On Density and Convergence of Fuzzy Cellular Automata

Fuzzy cellular automata (FCA) are cellular automata where the local rule is defined as the “fuzzification" of the local rule of the corresponding Boolean cellular automaton in disjunctive normal form. In this paper we are interested in two separate issues: the relationship between Boolean and fuzzy models, and the asymptotic behavior of elementary fuzzy cellular automata. On the first issue, we show that the number conservation property, extensively studied in the Boolean domain, is preserved in the fuzzy domain: a Boolean CA is number conserving if and only if the corresponding FCA is sum preserving; a similar result is established for another density property. This is the first formal proof of a relationship between the fuzzy and the Boolean model. As for the second issue, we analyse the asymptotic behavior of circular elementary FCA for the first time, and provide a complete analytical study of Weighted Average FCA, a class that includes the density conserving rules and that encompasses most of the observable behaviours.