Shuffles of Context-Free Languages along Regular Trajectories

2026-03-27

AI Summary

This work investigates the closure properties of context-free languages (CFLs) under shuffle operations constrained by regular trajectories, with a particular focus on non-regular CFLs and deterministic CFLs (DCFLs). By leveraging formal language theory and trajectory automaton modeling, the paper proposes a general method to decide whether a given trajectory preserves the CFL or DCFL structure. Building upon the expressiveness lemma of Jančar and Šíma (MFCS’2021), it develops key analytical tools to establish sufficient conditions under which CFLs fail to be closed under such shuffles. The study fully characterizes three distinct behavioral patterns of DCFLs under regular-trajectory shuffling and uncovers their deep connections to scheduling semantics, thereby advancing the theoretical foundations at the intersection of concurrency theory and formal languages.

Abstract

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).

Problem

• shuffle
• context-free languages
• regular trajectories
• deterministic CFLs
• closure properties

Innovation

• shuffle
• context-free languages
• regular trajectories
• deterministic CFL
• closure properties