On the Dynamics of Some Exceptional Fuzzy Cellular Automata

Abstract

Over the past 20 years, the study of cellular automata has emerged as one of the most interesting and popular forms of “new mathematics”. The study of cellular automata has broadened into many variations of the original concepts. One such variation is the study of one-dimensional fuzzy cellular automata. The evolution and dynamics of the majority of one-dimensional fuzzy cellular automata rules can be determined analytically using techniques devised by the second author. It turns out that only 9 rules (out of 256), three of which are trivial, fail to comply with the techniques given. We give a brief overview of finite cellular automata and their fuzzification. We summarize the method used to study the majority of fuzzy rules and give some examples of its application. We analyze and uncover the dynamics of those few rules which do not conform to such techniques. Using new techniques, combined with direct analysis, we determine the long term evolution of the 4 remaining rules (since two of them were treated in detail elsewhere). We specifically analyze rules 172 and 202 and then, by deriving equivalences to the final two rules, we complete the program, initiated in 2003, of determining the long term dynamics of all 256 one-dimensional fuzzy cellular automata, thereby showing that chaotic dynamics are incompatible with this type of fuzziness, in sharp contrast with boolean cellular automata.