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On the Expressive Power of Quasiperiodic SFT

Authors Bruno Durand, Andrei Romashchenko

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Keywords
  • 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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Abstract

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).

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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) https://doi.org/10.4230/LIPIcs.MFCS.2017.5

Author Details

Bruno Durand
Andrei Romashchenko

References

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