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A Uniform Framework for Language Inclusion Problems

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Taming the Infinities of Concurrency

Abstract

We present a uniform approach for solving language inclusion problems. Our approach relies on a least fixpoint characterization and a quasiorder to compare words of the “smaller” language, reducing the inclusion check to a finite number of membership queries in the “larger” language. We present our approach in detail on the case of inclusion of a context-free language given by a grammar into a regular language. We then explore other inclusion problems and discuss how to apply our approach.

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Notes

  1. 1.

    Straight-line program are context-free grammars where at most one word is derived from each grammar variable.

  2. 2.

    \(f\) is continuous iff \(f\) preserves least upper bounds of nonempty increasing chains.

  3. 3.

    We can even relax the inclusion \(S\subseteq L({\mathcal {G}})\) to the weaker condition \(S \sqsubseteq _{\ltimes } L({\mathcal {G}})\).

  4. 4.

    This is equivalent to saying that M is upward-closed w.r.t. the quasiorder \(\mathord {\ltimes } \).

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Acknowledgments

Pierre visited Javier during his first year of PhD, a visit that turned out to be a milestone in Pierre’s career and has had influence up to this day. This visit also got Pierre a new colleague, a mentor and, most importantly, a friend. Pierre wishes to thank Javier from the bottom of his heart for all the good memories throughout the years. Chana is grateful to have had Javier as a PhD advisor and as an academic role model. She learned a lot from him, and benefited from the kind and studious atmosphere that he has established in his Chair at the Technical University of Munich. We also are thankful to the reviewers for their valuable feedback. This publication is part of the grant PID2022-138072OB-I00, funded by MCIN, FEDER, UE and has been partially supported by PRODIGY Project (TED2021-132464B-I00) funded by MCIN and the European Union NextGeneration.

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Doveri, K., Ganty, P., Weil-Kennedy, C. (2024). A Uniform Framework for Language Inclusion Problems. In: Kiefer, S., Křetínský, J., Kučera, A. (eds) Taming the Infinities of Concurrency. Lecture Notes in Computer Science, vol 14660. Springer, Cham. https://doi.org/10.1007/978-3-031-56222-8_9

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