Criterion for Stability in Random Boolean Cellular

Random boolean cellular automata are investigated, where each gate has two randomly chosen inputs and is randomly assigned a boolean function of its inputs. The e ect of non-uniform distributions on the choice of the boolean functions is considered. The main results are that if the gates are more likely to be assigned constant functions than non-canalyzing functions, then with very high probability, the automaton will exhibit very stable behavior: most of the gates will stabilize, and the state cycles will be bounded in size. Introduction Boolean cellular automata are models of parallel computation that have attracted much attention from researchers in complex systems and arti cial life. Computer simulation of these automata have shown that they often exhibit stable and robust behavior, even when randomly constructed. The implications of this evidence have been described in numerous articles by S. Kau man and others (see for example [1], which includes an extensive bibliography). Only recently, however, have rigorous mathematical methods been applied to the study of boolean cellular automata. This article is a continuation of the e orts begun in Luczak and Cohen [3] and Lynch [4]. We investigate randomly constructed boolean cellular automata, where each gate has two inputs, as in most of Kau man's simulations. However, instead of randomly assigning one of the 16 boolean functions of two arguments to each gate with equal probability, we consider the e ect of nonuniform distributions, using a classi cation of boolean functions introduced by Kau man (op. cit.). (We still require some mild symmetry conditions on the probabilities.) 1Research supported by NSF Grant CCR-9006303.