Decidable problems in substitution shifts

TL;DR

This paper studies the structure of substitution shifts, including non-minimal and erasing morphisms, and proves the decidability of key properties like aperiodicity, recognizability, and minimality within these systems.

Contribution

It introduces a comprehensive analysis of general substitution shifts and establishes the decidability of several fundamental properties for these systems.

Findings

Decidability of aperiodicity in substitution shifts

Decidability of recognizability in substitution shifts

Decidability of minimality under certain conditions

Abstract

In this paper, we investigate the structure of the most general kind of substitution shifts, including non-minimal ones, and allowing erasing morphisms. We prove the decidability of many properties of these morphisms with respect to the shift space generated by iteration, such as aperiodicity, recognizability and (under an additional assumption) irreducibility, or minimality.