Impact of (a)Synchronism on ECA: Towards a New Classification

TL;DR

This paper investigates how different update schemes affect the dynamics of elementary cellular automata, classifying rules into regimes based on their complexity and analyzing their behavior through theoretical bounds and numerical simulations.

Contribution

It introduces a new classification of ECA rules based on their dynamic complexity under various update schemes and provides bounds and simulations to analyze their behavior.

Findings

Most rules fall into constant, linear, or superpolynomial regimes.

Some rules show behavior stability across update schemes, others vary significantly.

The study offers bounds for limit cycle periods and a simulation framework for analysis.

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.