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Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability (2312.08911v1)

Published 14 Dec 2023 in math.CO, cs.DM, math.DS, and math.GR

Abstract: We study the seeded domino problem, the recurring domino problem and the $k$-SAT problem on finitely generated groups. These problems are generalization of their original versions on $\mathbb{Z}2$ that were shown to be undecidable using the domino problem. We show that the seeded and recurring domino problems on a group are invariant under changes in the generating set, are many-one reduced from the respective problems on subgroups, and are positive equivalent to the problems on finite index subgroups. This leads to showing that the recurring domino problem is decidable for free groups. Coupled with the invariance properties, we conjecture that the only groups in which the seeded and recurring domino problems are decidable are virtually free groups. In the case of the $k$-SAT problem, we introduce a new generalization that is compatible with decision problems on finitely generated groups. We show that the subgroup membership problem many-one reduces to the $2$-SAT problem, that in certain cases the $k$-SAT problem many one reduces to the domino problem, and finally that the domino problem reduces to $3$-SAT for the class of scalable groups.

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References (47)
  1. The undecidability of the infinite ribbon problem: implications for computing by self-assembly. SIAM J. Comput., 38(6):2356–2381, 2009. doi:10.1137/080723971.
  2. About the domino problem for subshifts on groups. In Sequences, groups, and number theory, Trends Math., pages 331–389. Birkhäuser/Springer, Cham, 2018. doi:10.1007/978-3-319-69152-7_9.
  3. Domino snake problems on groups. In International Symposium on Fundamentals of Computation Theory, pages 46–59. Springer Nature Switzerland, 2023. doi:10.1007/978-3-031-43587-4_4.
  4. Strongly aperiodic SFTs on generalized Baumslag–Solitar groups. Ergodic Theory and Dynamical Systems, page 1–30, 2023. doi:10.1017/etds.2023.44.
  5. Tiling problems on Baumslag-Solitar groups. In Proceedings: Machines, Computations and Universality 2013, volume 128 of Electron. Proc. Theor. Comput. Sci. (EPTCS), pages 35–46. EPTCS, 2013. doi:10.4204/EPTCS.128.12.
  6. Tilings and model theory. In First Symposium on Cellular Automata "Journées Automates Cellulaires" (JAC 2008), Uzès, France, April 21-25, 2008. Proceedings, pages 29–39. MCCME Publishing House, Moscow, 2008.
  7. The domino problem on groups of polynomial growth. Groups Geom. Dyn., 12(1):93–105, 2018. doi:10.4171/GGD/439.
  8. Laurent Bartholdi. Monadic second-order logic and the domino problem on self-similar graphs. Groups Geom. Dyn., 16(4):1423–1459, 2022. doi:10.4171/ggd/689.
  9. Laurent Bartholdi. The domino problem for hyperbolic groups. arXiv preprint arXiv:2305.06952, 2023.
  10. Robert Berger. The undecidability of the domino problem. Mem. Amer. Math. Soc., 66:72, 1966.
  11. J. Richard Büchi. Turing-machines and the Entscheidungsproblem. Math. Ann., 148:201–213, 1962. doi:10.1007/BF01470748.
  12. Antonin Callard and Benjamin Hellouin de Menibus. The aperiodic Domino problem in higher dimension. In 39th International Symposium on Theoretical Aspects of Computer Science, volume 219 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 19, 15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2022.
  13. Post’s correspondence problem for hyperbolic and virtually nilpotent groups. arXiv preprint arXiv:2211.12158, 2022.
  14. Variations on the Post correspondence problem for free groups. In International Conference on Developments in Language Theory, pages 90–102. Springer, 2021.
  15. Undecidability of the spectral gap. Forum Math. Pi, 10:Paper No. e14, 102, 2022. doi:10.1017/fmp.2021.15.
  16. Michael H. Freedman. Limit, logic, and computation. Proc. Natl. Acad. Sci. USA, 95(1):95–97, 1998. doi:10.1073/pnas.95.1.95.
  17. Michael H. Freedman. k𝑘kitalic_k-SAT on groups and undecidability. In STOC ’98 (Dallas, TX), pages 572–576. ACM, New York, 1999.
  18. Aperiodic point in ℤ2superscriptℤ2\mathbb{Z}^{2}blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-subshifts. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 128, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018.
