On the Asymptotic Behavior of Fuzzy Cellular Automata

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 behavior has been observed through simulations and FCA have been empirically classified accordingly. No analytical study exists so far to support those observations. We now start the analytical study of circular FCA’s dynamics by considering a particular set of FCA (Weighted Average rules) which includes rules displaying most of the observed dynamics, and we precisely derive their behavior. We confirm the empirical observations proving that all weighted average rules are periodic in time and space, and we derive their periods.