Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability

Author Nicolás Bitar

Thumbnail PDF


  • Filesize: 0.78 MB
  • 18 pages

Document Identifiers

Author Details

Nicolás Bitar
  • Université Paris-Saclay, CNRS, LISN, 91190 Gif-sur-Yvette, France


I want to thank Nathalie Aubrun for her helpful comments, reviews and discussions.

Cite AsGet BibTex

Nicolás Bitar. Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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 ℤ² 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Tilings
  • Domino problem
  • SAT
  • Computability
  • Finitely generated groups


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


  1. Leonard Adleman, Jarkko Kari, Lila Kari, and Dustin Reishus. The undecidability of the infinite ribbon problem: implications for computing by self-assembly. SIAM J. Comput., 38(6):2356-2381, 2009. URL:
  2. Nathalie Aubrun, Sebastián Barbieri, and Emmanuel Jeandel. About the domino problem for subshifts on groups. In Sequences, groups, and number theory, Trends Math., pages 331-389. Birkhäuser/Springer, Cham, 2018. URL:
  3. Nathalie Aubrun and Nicolás Bitar. Domino snake problems on groups. In International Symposium on Fundamentals of Computation Theory, pages 46-59. Springer Nature Switzerland, 2023. URL:
  4. Nathalie Aubrun, Nicolás Bitar, and Sacha Huriot-Tattegrain. Strongly aperiodic SFTs on generalized Baumslag–Solitar groups. Ergodic Theory and Dynamical Systems, pages 1-30, 2023. URL:
  5. Nathalie Aubrun and Jarkko Kari. 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. URL:
  6. Alexis Ballier and Emmanuel Jeandel. 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. Google Scholar
  7. Alexis Ballier and Maya Stein. The domino problem on groups of polynomial growth. Groups Geom. Dyn., 12(1):93-105, 2018. URL:
  8. Laurent Bartholdi. Monadic second-order logic and the domino problem on self-similar graphs. Groups Geom. Dyn., 16(4):1423-1459, 2022. URL:
  9. Laurent Bartholdi. The domino problem for hyperbolic groups. arXiv preprint arXiv:2305.06952, 2023. Google Scholar
  10. Robert Berger. The undecidability of the domino problem. Mem. Amer. Math. Soc., 66:72, 1966. Google Scholar
  11. J. Richard Büchi. Turing-machines and the Entscheidungsproblem. Math. Ann., 148:201-213, 1962. URL:
  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. Google Scholar
  13. Laura Ciobanu, Alex Levine, and Alan D Logan. Post’s correspondence problem for hyperbolic and virtually nilpotent groups. arXiv preprint arXiv:2211.12158, 2022. Google Scholar
  14. Laura Ciobanu and Alan D Logan. Variations on the Post correspondence problem for free groups. In International Conference on Developments in Language Theory, pages 90-102. Springer, 2021. Google Scholar
  15. Toby Cubitt, David Perez-Garcia, and Michael M. Wolf. Undecidability of the spectral gap. Forum Math. Pi, 10:Paper No. e14, 102, 2022. URL:
  16. Michael H. Freedman. Limit, logic, and computation. Proc. Natl. Acad. Sci. USA, 95(1):95-97, 1998. URL:
  17. Michael H. Freedman. k-SAT on groups and undecidability. In STOC '98 (Dallas, TX), pages 572-576. ACM, New York, 1999. Google Scholar
  18. Anael Grandjean, Benjamin Hellouin de Menibus, and Pascal Vanier. Aperiodic point in ℤ²-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. Google Scholar
  19. Rachel Greenfeld and Terence Tao. Undecidability of translational monotilings. arXiv preprint arXiv:2309.09504, 2023. Google Scholar
  20. Ju. Š. Gurevič and I. O. Korjakov. A remark on R. Berger’s work on the domino problem. Sibirsk. Mat. Ž., 13:459-463, 1972. Google Scholar
  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. Google Scholar
  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. URL:
  23. Benjamin Hellouin de Menibus, Victor H. Lutfalla, and Camille Noûs. 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. URL:
  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. URL:
  25. Emmanuel Jeandel. The periodic domino problem revisited. Theoret. Comput. Sci., 411(44-46):4010-4016, 2010. URL:
  26. Emmanuel Jeandel. Aperiodic subshifts on polycyclic groups. arXiv preprint arXiv:1510.02360, 2015. Google Scholar
  27. Emmanuel Jeandel. Translation-like actions and aperiodic subshifts on groups. arXiv preprint arXiv:1508.06419, 2015. Google Scholar
  28. Emmanuel Jeandel and Pascal Vanier. 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. URL:
  29. A. S. Kahr, Edward F. Moore, and Hao Wang. Entscheidungsproblem reduced to the ∀ ∃ ∀ case. Proc. Nat. Acad. Sci. U.S.A., 48:365-377, 1962. URL:
  30. Jarkko Kari. Reversibility of 2d cellular automata is undecidable. Physica D: Nonlinear Phenomena, 45(1-3):379-385, 1990. URL:
  31. Jarkko Kari. Reversibility and surjectivity problems of cellular automata. J. Comput. System Sci., 48(1):149-182, 1994. URL:
  32. Dietrich Kuske and Markus Lohrey. Logical aspects of Cayley-graphs: the group case. Ann. Pure Appl. Logic, 131(1-3):263-286, 2005. URL:
  33. Douglas Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2021. URL:
  34. Markus Lohrey. Subgroup membership in GL(2,Z). Theory of Computing Systems, pages 1-26, 2023. Google Scholar
  35. Roger C. Lyndon and Paul E. Schupp. 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. Google Scholar
  36. K. A. Mihaĭlova. The occurrence problem for free products of groups. Mat. Sb. (N.S.), 75(117):199-210, 1968. Google Scholar
  37. David E. Muller and Paul E. Schupp. The theory of ends, pushdown automata, and second-order logic. Theoret. Comput. Sci., 37(1):51-75, 1985. URL:
  38. Alexei Myasnikov, Andrey Nikolaev, and Alexander Ushakov. The Post correspondence problem in groups. Journal of Group Theory, 17(6):991-1008, 2014. Google Scholar
  39. D Myers. Decidability of the tiling connectivity problem. abstract 79t-e42. Notices Amer. Math. Soc, 195(26):177-209, 1979. Google Scholar
  40. Volodymyr Nekrashevych and Gábor Pete. Scale-invariant groups. Groups Geom. Dyn., 5(1):139-167, 2011. URL:
  41. Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete Contin. Dyn. Syst., 20(3):725-738, 2008. URL:
  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. Google Scholar
  43. E. Rips. Subgroups of small cancellation groups. Bull. London Math. Soc., 14(1):45-47, 1982. URL:
  44. Neil Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B, 41(1):92-114, 1986. URL:
  45. Hartley Rogers, Jr. Theory of recursive functions and effective computability. McGraw-Hill Book Co., New York-Toronto-London, 1967. Google Scholar
  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. Google Scholar
  47. Hao Wang. Proving theorems by pattern recognition—II. Bell system technical journal, 40(1):1-41, 1961. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail