The chain relation in sofic subshifts

TL;DR

This paper characterizes the chain relation in sofic subshifts using graph factorization, providing insights into their dynamics and algorithms to decide properties like chain-transitivity and chain-mixing.

Contribution

It introduces a graph-based method to analyze the chain relation in sofic subshifts and offers algorithms for property decision and attractor listing.

Findings

Characterization of the chain relation via graph factorization.

Algorithms for deciding chain-transitivity and chain-mixing.

Method to list all attractors of the subshift system.

Abstract

The paper gives a characterisation of the chain relation of a sofic subshift. Every sofic subshift $\Sigma$ can be described by a labelled graph $G$. Factorising $G$ in a suitable way we obtain the graph $G/_\approx$ that offers insight into some properties of the original subshift. Using $G/_\approx$ we describe first the chain relation in $\Sigma$, then characterise chain-transitive sofic subshifts, chain-mixing sofic subshifts and finally the attractors of the subshift dynamic system. At the end we present (straightforward) algorithms deciding chain-transitivity and chain-mixing properties of a sofic subshift and listing all the attractors of the subshift system.