Sofic measures

TL;DR

This paper surveys the concept of sofic measures, also known as hidden Markov measures, and improves bounds for determining their Markov properties based on linear representations.

Contribution

It provides a survey of equivalent definitions of sofic measures and establishes a new upper bound for the Markov order of such measures with linear representations.

Findings

Improved the bound for deciding if a sofic measure is k-step Markov.

Proved that if a sofic measure with a linear representation of dimension n is k-step Markov, then k ≤ 2^{n^2-1}.

Surveyed equivalent definitions of sofic measures.

Abstract

Sofic measures, also known as hidden Markov measures, have been extensively studied. In this paper, we survey some equivalent definitions of this notion and improve a bound for deciding whether a sofic measure is~$k$-step Markov. We prove that if an invariant sofic measure with a linear representation of dimension~$n$ is a~$k$-step Markov chain, then~$k$ can be chosen at most equal to~$2^{n^2-1}$.