On the decidability of the evolution of the fuzzy cellular automaton 184

Abstract

In the previous paper [1] we presented general methods for detecting the evolution and dynamics of any one of the 255 fuzzy cellular automata (FCA) and showed that the method was applicable to all but nine of the 255 FCA. The main result there was that the limiting behavior of these FCA is decidable, except possibly for these nine, for finite initial configurations in a homogeneous background of zeros. Only six of these nine so called exceptional CA namely, FCA 172, 184, 202, 216, 226, and 228, appear to be interesting enough to warrant separate study, the other three, namely FCA 204, 228, and 240 being trivial. In this paper we study the exceptional FCA 184, a cellular automaton that admits a continuum of fixed points, namely the interval [0,1], This FCA is of interest because the general technique developed in [1] fails for the determination of its asymptotics. We show, in particular, that the asymptotic evolution of FCA 184 from any finite initial including random configuration of non-zero cells is decidable. © Springer-Verlag Berlin Heidelberg 2006.