Convergence and aperiodicity in fuzzy cellular automata: revisiting rule 90

In this paper we consider a continuous version of cellular automata (Fuzzy CA) obtained by “fuzzification ” of the disjunctive normal form which describes the corresponding Boolean rule. We concentrate on Fuzzy rule 90, whose Boolean version has deserved some attention for the com-plex patterns it generates. We show that the behavior of Fuzzy rule 90 is very simple, in that the system always converges to a fixed point. In the case of finite support configurations, we also show aperiodicity of every temporal sequences, extending and complementing Jen’s result on aperi-odicity of Boolean rule 90. We finally show and analyze the remarkable fact that, depending on the level of state-discreteness used to visualize the dynamics of Fuzzy 90, the display might show (after a transient) the well known complex Boolean behavior instead of the (correct) convergence to a fixed point. The results of the analysis lead not only to a caveat on the dangers of visualization, but also an unexpected explanation of the dynamics of Boolean rule 90.