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)


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.


Random Walk Cellular Automaton Convergence Time Probabilistic Cellular Automaton Synchronous Dynamic 
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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 

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