Coalescing Cellular Automata -- Synchronizing CA by Common Random Source and Varying Asynchronicity

TL;DR

This paper investigates the coalescence behavior of cellular automata under asynchronous updates, proving coalescence for some rules, non-coalescence for others, and identifying phase transitions in elementary CA related to directed percolation.

Contribution

It provides the first rigorous proofs of coalescence for certain CA rules and an extensive experimental analysis revealing phase transitions and universality classes.

Findings

Proved coalescence for two elementary CA rules.

Established non-coalescence for two other rules.

Discovered phase transitions in elementary CA related to directed percolation.

Abstract

We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We prove coalescence for two elementary rules, non coalescence for two other, and show that there exists infinitely many coalescing CA. We then conduct an experimental study on all elementary CA and show that some rules exhibit a phase transition, which belongs to the universality class of directed percolation.