Networks of Binary Necklaces Induced by Elementary Cellular Automata Rules

TL;DR

This paper explores the structure of networks formed by binary necklaces under elementary cellular automata rules, revealing how these structures evolve with increasing necklace length and providing insights into their rotational invariance.

Contribution

It introduces a method to analyze cellular automata networks using binary necklaces, simplifying the visualization and understanding of rule-induced structures.

Findings

Network structures vary with necklace length

Rotational invariance reduces complexity

General rule patterns are revealed through necklaces

Abstract

Elementary cellular automata deterministically map a binary sequence to another using simple local rules. Visualizing the structure of this mapping is difficult because the number of nodes (i.e. possible binary sequences) grows exponentially. If periodic boundary conditions are used, rotation of a sequence and rule application to that sequence commute. This allows us to recover the rotational invariance property of loops and to reduce the number of nodes by only considering binary necklaces, the equivalence class of n-character strings taking all rotations as equivalent. Combining together many equivalent histories reveals the general structure of the rule, both visually and computationally. In this work, we investigate the structure of necklace-networks induced by the 256 Elementary Cellular Automata rules and show how their network structure change as the length of necklaces grow.