Advertisement

Fully Asynchronous Behavior of Double-Quiescent Elementary Cellular Automata

  • Nazim Fatés
  • Michel Morvan
  • Nicolas Schabanel
  • Éric Thierry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

In this paper we propose a probabilistic analysis of the fully asynchronous behavior (i.e., two cells are never simultaneously updated, as in a continuous time process) of elementary finite cellular automata (i.e., {0,1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0,0,0) ↦ 0 and (1,1,1) ↦ 1). It has been experimentally shown in previous works that introducing asynchronism in the global function of a cellular automaton may perturb its behavior, but as far as we know, only few theoretical work exist on the subject. The cellular automata we consider live on a ring of size n and asynchronism is introduced as follows: at each time step one cell is selected uniformly at random and the transition rule is applied to this cell while the others remain unchanged. Among the sixty-four cellular automata belonging to the class we consider, we show that fifty-five other converge almost surely to a random fixed point while nine of them diverge on all non-trivial configurations. We show that the convergence time of these fifty-five automata can only take the following values: either 0, Θ(n ln n), Θ(n 2), Θ(n 3), or Θ(n2 n ). Furthermore, the global behavior of each of these cellular automata can be guessed by simply reading its code.

Keywords

Random Walk Cellular Automaton Convergence Time Probabilistic Cellular Automaton Synchronous Dynamic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bersini, H., Detours, V.: Asynchrony induces stability in cellular automata based models. In: Brooks, Maes, Pattie (eds.) Proceedings of the 4th International Workshop on the Synthesis and Simulation of Living Systems (Artificial Life IV, July 1994, pp. 382–387. MIT Press, Cambridge (1994)Google Scholar
  2. 2.
    Brémaud, P.: Markov chains, Gibbs fileds, Monte Carlo simulation, and queues. Springer, Heidelberg (1999)Google Scholar
  3. 3.
    Buvel, R.L., Ingerson, T.E.: Structure in asynchronous cellular automata. Physica D 1, 59–68 (1984)MathSciNetGoogle Scholar
  4. 4.
    Fatés, N., Morvan, M.: An experimental study of robustness to asynchronism for elementary cellular automata. arxiv:nlin.CG/0402016 (2004) (Submitted)Google Scholar
  5. 5.
    Fatès, N., Morvan, M.: Perturbing the topology of the game of life increases its robustness to asynchrony. In: Sloot, P.M.A., Chopard, B., Hoekstra, A.G. (eds.) ACRI 2004. LNCS, vol. 3305, pp. 111–120. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Fatès, N., Morvan, M., Schabanel, N., Thierry, E.: Fully asynchronous behavior of double-quiescent elementary cellular automata. Research report LIP RR2005-04, ENS Lyon (2005)Google Scholar
  7. 7.
    Grimmet, G., Stirzaker, D.: Probability and Random Process, 3rd edn. Oxford University Press, Oxford (2001)Google Scholar
  8. 8.
    Gács, P.: Deterministic computations whose history is independent of the order of asynchronous updating (2003), http://arXiv.org/abs/cs/0101026
  9. 9.
    Huberman, B.A., Glance, N.: Evolutionary games and computer simulations. In: Proceedings of the National Academy of Sciences, USA, August 1993, vol. 90, pp. 7716–7718 (1993)Google Scholar
  10. 10.
    Louis, P.-Y.: Automates Cellulaires Probabilistes: mesures stationnaires, mesures de Gibbs associées et ergodicité. PhD thesis, Université de Lille I (September 2002)Google Scholar
  11. 11.
    Mattera, M.: Annihilating random walks and perfect matchings of planar graphs. In: Discrete Mathematics and Theoretical Computer Science, AC, pp. 173–180 (2003)Google Scholar
  12. 12.
    Nowak, M.A., May, R.M.: Evolutionary games and spatial chaos. Nature (London) 359, 826–829 (1992)CrossRefGoogle Scholar
  13. 13.
    Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. BioSystems 51, 123–143 (1999)CrossRefGoogle Scholar
  14. 14.
    Wolfram, S.: Universality and complexity in cellular automata. Physica D 10, 1–35 (1984)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Nazim Fatés
    • 1
  • Michel Morvan
    • 1
    • 2
  • Nicolas Schabanel
    • 1
  • Éric Thierry
    • 1
  1. 1.ENS Lyon – LIP (UMR CNRS – ENS Lyon – UCB Lyon – INRIA 5668)Lyon Cedex 07France
  2. 2.Institut universitaire de FranceÉcole des hautes études en sciences sociales and Santa Fe Insitute 

Personalised recommendations