Some notes on the classification of shift spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts

TL;DR

This paper generalizes the definitions of shifts of finite type and sofic shifts to infinite alphabets and other monoids, introducing finitely defined shifts and connecting them to variable length Markov chains.

Contribution

It proposes new, adaptable definitions for shifts of finite type and sofic shifts applicable in broader symbolic dynamics contexts, including infinite alphabets.

Findings

Classical definitions do not extend to infinite alphabets or other monoids.

Introduces finitely defined shifts as a new class intersecting with sofic shifts.

Links shifts of variable length to topological variable length Markov chains.

Abstract

The aim of this article is to find appropriate definitions for shifts of finite type and sofic shifts in a general context of symbolic dynamics. We start showing that the classical definitions of shifts of finite type and sofic shifts, as they are given in the context of finite-alphabet shift spaces on the one-dimensional monoid $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, do not fit for shift spaces over infinite alphabet or on other monoids. Therefore, by examining the core features in the classical definitions of shifts of finite type and sofic shifts, we propose general definitions that can be used in any context. The alternative definition given for shifts of finite type inspires the definition of a new class of shift spaces which intersects with the class of sofic shifts and includes shifts of finite type. This new class is named finitely defined shifts, and the non-finite-type shifts in it are named shifts of variable length. For the specific case of infinite-alphabet shifts on the lattice $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, shifts of variable length can be interpreted as the topological version of variable length Markov chains.