Progresses in the Analysis of Stochastic 2D Cellular Automata: a Study of Asynchronous 2D Minority

TL;DR

This paper analyzes the complex behavior of a simple 2D asynchronous cellular automaton, the Minority rule, revealing its asymptotic dynamics and proposing methods potentially extendable to other threshold automata.

Contribution

It provides the first comprehensive analysis of the 2D Minority automaton under asynchronous updates, including a complete description of its asymptotic behavior.

Findings

The 2D Minority automaton exhibits complex responses to asynchronism.

Asymptotic behavior can be fully characterized under certain initial conditions.

Energy-based techniques may extend to broader classes of threshold automata.

Abstract

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on one-dimensional probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from one- to two-dimensional systems is a long lasting challenging problem. In this paper, we address the problem of analysing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our experiments reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.