Role of atomicity in phase transitions under asynchronous updates

Ascertain whether the atomicity property—prohibiting simultaneous enablement of neighboring cells during state updates—constitutes a causal factor in the phase transition behavior observed for elementary cellular automata rules 26, 38, 58, and 134 when comparing α‑asynchronous updating to skew‑asynchronous updating, and, if so, characterize the mechanism by which atomicity influences convergence/divergence phases.

Background

The paper compares fully asynchronous, α‑asynchronous, and skew‑asynchronous update schemes for elementary cellular automata (ECAs), focusing on how breaking the atomicity property (i.e., allowing adjacent cells to update together) impacts dynamics. The authors report that ECAs 26, 38, 58, and 134 exhibit phase changes (convergence ↔ divergence) depending on the presence or absence of atomicity.

They reference prior work showing phase transitions for α‑asynchronous updates in these rules and suggest that atomicity may be involved, but explicitly state that determining its role remains unresolved.