Universality Frontier for Asynchronous Cellular Automata

TL;DR

This paper explores the computational capabilities of asynchronous cellular automata, introducing flip automata networks that are robust under asynchronous updates and identifying the minimal universal automata in various dimensions.

Contribution

It introduces flip automata networks for asynchronous robustness and establishes new minimal universal automata in one and two dimensions.

Findings

Asynchronous automata can simulate synchronous ones with linear memory overhead.

All one-way asynchronous automata are non-universal.

Existence of smallest known universal automata in 1D and 2D.

Abstract

In this work, we investigate the computational aspects of asynchronous cellular automata (ACAs), a modification of cellular automata in which cells update independently, following an asynchronous schedule. We introduce flip automata networks (FAN), a simple modification of automata networks that remain robust under any asynchronous update schedule. We show that asynchronous automata can efficiently simulate their synchronous counterparts with a linear memory overhead, which improves upon the previously established quadratic bound. Additionally, we address the universality gap for (a)synchronous cellular automata -- the boundary separating universal and non-universal automata, which is still not fully understood. We tighten this boundary by proving that all one-way asynchronous automata lack universal computational power. Conversely, we establish the existence of a universal 6-state first-neighbor automaton in one dimension and a 3-state von Neumann automaton in two dimensions, which represent the smallest known universal constructions to date.