Complete Characterization of Structure of Rule 54

TL;DR

This paper thoroughly characterizes the structure and dynamics of Rule 54 cellular automaton, including glider types, regular expression encoding, and collision behaviors, revealing its complex self-organizing properties.

Contribution

It introduces a comprehensive catalog of gliders, regular expression encoding, and an abstract recognition machine for Rule 54, advancing understanding of its computational and self-organizing capabilities.

Findings

Catalog of primitive gliders and their tiles

Regular expressions encoding gliders derived from de Bruijn diagrams

Demonstration of soliton-like glider interactions and self-organization

Abstract

The dynamics of rule 54 one-dimensional two-state cellular automaton (CA) are a discrete analog of a space-time dynamics of excitations in nonlinear active medium with mutual inhibition. A cell switches its state 0 to state 1 if one of its two neighbors is in state 1 (propagation of a perturbation) and a cell remains in state 1 only if its two neighbors are in state 0. A lateral inhibition is because a 1-state neighbor causes a 1-state cell to switch to state 0. The rule produces a rich spectrum of space-time dynamics, including gliders and glider guns just from four primitive gliders. We construct a catalogue of gliders and describe them by tiles. We calculate a subset of regular expressions $\Psi_{R54}$ to encode gliders. The regular expressions are derived from de Bruijn diagrams, tile-based representation of gliders, and cycle diagrams sometimes. We construct an abstract machine that recognizes regular expressions of gliders in rule 54 and validate $\Psi_{R54}$. We also propose a way to code initial configurations of gliders to depict any type of collision between the gliders and explore self-organization of gliders, formation of larger tiles, and soliton-like interactions of gliders and computable devices.