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 deserves some attention for the complex 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 aperiodicity 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 rule 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.