On the Reversibility of ECAs with Fully Asynchronous Updating: The Recurrence Point of View
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).
The authors have benefitted from the careful reading of the manuscript by Jordina Francès de Mas and from helpful remarks of the reviewers.
- 2.Das, S., Sarkar, A., Sikdar, B.K.: Synthesis of reversible asynchronous cellular automata for pattern generation with specific hamming distance. In: Proceedings of ACRI’12, pp. 643–652. Springer (2012)Google Scholar
- 7.Sethi, B., Fatès, N., Das, S.: Reversibility of elementary cellular automata under fully asynchronous update. In: Gopal, T.V., Agrawal, M., Li, A., Barry Cooper, S. (eds.) Proceedings of TAMC 2014. Lecture Notes in Computer Science, vol. 8402, pp. 39–49. Springer (2014)Google Scholar