  12. Yuri Gurevich. Toward logic tailored for computational complexity. Computation and Proof Theory, pages 175-216, 1984. Google Scholar
  13. Jaroslav Nešetřil and Patrice Ossona De Mendez. Sparsity: graphs, structures, and algorithms, volume 28. Springer Science & Business Media, 2012. Google Scholar
  14. Benjamin Rossman. Homomorphism preservation theorems. Journal of the ACM (JACM), 55(3):1-53, 2008. Google Scholar
  15. W. W. Tait. A counterexample to a conjecture of Scott and Suppes. Journal of Symbolic Logic, 24(1):15-16, 1959. URL: https://doi.org/10.2307/2964569.
  16. Klaus Wagner. Über eine eigenschaft der ebenen komplexe. Mathematische Annalen, 114(1):570-590, 1937. Google Scholar
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