The n-ary Initial Literal and Literal Shuffle

TLDR: In this article, the authors extend the literal shuffle and the initial literal shuffle to multiple arguments and introduce iterated versions, much different from the iterated ones previously introduced for the binary literal and initial literal shuffles.

Abstract: The literal and the initial literal shuffle have been introduced to model the behavior of two synchronized processes. However, it is not possible to describe the synchronization of multiple processes. Furthermore, both restricted forms of shuffling are not associative. Here, we extend the literal shuffle and the initial literal shuffle to multiple arguments. We also introduce iterated versions, much different from the iterated ones previously introduced for the binary literal and initial literal shuffle. We investigate formal properties, and show that in terms of expressive power, in a full trio, they coincide with the general shuffle. Furthermore, we look at closure properties with respect to the regular, context-free, context-sensitive, recursive and recursively enumerable languages for all operations introduced. Then, we investigate various decision problems motivated by analogous problems for the (ordinary) shuffle operation. Most problems we look at are tractable, but we also identify one intractable decision problem.