A tutorial on Elementary cellular automata with fully asynchronous updating - General properties and convergence dynamics

International audienceWe present a picture of the convergence properties of the 256 Elementary Cellular Automata under the fully asynchronous updating , that is, when only one cell is updated at each time step. We regroup here the results which have been presented in different articles and expose a full analysis of the behaviour of finite systems with periodic boundary conditions. Our classification relies on the scaling properties of the average convergence time to a fixed point. We observe that different scaling laws can be found, which fall in one of the following classes: logarithmic, linear, quadratic, exponential and non-converging. The techniques for quantifying this behaviour rely mainly on Markov chain theory and martingales. Most behaviours can be studied analytically but there are still many rules for which obtaining a formal characterisation of their convergence properties is still an open problem