Dynamics and Resistance to Neighborhood Perturbations of Discrete-and Continuous-Time Cellular Automata
The dynamics of cellular automata rules are compared when simulated in discrete time and as continuous-time stochastic processes, where sites are updated asynchronously. Discrete-time dynamics are also predicted using local structure theory (pair approximations). A new statistic for lookup tables, the resistance to perturbations of the states of sites in the neighborhood, is proposed as one metric which may be applied to CA rules when studying the difference in dynamics between discrete and continuous-time simulations. Statistically, randomly-generated rules with either very small or large resistances tend to have uninteresting dynamics because they are either too disordered or too ordered, respectively. However, discrete-time rules with complex dynamics but large resistances are conjectured to be more likely to also produce interesting dynamics in continuous time.