Turing Degrees of Limit Sets of Cellular Automata

  • Alex Borello
  • Julien Cervelle
  • Pascal Vanier
Conference paper
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.


Cellular Automaton Turing Machine Sparse Grid Turing Degree Computation Zone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bennett, C.H.: Logical Reversibility of Computation. IBM J. Res. Dev. 17(6), 525–532 (1973)CrossRefzbMATHGoogle 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)zbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hooper, P.K.: The Undecidability of the Turing Machine Immortality Problem. Journal of Symbolic Logic 31(2), 219–234 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hurd, L.P.: Formal Language Characterization of Cellular Automaton Limit Sets. Complex Systems 1(1), 69–80 (1987)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hurd, L.P.: Nonrecursive Cellular Automata Invariant Sets. Complex Systems 4(2), 131–138 (1990)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hurd, L.P.: Recursive Cellular Automata Invariant Sets. Complex Systems 4(2), 131–138 (1990)zbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Meyerovitch, T.: Finite entropy for multidimensional cellular automata. Ergodic Theory and Dynamical Systems 28(04), 1243–1260 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Simpson, S.G.: Mass problems associated with effectively closed sets. Tohoku Mathematical Journal 63(4), 489–517 (2011)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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

Personalised recommendations