Rice’s Theorem for μ-Limit Sets of Cellular Automata

  • Martin Delacourt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)


Cellular automata are a parallel and synchronous computing model, made of infinitely many finite automata updating according to the same local rule. Rice’s theorem states that any nontrivial property over computable functions is undecidable. It has been adapted by Kari to limit sets of cellular automata [7], that is the set of configurations that can be reached arbitrarily late. This paper proves a new Rice theorem for μ-limit sets, which are sets of configurations often reached arbitrarily late.


Cellular Automaton Computable Function Local Rule Permanent State Bruijn Sequence 
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.
    Boyer, L., Delacourt, M., Sablik, M.: Construction of μ-limit sets. In: Kari, J. (ed.) JAC 2010, vol. 13, pp. 76–87. TUCS Lecture notes, Turku (2010)Google Scholar
  2. 2.
    Boyer, L., Poupet, V., Theyssier, G.: On the complexity of limit sets of cellular automata associated with probability measures. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 190–201. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Delacourt, M., Poupet, V., Sablik, M., Theyssier, G.: Directional dynamics along arbitrary curves in cellular automata. To be Pubished in Theoretical Computer Science (2011)Google Scholar
  4. 4.
    Fredricksen, H., Kessler, I.: Lexicographic compositions and debruijn sequences. Journal of Combinatorial Theory, Series A 22(1), 17–30 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Guillon, P., Richard, G.: Revisiting the rice theorem of cellular automata. In: Marion, J.Y., Schwentick, T. (eds.) STACS 2010, vol. 5, pp. 441–452. LIPIcs, Schloss Dagstuhl (2010)Google Scholar
  6. 6.
    Hurley, M.: Ergodic aspects of cellular automata. Ergodic Theory and Dynamical Systems 10, 671–685 (1990)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Kari, J.: Rice’s theorem for the limit sets of cellular automata. Theoretical Computer Science 127(2), 229–254 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kůrka, P., Maass, A.: Limit sets of cellular automata associated to probability measures. Journal of Statistical Physics 100, 1031–1047 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    von Neumann, J.: Theory of Self-Reproducing Automata. University of Illinois Press, Champaign (1966)Google Scholar
  10. 10.
    Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Transactions of the American Mathematical Society 74(2), 358–366 (1953)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Martin Delacourt
    • 1
  1. 1.Laboratoire d’Informatique Fondamentale de MarseilleUniversité de ProvenceFrance

Personalised recommendations