Pattern Formation without Favored Local Interactions

TL;DR

This paper analyzes the ensemble of all elementary cellular automata rules, revealing persistent patterns and collective behaviors that enable complex self-assembly without specific local interaction preferences.

Contribution

It provides the first analysis of the ensemble dynamics of all elementary cellular automata, uncovering collective phenomena and low-dimensional representations.

Findings

Persistent localized patterns in ensemble dynamics

Traveling peaks at fractional velocities not seen in individual rules

ECA rules can be represented by approximately 111 principal components

Abstract

Individual cellular automata rules are attractive models for a range of biological and physical self-assembling systems. While coexpression and coevolution are common in such systems, ensembles of cellular automata rules remain poorly understood. Here we report the first known analysis of the equally weighted ensemble of all elementary cellular automata (ECA) rules. Ensemble dynamics reveal persistent, localized, non-interacting patterns, rather than homogenization. The patterns are strongly correlated by velocity and have a quasi-linear dependence on initial conditions. Dispersion from a single initial site generates peaks traveling at low-denominator fractional velocities, some of which are not discernible in individual rules, suggesting collective excitation. Further analysis of the time-evolved rule space shows the 256 ECA rules can be represented by only approximately 111 principal components. These results suggest the rather surprising conclusion that rich self-assembly is possible without favoring particular local interactions.