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

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 filter automata, followed by solutions in elementary CA (ECA) domain with the famous universal ECA Rule 110, and reporting a number of new solitonic collisions in ECA 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 φR9maj: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.

The Spreading of Information in Online Social Networks through Cellular Automata

“…Cellular automata (CA) [34,35] proposed by Von Neumann can overcome the above drawbacks and have been used by several researchers as an alternative method of modeling epidemics. In this model every cell of the grid represents an individual, updating state of each cell by state transition criterion [36,37].…”

On the Dynamics of Cellular Automata with Memory

“…., etc [5,6]. Evolution rules representation for ECAM in this paper is given in [16,17,19] as follows: φ CARm:τ where CAR is the decimal notation of a particular ECA rule and m is the kind of memory used with a specific value of τ . This way, for example, the majority memory (maj) incorporated in ECA rule 30 employing five steps of a cell's history (τ = 5) is denoted simply as: φ R30maj:5 .…”

Complete Characterization of Structure of Rule 54

“…In [11] we show how a number of solitonic collisions can be simulated in rule 54. These solitons can be manipulated to develop some basic computable systems, such as simple substitution systems.…”