Diploid Cellular Automata: First Experiments on the Random Mixtures of Two Elementary Rules

Part 2: Regular PapersInternational audienceWe study a small part of the 8088 diploid cellular automata. These rules are obtained with a random mixture of two deterministic Elementary Cellular Automata. We use numerical simulations to study the mixtures obtained with three blind rules: the null rule, the identity rule and the inversion rule. As the mathematical analysis of such systems is a difficult task, we use numerical simulations to get insights into the dynamics of this class of stochastic cellular automata. We are particularly interested in studying phase transitions and various types of symmetry breaking

Proof of a phase transition in probabilistic cellular automata

International audienceCellular automata are a model of parallel computing. It is well known that simple deterministic cellular automata may exhibit complex behaviors such as Turing universality [3,13] but only few results are known about complex behaviors of probabilistic cellular automata. Several studies have focused on a specific probabilistic dynamics: α-asynchronism where at each time step each cell has a probability α to be updated. Experimental studies [5] followed by mathematical analysis [2,4,7,8] 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 [14] 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