The Commutative Closure of Shuffle Expressions over Group Languages is Regular

TL;DR

This paper proves that applying commutative closure and shuffle operations to group languages results in regular languages, providing bounds for automata size and extending to complex shuffle expressions.

Contribution

It establishes that the commutative closure combined with shuffle operations preserves regularity for group languages and extends this to complex shuffle expressions.

Findings

Commutative closure with shuffle preserves regularity in group languages.

Bounds are provided for the size of minimal recognizing automata.

Any shuffle expression over group languages yields a regular language.

Abstract

We show that the commutative closure combined with the iterated shuffle is a regularity-preserving operation on group languages. In particular, for commutative group languages, the iterated shuffle is a regularity-preserving operation. We also give bounds for the size of minimal recognizing automata. Then, we use these results to deduce that the commutative closure of any shuffle expression over group languages, i.e., expressions involving shuffle, iterated shuffle, concatenation, Kleene star and union in any order, starting with the group languages, always yields a regular language.