Turing Degrees of Limit Sets of Cellular Automata

  • Alex Borello
  • Julien Cervelle
  • Pascal Vanier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8573)


Cellular automata are discrete dynamical systems and a model of computation. The limit set of a cellular automaton consists of the configurations having an infinite sequence of preimages. It is well known that these always contain a computable point and that any non-trivial property on them is undecidable. We go one step further in this article by giving a full characterization of the sets of Turing degrees of limit sets of cellular automata: they are the same as the sets of Turing degrees of effectively closed sets containing a computable point.




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  1. 1.
    Bennett, C.H.: Logical Reversibility of Computation. IBM J. Res. Dev. 17(6), 525–532 (1973)CrossRefMATHGoogle Scholar
  2. 2.
    Ballier, A., Guillon, P., Kari, J.: Limit Sets of Stable and Unstable Cellular Automata. Fundam. Inform. 110(1-4), 45–57 (2011)MATHMathSciNetGoogle Scholar
  3. 3.
    Cenzer, D., Remmel, J.: \(\Pi_1^0\) classes in mathematics. In: Handbook of Recursive Mathematics - Volume 2: Recursive Algebra, Analysis and Combinatorics. Studies in Logic and the Foundations of Mathematics, ch. 13, vol. 139, pp. 623–821 (1998)Google Scholar
  4. 4.
    Formenti, E., Kůrka, P.: Subshift attractors of cellular automata. Nonlinearity 20, 105–117 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hooper, P.K.: The Undecidability of the Turing Machine Immortality Problem. Journal of Symbolic Logic 31(2), 219–234 (1966)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Hurd, L.P.: Formal Language Characterization of Cellular Automaton Limit Sets. Complex Systems 1(1), 69–80 (1987)MATHMathSciNetGoogle Scholar
  7. 7.
    Hurd, L.P.: Nonrecursive Cellular Automata Invariant Sets. Complex Systems 4(2), 131–138 (1990)MATHMathSciNetGoogle Scholar
  8. 8.
    Hurd, L.P.: Recursive Cellular Automata Invariant Sets. Complex Systems 4(2), 131–138 (1990)MATHMathSciNetGoogle Scholar
  9. 9.
    Jockusch, C.G., Soare, R.I.: Degrees of members of classes \(\Pi_1^0\). Pacific J. Math. 40(3), 605–616 (1972)Google Scholar
  10. 10.
    Jeandel, E., Vanier, P.: Hardness of Conjugacy, Embedding and Factorization of multidimensional Subshifts of Finite Type. In: STACS. LIPIcs, vol. 20, pp. 490–501 (2013)Google Scholar
  11. 11.
    Jeandel, E., Vanier, P.: Turing degrees of multidimensional SFTs. In: Theoretical Computer Science 505.0. Theory and Applications of Models of Computation, pp. 81–92 (2011)Google Scholar
  12. 12.
    Kari, J.: Reversibility of 2D cellular automata is undecidable. Physica D: Nonlinear Phenomena 45(1-3), 379–385 (1990)Google Scholar
  13. 13.
    Kari, J.: The Nilpotency Problem of One-Dimensional Cellular Automata. SIAM Journal on Computing 21(3), 571–586 (1992)Google Scholar
  14. 14.
    Kari, J.: Reversibility and surjectivity problems of cellular automata. Journal of Computer and System Sciences 48(1), 149–182 (1994)Google Scholar
  15. 15.
    Kari, J.: Rice’s theorem for the limit sets of cellular automata. Theoretical Computer Science 127(2), 229–254 (1994)Google Scholar
  16. 16.
    Kari, J., Ollinger, N.: Periodicity and Immortality in Reversible Computing. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 419–430. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Lena, P.D., Margara, L.: Undecidable Properties of Limit Set Dynamics of Cellular Automata. In: 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), vol. 3, pp. 337–348 (2009)Google Scholar
  18. 18.
    Maass, A.: On the sofic limit sets of cellular automata. Ergodic Theory and Dynamical Systems 15(04), 663–684 (1995)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Meyerovitch, T.: Finite entropy for multidimensional cellular automata. Ergodic Theory and Dynamical Systems 28(04), 1243–1260 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Simpson, S.G.: Mass problems associated with effectively closed sets. Tohoku Mathematical Journal 63(4), 489–517 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Čulik, K., Pachl, J., Yu, S.: On the limit sets of cellular automata. SIAM Journal on Computing 18(4), 831–842 (1989)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alex Borello
    • 1
  • Julien Cervelle
    • 1
  • Pascal Vanier
    • 1
  1. 1.Laboratoire d’algorithmiquecomplexité et logique Université de Paris-Est, LACL, UPECFrance

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