On Cellular Automata

TL;DR

This paper advances the theoretical understanding of cellular automata by introducing the concept of -cellular automata, establishing new theorems, and analyzing their properties on algebraic structures like circulant graphs.

Contribution

It introduces -cellular automata, generalizes the Uniform Curtis-Hedlund Theorem, and explores covering maps and quotient covers in algebraic graph contexts.

Findings

Generalized Uniform Curtis-Hedlund Theorem for -cellular automata

Defined and analyzed covering maps for -cellular automata

Derived results for quotient covers on circulant graphs

Abstract

Cellular automata are a fundamental computational model with applications in mathematics, computer science, and physics. In this work, we explore the study of cellular automata to cases where the universe is a group, introducing the concept of \( \phi \)-cellular automata. We establish new theoretical results, including a generalized Uniform Curtis-Hedlund Theorem and linear \( \phi \)-cellular automata. Additionally, we define the covering map for \( \phi \)-cellular automata and investigate its properties. Specifically, we derive results for quotient covers when the universe of the automaton is a circulant graph. This work contributes to the algebraic and topological understanding of cellular automata, paving the way for future exploration of different types of covers and their applications to broader classes of graphs and dynamical systems.