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A Characterization of Subshifts with Computable Language

Authors Emmanuel Jeandel , Pascal Vanier



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Author Details

Emmanuel Jeandel
  • LORIA, Campus Scientifique - BP 239, 54506 Vandoeuvre-les-Nancy, France
Pascal Vanier
  • Laboratoire d'Algorithmique, Complexité et Logique, Université de Paris-Est, LACL, UPEC, France

Acknowledgements

The authors wish to thanks the anonymous referees for many helpful remarks and improvements.

Cite AsGet BibTex

Emmanuel Jeandel and Pascal Vanier. A Characterization of Subshifts with Computable Language. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 40:1-40:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.40

Abstract

Subshifts are sets of colorings of Z^d by a finite alphabet that avoid some family of forbidden patterns. We investigate here some analogies with group theory that were first noticed by the first author. In particular we prove several theorems on subshifts inspired by Higman’s embedding theorems of group theory, among which, the fact that subshifts with a computable language can be obtained as restrictions of minimal subshifts of finite type.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • subshifts
  • computability
  • Enumeration degree
  • Turing degree
  • minimal subshifts

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References

  1. Nathalie Aubrun and Mathieu Sablik. An Order on Sets of Tilings Corresponding to an Order on Languages. In 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, February 26-28, 2009, Freiburg, Germany, Proceedings, pages 99-110, 2009. Google Scholar
  2. Nathalie Aubrun and Mathieu Sablik. Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type. Acta Applicandae Mathematicae, 126(1):35-63, 2013. URL: http://dx.doi.org/10.1007/s10440-013-9808-5.
  3. Alexis Ballier. Propriétés structurelles, combinatoires et logiques des pavages. PhD thesis, Aix-Marseille Université, 2009. Google Scholar
  4. Alexis Ballier and Emmanuel Jeandel. Computing (or not) Quasi-periodicity Functions of Tilings. In Jarkko Kari, editor, Second Symposium on Cellular Automata "Journées Automates Cellulaires", JAC 2010, Turku, Finland, December 15-17, 2010. Proceedings, pages 54-64. Turku Center for Computer Science, 2010. Google Scholar
  5. Robert Berger. The Undecidability of the Domino Problem. Number 66 in Memoirs of the American Mathematical Society. The American Mathematical Society, 1966. Google Scholar
  6. William W. Boone and Graham Higman. An algebraic characterization of groups with soluble word problem. Journal of the Australian Mathematical Society, 18(1):41-53, August 1974. URL: http://dx.doi.org/10.1017/S1446788700019108.
  7. Mike Boyle, Ronnie Pavlov, and Michael Schraudner. Multidimensional sofic shifts without separation and their factors. Transactions of the AMS, 362(9):4617-4653, September 2010. URL: http://dx.doi.org/10.1090/S0002-9947-10-05003-8.
  8. W. Craig and R. L. Vaught. Finite Axiomatizability Using Additional Predicates. The Journal of Symbolic Logic, 23(3):289-308, September 1958. Google Scholar
  9. Bruno Durand, Andrei Romashchenko, and Alexander Shen. Effective Closed Subshifts in 1D Can Be Implemented in 2D. In Fields of Logic and Computation, number 6300 in Lecture Notes in Computer Science, pages 208-226. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15025-8_12.
  10. Bruno Durand, Andrei Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. Journal of Computer and System Sciences, 78(3):731-764, May 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.11.001.
  11. Bruno Durand and Andrei E. Romashchenko. On the expressive power of quasiperiodic SFT. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 1-14, 2017. Google Scholar
  12. Gábor Elek and Nicolas Monod. On the Topological Full Group of a Minimal Cantor ℤ²-System. Proceedings of the American Mathematical Society, 141(10):3549-3552, October 2013. Google Scholar
  13. Richard M. Friedberg and Hartley Rogers. Reducibility and Completeness for Sets of Integers. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 5:117-125, 1959. Google Scholar
  14. Graham Higman. Subgroups of Finitely Presented Groups. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 262(1311):455-475, August 1961. Google Scholar
  15. Graham Higman and Elizabeth Scott. Existentially Closed Groups. Oxford University Press, 1988. Google Scholar
  16. Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Inventiones Mathematicae, 176(1):2009, April 2009. Google Scholar
  17. Michael Hochman and Tom Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011-2038, May 2010. URL: http://dx.doi.org/10.4007/annals.2010.171.2011.
  18. Emmanuel Jeandel. Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra. In ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-11, 2017. Google Scholar
  19. Emmanuel Jeandel and Pascal Vanier. Characterizations of periods of multidimensional shifts. Ergodic Theory and Dynamical Systems, 35(2):431-460, April 2015. URL: http://dx.doi.org/10.1017/etds.2013.60.
  20. Aimee Johnson and Kathleen Madden. Factoring higher-dimensional shifts of finite type onto the full shift. Ergodic Theory and Dynamical Systems, 25:811-822, 2005. Google Scholar
  21. S.C. Kleene. Two Papers on the Predicate Calculus, chapter Finite Axiomatizability of Theories in the Predicate Calculus Using Additional Predicate Symbols, pages 31-71. Number 10 in Memoirs of the American Mathematical Society. American Mathematical Society, 1952. Google Scholar
  22. R.C. Lyndon and P.E. Schupp. Combinatorial Group Theory. Classics in Mathematics. Springer Berlin Heidelberg, 2001. Google Scholar
  23. Daniele Marsibilio and Andrea Sorbi. Bounded Enumeration Reducibility and its degree structure. Archive for Mathematical Logic, 51:163-186, 2012. Google Scholar
  24. Mark Sapir. Combinatorial Algebra: Syntax and Semantics. Springer Monographs in Mathematics. Springer, 2014. Google Scholar
  25. Stephen G. Simpson. Mass problems associated with effectively closed sets. Tohoku Mathematical Journal, 63(4):489-517, 2011. Google Scholar
  26. Richard J. Thompson. Embeddings into Finitely Generated Simple Groups which Preserve the Word Problem. In Sergei I. Adian, William W. Boone, and Graham Higman, editors, Word Problems II, volume 95 of Studies in Logic and the Foundations of Mathematics, pages 401-441. North Holland, 1980. Google Scholar
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