A Max-Plus Model of Asynchronous Cellular Automata

TL;DR

This paper introduces a max-plus algebraic framework for asynchronous cellular automata, demonstrating that asynchronous and synchronous CA can produce equivalent long-term behaviors, with potential applications in modeling natural patterns.

Contribution

It establishes a bijective mapping between asynchronous and synchronous CA using max-plus algebra, providing a new, deterministic, and efficient timing model for cellular automata.

Findings

Asynchronous CA can replicate synchronous CA outputs when viewed over real-time contours.

The max-plus algebraic model simplifies the analysis of long-term behavior, which is deterministic and periodic.

The framework offers a more accurate and efficient timing mechanism for natural pattern modeling.

Abstract

This paper presents a new framework for asynchrony. This has its origins in our attempts to better harness the internal decision making process of cellular automata (CA). Thus, we show that a max-plus algebraic model of asynchrony arises naturally from the CA requirement that a cell receives the state of each neighbour before updating. The significant result is the existence of a bijective mapping between the asynchronous system and the synchronous system classically used to update cellular automata. Consequently, although the CA outputs look qualitatively different, when surveyed on "contours" of real time, the asynchronous CA replicates the synchronous CA. Moreover, this type of asynchrony is simple - it is characterised by the underlying network structure of the cells, and long-term behaviour is deterministic and periodic due to the linearity of max-plus algebra. The findings lead us to proffer max-plus algebra as: (i) a more accurate and efficient underlying timing mechanism for models of patterns seen in nature, and (ii) a foundation for promising extensions and applications.