Extended Cellular Automata

TL;DR

The paper extends one-dimensional cellular automata by introducing Zip-CA, a two-alphabet, two-rule framework that enables coupled evolution of two related phenomena using a transition map $\tau$ between alphabets $\mathcal{S}$ and $\mathcal{Z}$. It formalizes zip-shift spaces $\Sigma_{\mathcal{Z},\mathcal{S}}$ and the zip-shift map $\sigma_{\tau}$, and establishes a Zip-SBC/Zip-CA framework with a local rule that respects a two-sided symbolic-dynamical structure. The central contribution is the Extended Curtis–Hedlund–Lyndon Theorem, which characterizes Zip-CA as precisely the continuous maps that locally commute with zip-shift maps; this yields a constructive proof via a finite-block local rule of length $N=m+n+1$. A practical construction approach is provided by composing a classical CA $R_1$ with a second-layer map $R_2$, forming $R=R_2\circ R_1$, and several examples demonstrate how to merge ECA rules across two interrelated regimes, highlighting the potential for co-analysis of two interacting processes in complex systems.

Abstract

In this work, the one-dimensional Cellular Automaton is extended to one that involves two sets of symbols and two global rules. As a main result, the Extended Curtis-Hedlund-Lyndon Theorem is demonstrated. Such constructions can be useful in studying complex systems involving two related phenomena and provide a way to their co-study.

Figures (3)

• Figure 1: An example of a new generation of one-dimensional CA.
• Figure 2: Zip-CA: made from rules of classes III and IV.
• Figure 3: Some Zip-CAs made from different Wolfram classes

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7
• proof : Proof of Theorem \ref{['teo:local homo']}
• Remark 2.8
• Example 2.9