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Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2026.161

Out-Of-Order Membership in Regular Languages

Authors: Antoine Amarilli, Sebastien Labbe, and Charles Paperman

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

We introduce the task of out-of-order membership to a formal language L, where the letters of a word w are revealed one by one in an arbitrary order. The length |w| is known in advance, but the content of w is streamed as pairs (i, w[i]), received exactly once for each position i, in arbitrary order. We study efficient algorithms for this task when L is regular, seeking tight complexity bounds as a function of |w| for a fixed target language. Most of our results apply to an algebraically defined variant dubbed out-of-order evaluation: this problem is defined for a fixed finite monoid or semigroup S, and our goal is to compute the ordered product of the streamed elements of w. We show that, for any fixed regular language or finite semigroup, both problems can be solved in constant time per streamed symbol and in linear space. However, the precise space complexity strongly depends on the algebraic structure of the target language or evaluation semigroup. Our main contributions are therefore to show (deterministic) space complexity characterizations, which we do for out-of-order evaluation of monoids and semigroups. For monoids, we establish a trichotomy: the space complexity is either Θ(1), Θ(log n), or Θ(n), where n = |w|. More specifically, the problem admits a constant-space solution for commutative monoids, while all non-commutative monoids require Ω(log n) space. We further identify a class of monoids admitting an O(log n)-space algorithm, and show that all remaining monoids require Ω(n) space. For general semigroups, the situation is more intricate. We characterize a class of semigroups admitting constant-space algorithms for out-of-order evaluation, and show that semigroups outside this class require at least Ω(log n) space. At the same time, we exhibit semigroups for which specialized techniques yield intermediate bounds such as an O(√n)-space algorithm, suggesting that the landscape may be richer and less well-behaved than for the monoid setting.

Cite as

Antoine Amarilli, Sebastien Labbe, and Charles Paperman. Out-Of-Order Membership in Regular Languages. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 161:1-161:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{amarilli_et_al:LIPIcs.ICALP.2026.161,
  author =	{Amarilli, Antoine and Labbe, Sebastien and Paperman, Charles},
  title =	{{Out-Of-Order Membership in Regular Languages}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{161:1--161:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.161},
  URN =		{urn:nbn:de:0030-drops-265498},
  doi =		{10.4230/LIPIcs.ICALP.2026.161},
  annote =	{Keywords: Automata, Complexity, Algebra}
}

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DOI: 10.4230/LIPIcs.ICDT.2024.7

Skyline Operators for Document Spanners

Authors: Antoine Amarilli, Benny Kimelfeld, Sébastien Labbé, and Stefan Mengel

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract

When extracting a relation of spans (intervals) from a text document, a common practice is to filter out tuples of the relation that are deemed dominated by others. The domination rule is defined as a partial order that varies along different systems and tasks. For example, we may state that a tuple is dominated by tuples that extend it by assigning additional attributes, or assigning larger intervals. The result of filtering the relation would then be the skyline according to this partial order. As this filtering may remove most of the extracted tuples, we study whether we can improve the performance of the extraction by compiling the domination rule into the extractor. To this aim, we introduce the skyline operator for declarative information extraction tasks expressed as document spanners. We show that this operator can be expressed via regular operations when the domination partial order can itself be expressed as a regular spanner, which covers several natural domination rules. Yet, we show that the skyline operator incurs a computational cost (under combined complexity). First, there are cases where the operator requires an exponential blowup on the number of states needed to represent the spanner as a sequential variable-set automaton. Second, the evaluation may become computationally hard. Our analysis more precisely identifies classes of domination rules for which the combined complexity is tractable or intractable.

Cite as

Antoine Amarilli, Benny Kimelfeld, Sébastien Labbé, and Stefan Mengel. Skyline Operators for Document Spanners. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{amarilli_et_al:LIPIcs.ICDT.2024.7,
  author =	{Amarilli, Antoine and Kimelfeld, Benny and Labb\'{e}, S\'{e}bastien and Mengel, Stefan},
  title =	{{Skyline Operators for Document Spanners}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{7:1--7:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.7},
  URN =		{urn:nbn:de:0030-drops-197898},
  doi =		{10.4230/LIPIcs.ICDT.2024.7},
  annote =	{Keywords: Information Extraction, Document Spanners, Query Evaluation}
}

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Documents authored by Labbe, Sebastien

Out-Of-Order Membership in Regular Languages

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Skyline Operators for Document Spanners

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