Proving a phase transition in cellular automata under asynchronous dynamics
Résumé
Cellular automata are a model of parallel computing. It is well known that simple cellular automata may exhibit complex behaviors such as Turing universality. The underlying mechanisms of these rules are now rather well understood. Less results are known about probabilistic cellular automata. The most famous ones come from Toom and Gacs. They have shown that cellular automata are still able to perform reliable computation in presence of random faults even in one dimension. However, Gacs' automata is sophisticated and finding a simple rule with the same property is still an open question. Recently, Fatès has exhibited a family of simple one dimensional probabilistic rules which can solve the density classification problem with arbitrary precision. Several studies have focused on a specific probabilistic dynamics: alpha-asynchronism where at each time step each cell has a probability alpha to be updated. Experimental studies followed by mathematical analysis have permitted to exhibit simple rules with interesting behaviors. Among these behaviors, most of these studies conjectured that some cellular automata exhibit a polynomial/exponential phase transition on their convergence time, i.e. the time to reach a stable configuration. The study of these phase transitions is crucial to understand the behaviors which appear at low synchronicity. A first analysis proved the existence of the exponential phase in cellular automaton FLIP-IF-NOT-ALL-EQUAL but failed to prove the existence of the polynomial phase. In this paper, we prove the existence of a polynomial/exponential phase transition in a cellular automaton called FLIP-IF-NOT-ALL-0.
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