On Soliton Collisions between Localizations in Complex Elementary Cellular Automata: Rules 54 and 110 and Beyond

TL;DR

This paper explores soliton-like collisions in elementary cellular automata, especially Rules 54 and 110, introducing models with memory that exhibit stable mobile self-localizations and solitonic behaviors.

Contribution

It presents a novel cellular automata model with memory that reliably produces soliton collisions and self-localizations from various initial conditions, expanding understanding of solitonic phenomena in CA.

Findings

Identified new solitonic collisions in Rule 54

Developed a CA with memory that stabilizes solitons from random initial states

Demonstrated soliton behavior in Rule 110 and memory-based automata

Abstract

In this paper we present a single-soliton two-component cellular automata (CA) model of waves as mobile self-localizations, also known as: particles, waves, or gliders; and its version with memory. The model is based on coding sets of strings where each chain represents a unique mobile self-localization. We will discuss briefly the original soliton models in CA proposed with {\it filter automata}, followed by solutions in elementary CA (ECA) domain with the famous universal ECA {\it Rule 110}, and reporting a number of new solitonic collisions in ECA {\it Rule 54}. A mobile self-localization in this study is equivalent a single soliton because the collisions of these mobile self-localizations studied in this paper satisfies the property of solitonic collisions. We also present a specific ECA with memory (ECAM), the ECAM Rule $\phi_{R9maj:4}$, that displays single-soliton solutions from any initial codification (including random initial conditions) for a kind of mobile self-localization because such automaton is able to adjust any initial condition to soliton structures.