Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions

TL;DR

This paper investigates the entropy sensitivity of languages defined by infinite automata, demonstrating that forbidding certain subwords reduces the exponential growth rate, using Markov chains with forbidden transitions under general conditions.

Contribution

It proves that languages from infinite automata are growth-sensitive, employing Markov chains with forbidden transitions to establish this property.

Findings

Languages are growth-sensitive under general conditions

Markov chains with forbidden transitions are effective tools

Entropy decreases when subwords are forbidden

Abstract

A language L over a finite alphabet is growth-sensitive (or entropy sensitive) if forbidding any set of subwords F yields a sub-language L^F whose exponential growth rate (entropy) is smaller than that of L. Let (X, E, l) be an infinite, oriented, labelled graph. Considering the graph as an (infinite) automaton, we associate with any pair of vertices x,y in X the language consisting of all words that can be read as the labels along some path from x to y. Under suitable, general assumptions we prove that these languages are growth-sensitive. This is based on using Markov chains with forbidden transitions.