On the Reversibility of ECAs with Fully Asynchronous Updating: The Recurrence Point of View

Abstract

The reversibility of classical cellular automata is now a well-studied topic but what is reversibility when the evolution of the system is stochastic? In this context, we study a particular form of reversibility: the possibility of returning infinitely often to the initial condition after a random number of time steps. This corresponds to the recurrence property of the system. We analyse this property for the 256 elementary cellular automata with a finite size and a fully asynchronous updating, that is, we update only one cell, randomly chosen, at each time step. We show that there are 46 recurrent rules which almost surely come back to their initial condition. We analyse the structure of the communication graph of the system and find that the number of the communication classes may have different scaling laws, depending on the active transitions of the rules (those for which the state of the cell is modified when an update occurs).

A tutorial on elementary cellular automata with fully asynchronous updating

“…This suggests that the class of non-converging rules should be studied with more precise criteria and that for these rules, we need to understand the communication graph, that is, the graph where links between configurations model the possibility to go from one configuration to the other in one step. We refer to a recent work on this topic for a more detailed analysis [FSD18].…”

Reversibility of elementary cellular automata with fully asynchronous updating: An analysis of the rules with partial recurrence

“…To sum up, with this last series of results, the situation of the 256 ECA is now fully covered with regard to their main properties of recurrence. We extended the techniques presented for recurrent systems [19,8,7,18] to the case where the recurrence was only partial. We fully characterised the 21 remaining non-recurrent non-convergent ECA rules, counting their number of recurrent configurations and number of recurrent classes.…”

“…In our previous work [7,18], we have identified the ECAs which are recurrent (see Theorem 1 below). Here, we have R(n) = 2 n .…”

“…We will denote each ECA f with two complementary codes: a) its classical decimal code 7 . and, b) its transition code T(f ), which consists of the concatenation of letters in {A, • • • , H} which denote the active transitions of f according to the code presented in Table 1.…”

Asynchronous Cellular Automata

“…The problem then amounts to deciding the recurrence property of the Markov chain. This allows the authors to propose a partition of the Elementary Cellular Automata according to their recurrence properties and to show that among the 88 non-equivalent rules, there are 16 rules which are recurrent for any ring size greater than two and 2 rules which are recurrent for ring sizes greater than three [FSD17]. These rules are listed in Tab.…”