An Easily Checkable Algebraic Characterization of Positive Expansivity for Additive Cellular Automata over a Finite Abelian Group

TL;DR

This paper introduces an algebraic method to efficiently determine positive expansivity in additive cellular automata over finite abelian groups, simplifying analysis for both linear and general cases.

Contribution

It provides a straightforward algebraic characterization for positive expansivity, applicable to all additive cellular automata over finite abelian groups, including a specific approach for linear automata.

Findings

Algebraic criteria for positive expansivity in linear cellular automata

Extension of criteria to all additive cellular automata over finite abelian groups

Efficient decision procedure for positive expansivity

Abstract

We provide an easily checkable algebraic characterization of positive expansivity for Additive Cellular Automata over a finite abelian group. First of all, an easily checkable characterization of positive expansivity is provided for the non trivial subclass of Linear Cellular Automata over the alphabet $(\Z/m\Z)^n$. Then, we show how it can be exploited to decide positive expansivity for the whole class of Additive Cellular Automata over a finite abelian group.