Asynchronous Behavior of Double-Quiescent Elementary Cellular Automata

In this paper we propose a probabilistic analysis of the relaxation time of elementary finite cellular automata (i.e., {0,1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0,0,0) ↦ 0 and (1,1,1) ↦ 1), under α-asynchronous dynamics (i.e., each cell is updated at each time step independently with probability 0 < α ≤ 1). This work generalizes previous work in [1], in the sense that we study here a continuous range of asynchronism that goes from full asynchronism to full synchronism. We characterize formally the sensitivity to asynchronism of the relaxation times for 52 of the 64 considered automata. Our work relies on the design of probabilistic tools that enable to predict the global behaviour by counting local configuration patterns. These tools may be of independent interest since they provide a convenient framework to deal exhaustively with the tedious case analysis inherent to this kind of study. The remaining 12 automata (only 5 after symmetries) appear to exhibit interesting complex phenomena (such as polynomial/exponential/infinite phase transitions).