Computability of Countable Subshifts

  • Douglas Cenzer
  • Ali Dashti
  • Ferit Toska
  • Sebastian Wyman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6158)


The computability of countable subshifts and their members is examined. Results include the following. Subshifts of Cantor-Bendixson rank one contain only eventually periodic elements. Any rank one subshift, in which every limit point is periodic, is decidable. Subshifts of rank two may contain members of arbitrary Turing degree. In contrast, effectively closed (\(\Pi^0_1\)) subshifts of rank two contain only computable elements, but \(\Pi^0_1\) subshifts of rank three may contain members of arbitrary c. e. degree. There is no subshift of rank ω.


Computability Symbolic Dynamics \(\Pi^0_1\) Classes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bournez, O., Cosnard, M.: On the computational power of dynamical systems and hybrid systems. Theoretical Computer Science 168, 417–459 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Braverman, M., Yampolski, M.: Non-computable Julia sets. J. Amer. Math. Soc. 19, 551–578 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cenzer, D.: Effective dynamics. In: Crossley, J., Remmel, J., Shore, R., Sweedler, M. (eds.) Logical Methods in honor of Anil Nerode’s Sixtieth Birthday, pp. 162–177. Birkhauser, Basel (1993)Google Scholar
  4. 4.
    Cenzer, D., Dashti, A., King, J.L.F.: Computable Symbolic Dynamics. Math. Logic Quarterly 54, 524–533 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cenzer, D., Hinman, P.G.: Degrees of difficulty of generalized r. e. separating classes. Arch. for Math. Logic 45, 629–647 (2008)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cenzer, D., Remmel, J.B.: \(\Pi^0_1\) classes, in Handbook of Recursive Mathematics, Vol. 2: Recursive Algebra, Analysis and Combinatorics. In: Ersov, Y., Goncharov, S., Marek, V., Nerode, A., Remmel, J. (eds.). Elsevier Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 623–821 (1998)Google Scholar
  7. 7.
    Delvenne, J.-C., Kurka, P., Blondel, V.D.: Decidability and Universality in Symbolic Dynamical Systems. Fund. Informaticae (2005)Google Scholar
  8. 8.
    Ko, K.: On the computability of fractal dimensions and Julia sets. Ann. Pure Appl. Logic 93, 195–216 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Medvedev, Y.: Degrees of difficulty of the mass problem. Dokl. Akad. Nauk SSSR 104, 501–504 (1955)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Miller, J.: Two notes on subshifts (preprint)Google Scholar
  11. 11.
    Rettinger, R., Weihrauch, K.: The computational complexity of some Julia sets. In: Goemans, M.X. (ed.) Proc. 35th ACM Symposium on Theory of Computing, San Diego, June 2003, pp. 177–185. ACM Press, New York (2003)Google Scholar
  12. 12.
    Simpson, S.G.: Mass problems and randomness. Bull. Symbolic Logic 11, 1–27 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Simpson, S.G.: Subsystems of Second Order Arithmetic, 2nd edn. Cambridge U. Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  14. 14.
    Simpson, S.G.: Medvedev degrees of two-dimensional subshifts of finite type. Ergodic Theory and Dynamical Systems (to appear)Google Scholar
  15. 15.
    Sorbi, A.: The Medvedev lattice of degrees of difficulty. In: Cooper, S.B., et al. (eds.) Computability, Enumerability, Unsolvability: Directions in Recursion Theory. London Mathematical Society Lecture Notes, vol. 224, pp. 289–312. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  16. 16.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

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

Personalised recommendations