On the Self Shuffle Language

TL;DR

This paper investigates the properties of the self shuffle language, showing it is context sensitive but not context free, and that self shuffle uniquely determines the original word.

Contribution

It introduces the concept of self shuffle language, proves its complexity class, and establishes the uniqueness of the original word from its self shuffle.

Findings

Self shuffle language is context sensitive but not context free.

Self shuffle of a word uniquely determines the original word.

Self shuffle and shuffle of a language are different sets.

Abstract

The shuffle product \(u\shuffle v\) of two words \(u\) and \(v\) is the set of all words which can be obtained by interleaving \(u\) and \(v\). Motivated by the paper \emph{The Shuffle Product: New Research Directions} by Restivo (2015) we investigate a special case of the shuffle product. In this work we consider the shuffle of a word with itself called the \emph{self shuffle} or \emph{shuffle square}, showing first that the self shuffle language and the shuffle of the language are in general different sets. We prove that the language of all words arising as a self shuffle of some word is context sensitive but not context free. Furthermore, we show that the self shuffle \(w \shuffle w\) uniquely determines \(w\).