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On the Asymptotic Behaviour of Circular Fuzzy Cellular Automata
Heather Betel and Paola Flocchini

Fuzzy cellular automata (FCA) are continuous cellular automata where the local rule is defined as the “fuzzification” of the local rule of a corresponding Boolean cellular automaton in disjunctive normal form. In this paper, we consider circular FCA; their asymptotic behaviours had previously been observed through simulation and FCA had been empirically classified accordingly. However, no analytical study previously existed to support these observations.

We now begin the analytical study of circular FCA dynamics by considering a particular class of FCA (Weighted Average rules) which includes rules displaying most of the observed dynamics, and we precisely derive their behaviours. We confirm the empirical observations proving that all weighted average rules are periodic in time and space, and we derive their periods.

Keywords: Fuzzy cellular automata, asymptotic behaviour, periodic points.

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