Document Open Access Logo

Minimality and Computability of Languages of G-Shifts

Authors Djamel Eddine Amir , Benjamin Hellouin de Menibus

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.139.pdf
  • Filesize: 0.8 MB
  • 19 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Point-set topology
Keywords
  • shifts
  • subshifts
  • minimal shifts
  • computable language
  • computability
  • strong computable type
  • descriptive complexity

Metrics

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

Abstract

Motivated by the notion of strong computable type for sets in computable analysis, we define the notion of strong computable type for G-shifts, where G is a finitely generated group with decidable word problem. A G-shift has strong computable type if one can compute its language from the complement of its language. We obtain a characterization of G-shifts with strong computable type in terms of a notion of minimality with respect to properties with a bounded computational complexity. We provide a self-contained direct proof, and also explain how this characterization can be obtained from an existing similar characterization for sets by Amir and Hoyrup, and discuss its connexions with results by Jeandel on closure spaces. We apply this characterization to several classes of shifts that are minimal with respect to specific properties. This provides a unifying approach that not only generalizes many existing results but also has the potential to yield new findings effortlessly. In contrast to the case of sets, we prove that strong computable type for G-shifts is preserved under products. We conclude by discussing some generalizations and future directions.

Cite As Get BibTex

Djamel Eddine Amir and Benjamin Hellouin de Menibus. Minimality and Computability of Languages of G-Shifts. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 139:1-139:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.139

Author Details

Djamel Eddine Amir
  • Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France
Benjamin Hellouin de Menibus
  • Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France

Funding

We gratefully acknowledge funding from the ANR Project IZES-ANR-22-CE40-0011 (Inside Zero Entropy Systems) for this work.

Acknowledgements

The authors are grateful to anonymous reviewers, as well as to Pierre Guillon and Guillaume Theyssier for discussions and numerous remarks that helped to significantly improve this article. We particularly thank Pierre Guillon for pointing out links with results from [Jeandel, 2017].

