Convergence and Aperiodicity in Fuzzy Cellular Automata: Revisiting Rule 90 P. Flocchini, F. Geurts, A. Mingarelli, N. Santoro
Abstract
	Lifting Boolean to continuous CA by fuzzification of the disjunctive normal form local rule enables the analytical derivation of results for an interesting class of systems: Fuzzy CA, which include the Boolean evolution as a particular case. We concentrate on Fuzzy Rule 90, whose Boolean version has deserved some attention for the complex patterns it generates. We prove that aperiodicity is present in both versions, and a strong convergence to a fixed point is added in the fuzzy case, which extends and complement Jen's result on aperiodicity of Boolean CA. Moreover, we show that the complex pattern and the apparent chaotic behaviour observed in its Boolean evolution are not an inherent property of Rule 90 but rather an artifact of the fixed point being an extreme of one of the two value intervals.

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