Fully Asynchronous Behavior of Double-Quiescent Elementary Cellular Automata

• Published in: Lecture notes in computer science pp. 316 - 327
• Main Authors: Fatés, Nazim, Morvan, Michel, Schabanel, Nicolas, Thierry, Éric
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Cellular Automaton; Computer science; control theory; systems; Convergence Time; Exact sciences and technology; Probabilistic Cellular Automaton; Random Walk; Synchronous Dynamic; Theoretical computing; Continuous process; Cellular automaton; Fix point; Probabilistic approach; Asynchronism; Computer theory; Continuous time; Finite automaton; Modeling
• ISBN: 9783540287025; 3540287027
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11549345_28

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.