Generalized Linear Cellular Automata in Groups and Difference Galois Theory

TL;DR

This paper explores generalized linear cellular automata modeled as difference equations within groups, applying Galois theory and Fourier analysis to understand their structure and solutions.

Contribution

It introduces a Galois-theoretic framework for analyzing non-autonomous linear cellular automata on groups, linking difference equations with Fourier transforms.

Findings

Galois theory provides insights into the solvability of cellular automata systems.

Fourier transform simplifies the analysis of convolution equations in groups.

New connections between cellular automata and algebraic difference equations are established.

Abstract

Generalized non-autonomous linear celullar automata are systems of linear difference equations with many variables that can be seen as convolution equations in a discrete group. We study those systems from the stand point of the Galois theory of difference equations and discrete Fourier transform.