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Dynamic Membership for Regular Tree Languages

Authors Antoine Amarilli , Corentin Barloy , Louis Jachiet , Charles Paperman

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ACM Subject Classification
  • Theory of computation → Regular languages
Keywords
  • automaton
  • dynamic membership
  • incremental maintenance
  • forest algebra

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Abstract

We study the dynamic membership problem for regular tree languages under relabeling updates: we fix an alphabet Σ and a regular tree language L over Σ (expressed, e.g., as a tree automaton), we are given a tree T with labels in Σ, and we must maintain the information of whether the tree T belongs to L while handling relabeling updates that change the labels of individual nodes in T.
Our first contribution is to show that this problem admits an O(log n / log log n) algorithm for any fixed regular tree language, improving over known O(log n) algorithms. This generalizes the known O(log n / log log n) upper bound over words, and it matches the lower bound of Ω(log n / log log n) from dynamic membership to some word languages and from the existential marked ancestor problem. 
Our second contribution is to introduce a class of regular languages, dubbed almost-commutative tree languages, and show that dynamic membership to such languages under relabeling updates can be decided in constant time per update. Almost-commutative languages generalize both commutative languages and finite languages: they are the analogue for trees of the ZG languages enjoying constant-time dynamic membership over words. Our main technical contribution is to show that this class is conditionally optimal when we assume that the alphabet features a neutral letter, i.e., a letter that has no effect on membership to the language. More precisely, we show that any regular tree language with a neutral letter which is not almost-commutative cannot be maintained in constant time under the assumption that the prefix-U1 problem from [Antoine Amarilli et al., 2021] also does not admit a constant-time algorithm.

Cite As Get BibTex

Antoine Amarilli, Corentin Barloy, Louis Jachiet, and Charles Paperman. Dynamic Membership for Regular Tree Languages. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.8

Author Details

Antoine Amarilli
  • Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, France
Corentin Barloy
  • Ruhr University Bochum, Germany
Louis Jachiet
  • Télécom Paris, Institut Polytechnique de Paris, France
Charles Paperman
  • Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, France

Funding

  • Amarilli, Antoine: Partially supported by the ANR project EQUUS ANR-19-CE48-0019, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 431183758, and by the ANR project ANR-18-CE23-0003-02 (“CQFD”).
  • Barloy, Corentin: Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 532727578.

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