  19. Undecidability of translational monotilings. arXiv preprint arXiv:2309.09504, 2023.
  20. A remark on R. Berger’s work on the domino problem. Sibirsk. Mat. Ž., 13:459–463, 1972.
  21. David Harel. Recurring dominoes: making the highly undecidable highly understandable. In Topics in the theory of computation (Borgholm, 1983), volume 102 of North-Holland Math. Stud., pages 51–71. North-Holland, Amsterdam, 1985.
  22. David Harel. Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness. J. Assoc. Comput. Mach., 33(1):224–248, 1986. doi:10.1145/4904.4993.
  23. The Domino problem is undecidable on every rhombus subshift. In Developments in language theory, volume 13911 of Lecture Notes in Comput. Sci., pages 100–112. Springer, Cham, [2023] ©2023. doi:10.1007/978-3-031-33264-7_9.
  24. Benjamin Hellouin de Menibus and Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete Contin. Dyn. Syst., 40(4):2335–2346, 2020. doi:10.3934/dcds.2020116.
  25. Emmanuel Jeandel. The periodic domino problem revisited. Theoret. Comput. Sci., 411(44-46):4010–4016, 2010. doi:10.1016/j.tcs.2010.08.017.
  26. Emmanuel Jeandel. Aperiodic subshifts on polycyclic groups. arXiv preprint arXiv:1510.02360, 2015.
  27. Emmanuel Jeandel. Translation-like actions and aperiodic subshifts on groups. arXiv preprint arXiv:1508.06419, 2015.
  28. The undecidability of the Domino Problem. In Substitution and tiling dynamics: introduction to self-inducing structures, volume 2273 of Lecture Notes in Math., pages 293–357. Springer, Cham, [2020] ©2020. doi:10.1007/978-3-030-57666-0_6.
  29. Entscheidungsproblem reduced to the ∀∃∀for-allfor-all\forall\exists\forall∀ ∃ ∀ case. Proc. Nat. Acad. Sci. U.S.A., 48:365–377, 1962. doi:10.1073/pnas.48.3.365.
  30. Jarkko Kari. Reversibility of 2222d cellular automata is undecidable. volume 45, pages 379–385. 1990. Cellular automata: theory and experiment (Los Alamos, NM, 1989). doi:10.1016/0167-2789(90)90195-U.
  31. Jarkko Kari. Reversibility and surjectivity problems of cellular automata. J. Comput. System Sci., 48(1):149–182, 1994. doi:10.1016/S0022-0000(05)80025-X.
  32. Logical aspects of Cayley-graphs: the group case. Ann. Pure Appl. Logic, 131(1-3):263–286, 2005. doi:10.1016/j.apal.2004.06.002.
  33. An introduction to symbolic dynamics and coding. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2021. doi:10.1017/9781108899727.
  34. Markus Lohrey. Subgroup membership in GL(2,Z). Theory of Computing Systems, pages 1–26, 2023.
  35. Combinatorial group theory, volume Band 89 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin-New York, 1977.
  36. K. A. Mihaĭlova. The occurrence problem for free products of groups. Mat. Sb. (N.S.), 75(117):199–210, 1968.
  37. The theory of ends, pushdown automata, and second-order logic. Theoret. Comput. Sci., 37(1):51–75, 1985. doi:10.1016/0304-3975(85)90087-8.
  38. The Post correspondence problem in groups. Journal of Group Theory, 17(6):991–1008, 2014.
  39. D Myers. Decidability of the tiling connectivity problem. abstract 79t-e42. Notices Amer. Math. Soc, 195(26):177–209, 1979.
  40. Scale-invariant groups. Groups Geom. Dyn., 5(1):139–167, 2011. doi:10.4171/GGD/119.
  41. Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete Contin. Dyn. Syst., 20(3):725–738, 2008. doi:10.3934/dcds.2008.20.725.
  42. Yo’av Rieck. Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them. arXiv preprint arXiv:2202.00212, 2022.
  43. E. Rips. Subgroups of small cancellation groups. Bull. London Math. Soc., 14(1):45–47, 1982. doi:10.1112/blms/14.1.45.
  44. Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B, 41(1):92–114, 1986. doi:10.1016/0095-8956(86)90030-4.
  45. Hartley Rogers, Jr. Theory of recursive functions and effective computability. McGraw-Hill Book Co., New York-Toronto-London, 1967.
  46. Peter van Emde Boas. The convenience of tilings. In Complexity, logic, and recursion theory, volume 187 of Lecture Notes in Pure and Appl. Math., pages 331–363. Dekker, New York, 1997.
  47. Hao Wang. Proving theorems by pattern recognition—II. Bell system technical journal, 40(1):1–41, 1961.

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