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On the Expressive Power of Quasiperiodic SFT

Authors Bruno Durand, Andrei Romashchenko

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Bruno Durand
Andrei Romashchenko

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Bruno Durand and Andrei Romashchenko. On the Expressive Power of Quasiperiodic SFT. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 5:1-5:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in Z^d. The minimal shifts are those shifts in which all configurations contain exactly the same patterns. Two classes of shifts play a prominent role in symbolic dynamics, in language theory and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns). We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by 1) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E. Jeandel. We prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which Kolmogorov complexity of all patterns of size n\times n is \Omega(n).
  • minimal SFT
  • tilings
  • quasiperiodicityIn this paper we study the shifts
  • which are the shift-invariant and topologically closed sets of configurations


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  1. Nathalie Aubrun and Mathieu Sablik. Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta Applicandae Mathematicae, 128(1):35-63, 2013. Google Scholar
  2. Sergey V. Avgustinovich, Dmitrii G. Fon-Der-Flaass, and Anna E. Frid. Arithmetical complexity of infinite words. In 3rd Int. Colloq. on Words, Languages and Combinatorics, pages 51-62, 2003. Google Scholar
  3. Alexis Ballier and Emmanuel Jeandel. Computing (or not) quasiperiodicity functions of tilings. In 2nd Symposium on Cellular Automata Journées Automates Cellulaires (JAC 2010), pages 54-64, 2010. Google Scholar
  4. Bruno Durand. Tilings and quasiperiodicity. Theoretical Computer Science, 221(1):61-75, 1999. Google Scholar
  5. Bruno Durand, Leonid Levin, and Alexander Shen. Complex tilings. The Journal of Symbolic Logic, 73(2):593-613, 2008. Google Scholar
  6. 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. Google Scholar
  7. Brunourand Durand and Andrei Romashchenko. Quasiperiodicity and non-computability in tilings. In Proc. International Symposium on Mathematical Foundations of Computer Science (MFCS 2015), pages 218-230, 2015. Google Scholar
  8. Peter Gács. Reliable computation with cellular automata. Journal of Computer and System Sciences, 32(1):15-78, 1986. Google Scholar
  9. Gustav Hedlund and Marston Morse. Symbolic dynamics. American Journal of Mathematics, 60(4):815-866, 1938. Google Scholar
  10. Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Inventiones mathematicae, 176(1):131-167, 2009. Google Scholar
  11. Michael Hochman and Pascal Vanier. A note on turing degree spectra of minimal shifts. In The 12th International Computer Science Symposium in Russia, pages 154-161, 2017. Google Scholar
  12. Emmanuel Jeandel. Personal communication. private communication, 2015. Google Scholar
  13. Emmanuel Jeandel and Pascal Vanier. Turing degrees of multidimensional sfts. Theoretical Computer Science, 505:81-92, 2013. Google Scholar
  14. Andrey Rumyantsev and Maxim Ushakov. Forbidden substrings, kolmogorov complexity and almost periodic sequences. In Annual Symposium on Theoretical Aspects of Computer Science, pages 396-407, 2006. Google Scholar
  15. Pavel V. Salimov. On uniform recurrence of a direct product. Discrete Mathematics and Theoretical Computer Science, 12(4), 2010. Google Scholar
  16. Linda Brown Westrick. Seas of squares with sizes from a Π⁰₁ set. arXiv preprint arXiv:1609.07411, 2016. Google Scholar
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