Fuzzy automata and life

1. Introduction Cellular Automata have generated much interest [1,2,3] because of their diverse behavior and usefulness as a discrete model for many processes. Wolfram's 1984 paper on universality and complexity in cellular automata [3,4] described four classes of behavior for automata: class 1 for homogeneous stable behavior, class 2 for simple periodic patterns, class 3 for chaotic aperiodic behavior and class 4 for complex behavior which generates local structures. Recent work by Cattaneo, Flocchini, Mauri, Quaranta Vogliotti, and Santoro [5] introduced an alternate classification scheme that can be obtained by generalizing Boolean cellular automata to a fuzzy automata and then observing the qualitative behavior of a Boolean window embedded in a fuzzy domain. These automata can be classified according to whether the Boolean behavior dominates, a mixture of Boolean and Fuzzy behavior appears, or purely fuzzy behavior appears. The fuzzy behavior can further be divided into subclasses with homogeneous and nonhomogeneous behavior. In this paper we further investigate these fuzzy extensions of one-dimensional automata and we also consider two-dimensional automata. We investigate the impact of using several choices for the fuzzy logic functions including the unusual logic used in [5]. The choice of different fuzzy logics lead to significantly different behaviors. We also introduce and use the one-dimensional continuous dynamics of the response functions to explain much of the qualitative behavior of these fuzzy automata. Lastly, we consider two-dimensional automata giving the Game of Life. We will see the greatest common divisor/least common multiple trivalent logic seems to be remarkably able to nurture the generation of Life without dominating it; the Game of Life appears even more complex in this context. The situation is somewhat analogous to a wave tank with expanding walls generating waves with the interference controlled by the rules of the Game of Life. The authors of [5] defined a quantity they call rule entropy in a simple manner based upon the defining Boolean rule. While of limited capacity compared to more classical measures of entropy [3,6], the simplicity of this entropy comes from the fact that it does not depend upon the configuration of states. They observed that this entropy was bounded between 0 and 6 for their automata and that class 3 (chaotic) behavior appeared to be limited to their fuzzy classes and associated with intermediate or high values of rule entropy, thus connecting the qualitative behaviors with this simple quantitative measure. In the Appendix …