Decidability of irreducible tree shifts of finite type

TL;DR

This paper presents an algorithm to determine CPC-irreducibility of dyadic tree shifts of finite type by relating it to the connectivity of an extended graph representation, extending known results from one-dimensional shifts.

Contribution

It introduces a novel graph-based method for deciding CPC-irreducibility in tree shifts, expanding the understanding of their structural properties.

Findings

CPC-irreducibility is characterized by graph connectivity.

An extended directed graph representation effectively captures TSFT properties.

The algorithm provides a practical decision procedure for CPC-irreducibility.

Abstract

We reveal an algorithm for determining the complete prefix code irreducibility (CPC-irreducibility) of dyadic trees labeled by a finite alphabet. By introducing an extended directed graph representation of tree shift of finite type (TSFT), we show that the CPC-irreducibility of TSFTs is related to the connectivity of its graph representation, which is a similar result to one-dimensional shifts of finite type.