Revisiting the Notion of Independence in Asynchronous Cellular Automata — A Study on s-Skewed Automata

Abstract

In this study, we examine the notion of atomicity property in fully asynchronous cellular automata where two neighboring cells are not allowed to get updated together. In this direction, after breaking the assumption of atomicity property, we introduce the s-skewed asynchronous updating scheme where we allow s number of neighboring cells to get updated together for s∈[1,n]. As a special case, s=1 reports the fully asynchronous update, and s=n shows the traditional synchronous update. Here, we classify the dynamics of Elementary Cellular Automata (ECA) under s-skewed updating scheme following a qualitative (i.e. space–time diagram) and quantitative (i.e. density, activity, Kolmogorov–Sinai entropy) experimental approach. According to the results, some ECA rules show strong resistance against this s-skewed perturbation. However, most ECA rules report abrupt phase transition (i.e. discontinuity), continuous phase transition, and class transition dynamics. Next, based on the two-dimensional density surface, we classify the phase transition dynamics of the ECA system. To understand the abrupt phase change in the microscopic view, we further introduce the notion of a correlated s-skewed updating scheme. Moreover, we compare the phase transition dynamics of the s-skewed system with α-asynchronous cellular automata. Apart from the phase transition dynamics, here, the s-skewed system reports a variety of the following class transition dynamics: (a) There are situations where a simple locally chaotic rule can show chaotic dynamics with the effect of s-skewed perturbation

(b) Similarly, a chaotic rule is capable of generating simple periodic or fixed point dynamics with the effect of this s-skewed perturbation. © 2025 World Scientific Publishing Company.