On the Self Shuffle Language

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 \(w\).

m-Nearly k-Universal Words -- Investigating Simon Congruence

Determining the index of the Simon congruence is a long outstanding open problem. Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of $\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality, we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered factors of length $k$ are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for $m=1$. For $m\neq 1$, we show some results if in addition $w$ is $(k-1)$-universal as well as some further insights for different $m$.