Fully asynchronous behavior of double-quiescent elementary cellular automata

  • Authors:

  • Nazim Fatés;Michel Morvan;Nicolas Schabanel;Éric Thierry

  • Affiliations:

  • ENS Lyon – LIP (UMR CNRS – ENS Lyon – UCB Lyon – INRIA 5668), Lyon Cedex 07, France;ENS Lyon – LIP (UMR CNRS – ENS Lyon – UCB Lyon – INRIA 5668), Lyon Cedex 07, France;ENS Lyon – LIP (UMR CNRS – ENS Lyon – UCB Lyon – INRIA 5668), Lyon Cedex 07, France;ENS Lyon – LIP (UMR CNRS – ENS Lyon – UCB Lyon – INRIA 5668), Lyon Cedex 07, France

  • Venue:
  • MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
  • Year:
  • 2005

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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(n2), Θ(n3), or Θ(n2n). Furthermore, the global behavior of each of these cellular automata can be guessed by simply reading its code.