On the Problem of Computing the Probability of Regular Sets of Trees

TL;DR

This paper develops an algorithm to compute the probability of regular tree languages under a coin-flip measure, revealing properties like irrational probabilities and measure-zero sets, with implications for automata theory.

Contribution

It introduces a novel algorithm for calculating probabilities of regular tree languages, applicable to deterministic automata, and demonstrates new measure-theoretic properties of these languages.

Findings

Existence of regular sets with irrational probability

Comeager regular sets can have probability zero

Probability of game languages depends on parity of parameters

Abstract

We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game automata}. In particular this algorithm is applicable to all deterministic automata. We then use the algorithm to prove through examples three properties of measure: (1) there exist regular sets having irrational probability, (2) there exist comeager regular sets having probability $0$ and (3) the probability of \emph{game languages} $W_{i,k}$, from automata theory, is $0$ if $k$ is odd and is $1$ otherwise.