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Computability of Countable Subshifts in One Dimension

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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 2ℤ 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, FL, 32611, USA

    Douglas Cenzer, Ali Dashti, Ferit Toska & Sebastian Wyman

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  1. Douglas Cenzer
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  2. Ali Dashti
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  3. Ferit Toska
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  4. Sebastian Wyman
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Correspondence to Douglas Cenzer.

Additional information

This research was partially supported by NSF grants DMS 0532644 and 0554841 and 652372.

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Cenzer, D., Dashti, A., Toska, F. et al. Computability of Countable Subshifts in One Dimension. Theory Comput Syst 51, 352–371 (2012). https://doi.org/10.1007/s00224-011-9358-z

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