Regularity Conditions for Iterated Shuffle on Commutative Regular Languages

TL;DR

This paper characterizes when the iterated shuffle operation preserves regularity within certain subclasses of commutative regular languages, identifying conditions and classes where regularity is maintained.

Contribution

It introduces a subclass of commutative regular languages closed under iterated shuffle, providing a clear characterization and extending understanding of regularity preservation.

Findings

Identifies a subclass of commutative languages closed under iterated shuffle.

Provides a characterization for regularity preservation under iterated shuffle.

Shows closure properties of aperiodic and group languages under projection.

Abstract

We identify a subclass of the regular commutative languages that is closed under the iterated shuffle, or shuffle closure. In particular, it is regularity-preserving on this subclass. This subclass contains the commutative group languages and, for every alphabet $\Sigma$, the class $\textbf{Com}^+(\Sigma^*)$ given by the ordered variety $\textbf{Com}^+$. Then, we state a simple characterization when the iterated shuffle on finite commutative languages gives a regular language again and state partial results for aperiodic commutative languages. We also show that the aperiodic, or star-free, commutative languages and the commutative group languages are closed under projection.