Impact of (a)Synchronism on ECA: Towards a New Classification
Isabel Donoso-Leiva, Eric Goles, Martin Rios-Wilson, Sylvain Sene
TL;DR
This work addresses how (a)synchronism affects elementary cellular automata by analyzing five periodic update modes and introducing a maximal limit-cycle-length metric $L_{\max}$. The authors combine rigorous theoretical classification with a numerical density/energy framework to show three dynamical regimes: $\Theta(1)$, $\mathcal{O}(n)$, and dramatic $\Omega(2^{\sqrt{n\log n}})$ growth under certain update schemes, with BP and LC updates typically driving greater complexity. They demonstrate that many rules are update-schedule-invariant, while others (notably some class II–IV ECAs) exhibit strong sensitivity to update mode, sometimes yielding superpolynomial cycles. The findings have implications for understanding how asynchronous temporal organization influences distributed computation, signaling which update patterns can elevate dynamical complexity in simple rule-based systems.
Abstract
In this paper, we study the effect of (a)synchronism on the dynamics of elementary cellular automata. Within the framework of our study, we choose five distinct update schemes, selected from the family of periodic update modes: parallel, sequential, block-sequential, block-parallel, and local clocks. Our main measure of complexity is the maximum period of the limit cycles in the dynamics of each rule. In this context, we present a classification of the ECA rule landscape. We classified most elementary rules into three distinct regimes: constant, linear, and superpolynomial. Surprisingly, while some rules exhibit more complex behavior under a broader class of update schemes, others show similar behavior across all the considered update schemes. Although we are able to derive upper and lower bounds for the maximum period of the limit cycles in most cases, the analysis of some rules remains open. To complement the study of the 88 elementary rules, we introduce a numerical simulation framework based on two main measurements: the energy and density of the configurations. In this context, we observe that some rules exhibit significant variability depending on the update scheme, while others remain stable, confirming what was observed as a result of the classification obtained in the theoretical analysis.