On all things star-free

TL;DR

This paper studies the star-free closure of language classes, proving decidability of separation and covering properties for classes with certain algebraic and computational features, unifying previous results.

Contribution

It demonstrates that the star-free closure of finite and certain group language classes has decidable separation and covering, extending and unifying existing results.

Findings

Decidability of separation for star-free closures of specific language classes.

Decidability of covering property in the same context.

Equivalence of star-free closure with a restricted Kleene star closure.

Abstract

We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with decidable separation (plus mild additional properties) has decidable separation. We actually show decidability of a stronger property, called covering. This generalizes many results on the subject in a unified framework. A key ingredient is that star-free closure coincides with another closure operator where Kleene stars are also allowed in restricted contexts.