Document Open Access Logo

Computability of Extender Sets in Multidimensional Subshifts

Authors Antonin Callard , Léo Paviet Salomon , Pascal Vanier

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2025.21.pdf
  • Filesize: 1.04 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Models of computation
Keywords
  • Symbolic dynamics
  • subshifts
  • extender sets
  • extender entropy
  • computability
  • sofic shifts
  • tilings

Metrics

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

Abstract

Subshifts are sets of colorings of ℤ^d defined by families of forbidden patterns. Given a subshift and a finite pattern, its extender set is the set of admissible completions of this pattern. It has been conjectured that the behavior of extender sets, and in particular their growth called extender entropy [French and Pavlov, 2019], could provide a way to separate the classes of sofic and effective subshifts. We prove here that both classes have the same possible extender entropies: exactly the Π₃ real numbers of [0,+∞).

Cite As Get BibTex

Antonin Callard, Léo Paviet Salomon, and Pascal Vanier. Computability of Extender Sets in Multidimensional Subshifts. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 21:1-21:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.21

Author Details

Antonin Callard
  • Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000, Caen, France
Léo Paviet Salomon
  • Université de Lorraine, CNRS, Inria, LORIA, 54000, Nancy, France
Pascal Vanier
  • Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000, Caen, France

Funding

This research was partially funded by ANR JCJC 2019 19-CE48-0007-01.

Acknowledgements

We are thankful to the referees for their many helpful remarks and suggestions.

References

  1. Nathalie Aubrun, Sebastián Barbieri, and Emmanuel Jeandel. About the Domino Problem for Subshifts on Groups, pages 331-389. Springer International Publishing, Cham, 2018. URL: https://doi.org/10.1007/978-3-319-69152-7_9.
  2. Nathalie Aubrun and Mathieu Sablik. Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta applicandae mathematicae, 126:35-63, 2013. URL: https://doi.org/10.1007/s10440-013-9808-5.
  3. Sebastián Barbieri, Mathieu Sablik, and Ville Salo. Soficity of free extensions of effective subshifts. Discrete and Continuous Dynamical Systems, 45(4):1117-1149, 2025. URL: https://doi.org/10.3934/dcds.2024125.
  4. Jérôme Buzzi. Subshifts of quasi-finite type. Inventiones mathematicae, 2003. URL: https://doi.org/10.1007/s00222-004-0392-1.
  5. Silvio Capobianco. Multidimensional cellular automata and generalization of Fekete’s lemma. Discrete Mathematics & Theoretical Computer Science (DMTCS), 10(3):95-104, 2008. URL: https://doi.org/10.46298/dmtcs.442.
  6. Julien Destombes. Algorithmic Complexity and Soficness of Shifts in Dimension Two. PhD thesis, Université de Montpellier, 2023. https://arxiv.org/abs/2309.12241, URL: https://doi.org/10.48550/arXiv.2309.12241.
  7. Julien Destombes and Andrei Romashchenko. Resource-bounded Kolmogorov complexity provides an obstacle to soficness of multidimensional shifts. Journal of Computer and System Sciences, 128:107-134, 2022. URL: https://doi.org/10.1016/j.jcss.2022.04.002.
  8. Bruno Durand, Andrei Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. Journal of Computer and System Sciences, 78(3):731-764, 2012. URL: https://doi.org/10.1016/j.jcss.2011.11.001.
  9. Thomas French. Characterizing follower and extender set sequences. Dynamical Systems, 31(3):293-310, 2016. URL: https://doi.org/10.1080/14689367.2015.1111865.
  10. Thomas French and Ronnie Pavlov. Follower, predecessor, and extender entropies. Monatshefte für Mathematik, 188:495-510, 2019. URL: https://doi.org/10.1007/s00605-018-1224-5.
  11. Silvère Gangloff. Algorithmic complexity of growth-type invariants of SFT under dynamical constraints. PhD thesis, Aix-Marseille Université, 2018. URL: http://www.theses.fr/2018AIXM0231/document.
  12. Pierre Guillon and Emmanuel Jeandel. Infinite communication complexity, 2015. URL: https://arxiv.org/abs/1501.05814.
  13. Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Inventiones Mathematicae, 176(1):2009, April 2009. URL: https://doi.org/10.1007/s00222-008-0161-7.
  14. Michael Hochman and Tom Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011-2038, 2010. URL: https://doi.org/10.4007/annals.2010.171.2011.
  15. Steve Kass and Kathleen Madden. A sufficient condition for non-soficness of higher-dimensional subshifts. Proceedings of the American Mathematical Society, 141(11):3803-3816, 2013. URL: https://doi.org/10.1090/S0002-9939-2013-11646-1.
  16. Petr Kůrka. Topological and Symbolic Dynamics. Société mathématique de France, 2003. Google Scholar
  17. Douglas Lind and Brian Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge university press, 2021. Google Scholar
  18. Tom Meyerovitch. Growth-type invariants for ℤ^d subshifts of finite type and arithmetical classes of real numbers. Inventiones Mathematicae, 184(3), 2010. URL: https://doi.org/10.1007/s00222-010-0296-1.
  19. Harold Marston Morse and Gustav Arnold Hedlund. Symbolic Dynamics. American Journal of Mathematics, 60(4):815-866, October 1938. URL: https://doi.org/10.2307/2371264.
  20. Shahar Mozes. Tilings, substitution systems and dynamical systems generated by them. Journal d'Analyse Mathématique, 53(1):139-186, 1989. URL: https://doi.org/10.1007/bf02793412.
  21. Nic Ormes and Ronnie Pavlov. Extender sets and multidimensional subshifts. Ergodic Theory and Dynamical Systems, 36(3):908-923, 2016. URL: https://doi.org/10.1017/etds.2014.71.
  22. Ronnie Pavlov. A class of nonsofic multidimensional shift spaces. Proceedings of the American Mathematical Society, 141(3):987-996, 2013. URL: https://doi.org/10.1090/S0002-9939-2012-11382-6.
  23. Anthony N. Quas and Paul B. Trow. Subshifts of multi-dimensional shifts of finite type. Ergodic Theory and Dynamical Systems, 20(3):859-874, 2000. URL: https://doi.org/10.1017/S0143385700000468.
  24. Robert I Soare. Turing computability: Theory and applications, volume 300. Springer, 2016. URL: https://doi.org/10.1007/978-3-642-31933-4.
  25. Hao Wang. Proving theorems by Pattern Recognition II. Bell Systems technical journal, 40:1-41, 1961. URL: https://doi.org/10.1002/j.1538-7305.1961.tb03975.x.
  26. Benjamin Weiss. Subshifts of finite type and sofic systems. Monatshefte für Mathematik, 77:462-474, 1973. URL: https://doi.org/10.1007/BF01295322.
  27. Xizhong Zheng and Klaus Weihrauch. The arithmetical hierarchy of real numbers. Mathematical Logic Quarterly: Mathematical Logic Quarterly, 47(1):51-65, 2001. URL: https://doi.org/10.1002/1521-3870(200101)47:1<51::AID-MALQ51>3.0.CO;2-W.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact