Branching Frequency and Markov Entropy of Repetition-Free Languages

TL;DR

This paper introduces Markov entropy as a new measure for factorial languages, relating it to language growth and branching, with methods for computation and experimental analysis on specific language classes.

Contribution

It defines Markov entropy for factorial languages, links it to growth and branching, and provides computational and experimental frameworks for its analysis.

Findings

Markov entropy can be computed for regular languages.

Experimental results on power-free languages.

Framework for studying entropy in repetition-free languages.

Abstract

We define a new quantitative measure for an arbitrary factorial language: the entropy of a random walk in the prefix tree associated with the language; we call it Markov entropy. We relate Markov entropy to the growth rate of the language and to the parameters of branching of its prefix tree. We show how to compute Markov entropy for a regular language. Finally, we develop a framework for experimental study of Markov entropy by modelling random walks and present the results of experiments with power-free and Abelian-power-free languages.