Theory of Computing Systems

, Volume 51, Issue 3, pp 352–371 | Cite as

Computability of Countable Subshifts in One Dimension

  • Douglas Cenzer
  • Ali Dashti
  • Ferit Toska
  • Sebastian Wyman
Article

Abstract

We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of Cantor–Bendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In contrast, effectively closed (\(\Pi^{0}_{1}\)) subshifts of rank three contain only computable elements, but \(\Pi^{0}_{1}\) subshifts of rank four may contain members of arbitrary \(\Delta^{0}_{2}\) degree. There is no subshift of rank ω+1.

Keywords

Computability Symbolic dynamics \(\Pi^{0}_{1}\) classes 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Douglas Cenzer
    • 1
  • Ali Dashti
    • 1
  • Ferit Toska
    • 1
  • Sebastian Wyman
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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