Shuffles of Context-Free Languages along Regular Trajectories

In single-core processors, when multiple processes execute concurrently, they are, in practice, intertwined by a scheduler as a single thread of execution. The language-theoretic operation that corresponds to this is the shuffle of two languages: in general, this is defined as the set of words obtained by interleaving words from the first and second language in an arbitrary fashion. It is well known that regular languages are closed under shuffles, while context-free languages (CFL) are not. Following an established line of research, this paper considers shufflings according to regular "trajectories", that is, subject to scheduling constraints expressed by an automaton. Unsurprisingly, some trajectories, such as "a word from the first language first, then a word from the second", allow for CFLs to be shuffled into CFLs, while some other trajectories do not. This paper provides a robust toolset to show that a given trajectory would always shuffle two nonregular CFLs into a nonCFL. In the case of deterministic CFLs (DCFLs), a salient trichotomy of trajectories depending on how they shuffle DCFLs is provided. These results are based on intricate expressiveness lemmas for CFLs and DCFLs of independent interest, the latter lemma relying on a recent result of Jančar and Šíma (MFCS'2021).