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Stability in One-dimensional Random Boolean Cellular Automata
F. Michel Dekking, Leonard Van Driel and Anne Fey

We consider one-dimensional random Boolean cellular automata, where the cells are identified with the integers from 1 to N. The behavior of the automaton is mainly determined by the support of the random variable that selects one of the sixteen possible Boolean rules, independently for each cell. A cell is said to stabilize if it will not change its state anymore after some time. We classify the one-dimensional random Boolean automata according to the positivity of their probability of stabilization. Here is an example of a consequence of our results: if the support contains at least 5 rules, then asymptotically as N → ∞the probability of stabilization is positive, whereas there exist random Boolean cellular automata with 4 rules in their support for which this probability is 0.

Keywords: random cellular automata, random Boolean networks, stability.

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