On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures

  • Laurent Boyer
  • Victor Poupet
  • Guillaume Theyssier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


We study the notion of limit sets of cellular automata associated with probability measures (μ-limit sets). This notion was introduced by P. Kůrka and A. Maass in [1]. It is a refinement of the classical notion of ω-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterization of the persistent language for non sensitive cellular automata associated with Bernoulli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their μ-limit set) is neither recursively enumerable nor co-recursively enumerable.


Cellular Automaton Turing Machine Left Border Term Behavior Turing Degree 
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  1. 1.
    Kůrka, P., Maass, A.: Limit Sets of Cellular Automata Associated to Probability Measures. Journal of Statistical Physics 100, 1031–1047 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Wolfram, S.: Universality and complexity in cellular automata. Physica D 10, 1–35 (1984)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Kůrka, P.: Languages, equicontinuity and attractors in cellular automata. Ergodic Theory and Dynamical Systems 17, 417–433 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Mazoyer, J., Rapaport, I.: Inducing an Order on Cellular Automata by a Grouping Operation. In: Symposium on Theoretical Aspects of Computer Science. LNCS (1998)Google Scholar
  5. 5.
    Čulik II, K., Pachl, J., Yu, S.: On the limit sets of cellular automata. SIAM Journal on Computing 18, 831–842 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kari, J.: Rice’s theorem for the limit sets of cellular automata. Theoretical Computer Science 127, 229–254 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nagel, K., Schreckenberg, M.: A cellular automaton model for freeway traffic. J. Phys. 2, 2221–2229 (1992)CrossRefGoogle Scholar
  8. 8.
    Kari, J.: The Nilpotency Problem of One-dimensional Cellular Automata. SIAM Journal on Computing 21, 571–586 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Sutner, K.: Cellular automata and intermediate degrees. Theoretical Computer Science 296 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Laurent Boyer
    • 1
  • Victor Poupet
    • 1
  • Guillaume Theyssier
    • 2
  1. 1.LIP (UMR 5668 — CNRS, ENS Lyon, UCB Lyon, INRIA)ENS LyonLYON cedex 07France
  2. 2.LAMA (UMR 5127 — CNRS, Université de Savoie)Université de SavoieLe Bourget-du-lac cedexFrance

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