Closing star-free closure

TL;DR

This paper introduces the star-free closure operator on regular language classes, providing multiple characterizations and decidability results for membership, separation, and covering problems within this framework.

Contribution

It defines and characterizes the star-free closure operator, linking it to various logical and algebraic frameworks, and establishes decidability results for membership, separation, and covering problems.

Findings

Star-free closure coincides with a closure operator defined via regular operations with restricted Kleene stars.

Membership in the star-free closure is decidable for classes with decidable separation.

Separation and covering problems are decidable for the star-free closure of finite classes and certain group language classes.

Abstract

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class $C$, the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of $C$ as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted~contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.