On Computing the Measures of First-Order Definable Sets of Trees

TL;DR

This paper investigates the computability of measures for regular languages of infinite trees, showing that certain classes have rational, computable measures, while others can have irrational measures.

Contribution

It demonstrates that measures of languages defined by specific first-order formulas are rational and computable, and provides an example of an irrational measure.

Findings

Measures of languages defined by first-order formulas without descendant relation are rational and computable.

Languages defined by Boolean combinations of conjunctive queries with descendant relation are rational and computable.

An example of a language with an irrational measure is provided.

Abstract

We consider the problem of computing the measure of a regular language of infinite binary trees. While the general case remains unsolved, we show that the measure of a language defined by a first-order formula with no descendant relation or by a Boolean combination of conjunctive queries (with descendant relation) is rational and computable. Additionally, we provide an example of a first-order formula that uses descendant relation and defines a language of infinite trees having an irrational measure.