Diagrammatic approach to cellular automata and the emergence of form with inner structure

TL;DR

This paper introduces a diagrammatic method for constructing and classifying cellular automata, demonstrating how complex structures and order can emerge from disorder through layered modeling.

Contribution

The paper presents a novel diagrammatic approach to build and classify cellular automata, applicable to any dimension or topology, enhancing understanding of emergent order in complex systems.

Findings

Diagrammatic method effectively classifies CA rule space.

Majority rule as a building block for complex CA structures.

Emergence of stable, coexisting domains from disorder.

Abstract

We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allow CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time, arbitrary neighborhood ranges and topology. We provide several examples of the method, illustrating how it can be applied to the mathematical modeling of the emergence of order out of disorder. Specifically, we show how the the majority CA rule can be used as a building block to construct more complex cellular automata in which separate domains (with substructures having different dynamical properties) are able to emerge out of disorder and coexist in a stable manner.