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Shuffles of Context-Free Languages Along Regular Trajectories

Authors Corentin Barloy , Michaël Cadilhac , Kyle Ockerlund

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LIPIcs.ICALP.2026.163.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Grammars and context-free languages
Keywords
  • Context-free languages
  • shuffles
  • concurrency
  • non-regularity

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Abstract

In single-core processors, concurrency requires that multiple processes be interleaved into a single thread of execution by a scheduler. The language-theoretic operation that corresponds to this is the shuffle of two languages: the set of words obtained by interleaving a word from each language in an arbitrary, letter-wise fashion. It is well known that regular languages are closed under shuffles, while context-free languages (CFLs) are not.
Following an established line of research, this paper considers shuffles according to regular "trajectories," that is, subject to scheduling constraints expressed by an automaton. Unsurprisingly, some trajectories allow for CFLs to be shuffled into CFLs (e.g., simple concatenation of the two words), while others 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 lemmata of independent interest regarding how pushdown automata (PDA) must invoke the stack when accepting a nonregular CFL or DCFL. The latter case relies on a recent result of Jančar and Šíma (MFCS'2021); answering an open question therein, it is demonstrated that said result cannot be generalized to arbitrary CFLs, leading to dedicated machinery for both cases.

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Corentin Barloy, Michaël Cadilhac, and Kyle Ockerlund. Shuffles of Context-Free Languages Along Regular Trajectories. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 163:1-163:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.163

Author Details

Corentin Barloy
  • University of Bochum, Germany
Michaël Cadilhac
  • DePaul University, Chicago, IL, USA
Kyle Ockerlund
  • Google, Sunnyvale, CA, USA

Funding

  • Barloy, Corentin: is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 532727578.

Acknowledgements

Proposition 17 is due to Georg Zetzsche. We thank Arka Ghosh and Petra Wolf for stimulating discussions on this topic.

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