References

  1. Djamel Eddine Amir. Computability of Topological Spaces. PhD thesis, Université de Lorraine, 2023. Google Scholar
  2. Djamel Eddine Amir. Spaces with unfixable type, invited speaker, CCA 2024. Computability and Complexity in Analysis (CCA). Google Scholar
  3. Djamel Eddine Amir and Mathieu Hoyrup. Computability of finite simplicial complexes. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 111:1-111:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.111.
  4. Djamel Eddine Amir and Mathieu Hoyrup. Comparing computability in two topologies. The Journal of Symbolic Logic, pages 1-19, 2023. URL: https://doi.org/10.1017/jsl.2023.17.
  5. Djamel Eddine Amir and Mathieu Hoyrup. Strong computable type. Computability, pages 227-269, january 1 2023. URL: https://doi.org/10.3233/COM-220430.
  6. Djamel Eddine Amir and Mathieu Hoyrup. The surjection property and computable type. Topology and its Applications, 355:109020, 2024. URL: https://doi.org/10.1016/j.topol.2024.109020.
  7. Djamel Eddine Amir and Mathieu Hoyrup. Descriptive complexity of topological invariants. Annals of Pure and Applied Logic, 176(8):103611, 2025. URL: https://doi.org/10.1016/j.apal.2025.103611.
  8. Nathalie Aubrun, Sebastián Barbieri, and Emmanuel Jeandel. About the domino problem for subshifts on groups. Sequences, groups, and number theory, pages 331-389, 2018. Google Scholar
  9. Nathalie Aubrun, Sebastián Barbieri, and Mathieu Sablik. A notion of effectiveness for subshifts on finitely generated groups. Theoretical Computer Science, 661:35-55, 2017. URL: https://doi.org/10.1016/j.tcs.2016.11.033.
  10. Alexis Ballier, Bruno Durand, and Emmanuel Jeandel. Structural aspects of tilings. In STACS 2008, pages 61-72. IBFI Schloss Dagstuhl, 2008. URL: https://doi.org/10.4230/LIPICS.STACS.2008.1334.
  11. Sebastián Barbieri, Nicanor Carrasco-Vargas, and Cristóbal Rojas. Effective dynamical systems beyond dimension zero and factors of sfts. Ergodic Theory and Dynamical Systems, pages 1-41, 2024. Google Scholar
  12. Nicolás Bitar. Contributions to the domino problem: Seeding, recurrence and satisfiability. arXiv preprint arXiv:2312.08911, 2023. URL: https://doi.org/10.48550/arXiv.2312.08911.
  13. Antonin Callard and Benjamin Hellouin de Menibus. The aperiodic domino problem in higher dimension. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2022. Google Scholar
  14. Nicanor Carrasco-Vargas. Subshifts on groups and computable analysis. PhD thesis, Pontifical Catholic University of Chile, 2024. Google Scholar
  15. T. Ceccherini-Silberstein and M. Coornaert. Cellular Automata and Groups. Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2010. URL: https://books.google.fr/books?id=N-LSFFaHTKwC.
  16. Matea Čelar and Zvonko Iljazović. Computability of products of chainable continua. Theory of Computing Systems, 65(2):410-427, February 2021. URL: https://doi.org/10.1007/s00224-020-10017-6.
  17. Douglas Cenzer. Chapter 2 - pi_o¹ classes in computability theory. In Edward R. Griffor, editor, Handbook of Computability Theory, volume 140 of Studies in Logic and the Foundations of Mathematics, pages 37-85. Elsevier, 1999. URL: https://doi.org/10.1016/S0049-237X(99)80018-4.
  18. Jean-Charles Delvenne, Petr Kurka, and Vincent Blondel. Decidability and universality in symbolic dynamical systems. Fundamenta Informaticae, 74(4):463-490, 2006. URL: http://content.iospress.com/articles/fundamenta-informaticae/fi74-4-06.
  19. Rodney G. Downey and Alexander G. Melnikov. Computably compact metric spaces. The Bulletin of Symbolic Logic, 29(2):170-263, 2023. URL: https://doi.org/10.1017/bsl.2023.16.
  20. Francesca Fiorenzi. Semi-strongly irreducible shifts. Advances in Applied Mathematics, 32(3):421-438, 2004. URL: https://doi.org/10.1016/S0196-8858(03)00053-8.
  21. Silvère Gangloff and Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. arXiv preprint arXiv:1602.06166, 2016. Google Scholar
  22. Graham Higman. Subgroups of finitely presented groups. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 262(1311):455-475, 1961. Google Scholar
  23. Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Inventiones mathematicae, 176(1):131-167, April 2009. URL: https://doi.org/10.1007/s00222-008-0161-7.
  24. Zvonko Iljazović and Takayuki Kihara. Computability of subsets of metric spaces. In Vasco Brattka and Peter Hertling, editors, Handbook of Computability and Complexity in Analysis, pages 29-69. Springer International Publishing, Cham, 2021. URL: https://doi.org/10.1007/978-3-030-59234-9_2.
  25. Zvonko Iljazović and Igor Sušić. Semicomputable manifolds in computable topological spaces. J. Complex., 45:83-114, 2018. URL: https://doi.org/10.1016/j.jco.2017.11.004.
  26. Emmanuel Jeandel. Enumeration reducibility in closure spaces with applications to logic and algebra. In 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-11. IEEE, 2017. URL: https://doi.org/10.1109/LICS.2017.8005086.
  27. Emmanuel Jeandel and Pascal Vanier. A characterization of subshifts with computable language. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  28. Andrew S Kahr, Edward F Moore, and Hao Wang. Entscheidungsproblem reduced to the aea case. Proceedings of the National Academy of Sciences, 48(3):365-377, 1962. Google Scholar
  29. D. Kerr and H. Li. Ergodic Theory: Independence and Dichotomies. Springer Monographs in Mathematics. Springer International Publishing, 2017. Google Scholar
  30. Bruce Kitchens and Klaus Schmidt. Periodic points, decidability and markov subgroups. In Dynamical Systems: Proceedings of the Special Year held at the University of Maryland, College Park, 1986-87, pages 440-454. Springer, 2006. Google Scholar
  31. D. A. Lind. A zeta function for ℤ^d-actions, pages 433-450. London Mathematical Society Lecture Note Series. Cambridge University Press, 1996. Google Scholar
  32. M Lothaire. Algebraic Combinatorics on Words. Cambridge University Press, 2002. Google Scholar
  33. Roger C Lyndon and Paul E Schupp. Combinatorial group theory, volume 89. Springer, 1977. Google Scholar
  34. Joseph S. Miller. Effectiveness for embedded spheres and balls. Electronic Notes in Theoretical Computer Science, 66(1):127-138, 2002. CCA 2002, Computability and Complexity in Analysis (ICALP 2002 Satellite Workshop). URL: https://doi.org/10.1016/S1571-0661(04)80384-0.
  35. Ville Salo. Decidability and universality of quasiminimal subshifts. Journal of Computer and System Sciences, 89:288-314, 2017. URL: https://doi.org/10.1016/j.jcss.2017.05.017.
  36. Alan L. Selman. Arithmetical reducibilities i. Mathematical Logic Quarterly, 17(1):335-350, 1971. URL: https://doi.org/10.1002/malq.19710170139.
  37. Hao Wang. Proving theorems by pattern recognition—ii. Bell system technical journal, 40(1):1-41, 1961. Google Scholar
  38. Vedran Čačić, Matea Čelar, Marko Horvat, and Zvonko Iljazović. Computable approximations of semicomputable graphs, 2024. URL: https://doi.org/10.48550/arXiv.2411.13672.
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