N-block presentations and decidability of direct conjugacy between Subshifts of Finite Type

TL;DR

This paper studies the invertibility of N-block presentations of subshifts of finite type, providing a combinatorial characterization of maximal preimages and proving the decidability of conjugacy between such subshifts.

Contribution

It introduces a combinatorial characterization of maximal preimages and establishes the decidability of direct conjugacy between subshifts of finite type.

Findings

Existence of a maximal preimage characterized combinatorially.

Decidability of conjugacy between subshifts of finite type.

Algorithmic approach to determine conjugacy via block presentations.

Abstract

We consider the problem of inverting the transformation which consists in replacing a word by the sequence of its blocks of length N, i.e. its so-called N-block presentation. It was previously shown that among all the possible preimages of an N-block presentation, there exists a particular one which is maximal in the sense that all the other preimages can be obtained from it by letter to letter applications. We give here a combinatorial characterization of the maximal preimages of N-block presentations. Using this characterization, we show that, being given two subshifts of finite type X and Y, the existence of two numbers N and M such that the N-block presentation of X is similar to the M-block presentation of Y, which implies that X and Y are conjugate, is decidable.