A Characterization of Cellular Automata Generated by Idempotents on the Full Shift

TL;DR

This paper characterizes cellular automata on the full shift generated by idempotent automata, providing a decomposition criterion, exploring examples, and addressing decidability questions within this class.

Contribution

It offers a new characterization of products of idempotent cellular automata and investigates their properties and decidability issues.

Findings

Characterization of products of idempotent CA

Examples of CA satisfying the characterization

Decidability questions for the class of products of idempotent CA

Abstract

In this article, we discuss the family of cellular automata generated by so-called idempotent cellular automata (CA G such that G^2 = G) on the full shift. We prove a characterization of products of idempotent CA, and show examples of CA which are not easy to directly decompose into a product of idempotents, but which are trivially seen to satisfy the conditions of the characterization. Our proof uses ideas similar to those used in the well-known Embedding Theorem and Lower Entropy Factor Theorem in symbolic dynamics. We also consider some natural decidability questions for the class of products of idempotent CA.