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

DOI: 10.4230/LIPIcs.ICALP.2026.174

The Role of Counting Quantifiers in Laminar Set Systems

Authors: Rutger Campbell and Noleen Köhler

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

Abstract

Laminar set systems consist of non-crossing subsets of a universe with set inclusion essentially corresponding to the descendant relationship of a tree, the so-called laminar tree. Laminar set systems lie at the core of many graph decompositions such as modular decompositions, split decompositions, and bi-join decompositions. We show that from a laminar set system we can obtain the corresponding laminar tree by means of a monadic second order logic (MSO) transduction. This resolves an open question originally asked by Courcelle and is a satisfying resolution as MSO is the natural logic for set systems and is sufficient to define the property "laminar". Using results from Campbell et al. [STACS 2025], we can now obtain transductions for obtaining modular decompositions, co-trees, split decompositions and bi-join decompositions using MSO instead of CMSO. We further gain some insight into the expressive power of counting quantifiers and provide some results towards determining when counting quantifiers can be simulated in MSO in laminar set systems and when they cannot.

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Rutger Campbell and Noleen Köhler. The Role of Counting Quantifiers in Laminar Set Systems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 174:1-174:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{campbell_et_al:LIPIcs.ICALP.2026.174,
  author =	{Campbell, Rutger and K\"{o}hler, Noleen},
  title =	{{The Role of Counting Quantifiers in Laminar Set Systems}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{174:1--174:18},
  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.2026.174},
  URN =		{urn:nbn:de:0030-drops-265625},
  doi =		{10.4230/LIPIcs.ICALP.2026.174},
  annote =	{Keywords: MSO-transductions, simulating counting quantifiers, laminar set systems, graph decompositions}
}

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DOI: 10.4230/LIPIcs.STACS.2025.22

CMSO-Transducing Tree-Like Graph Decompositions

Authors: Rutger Campbell, Bruno Guillon, Mamadou Moustapha Kanté, Eun Jung Kim, and Noleen Köhler

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract

We show that given a graph G we can CMSO-transduce its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant MSO, a strictly more expressive logic than CMSO. Our methods more generally yield C_{2}MSO-transductions of the canonical decomposition of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

Cite as

Rutger Campbell, Bruno Guillon, Mamadou Moustapha Kanté, Eun Jung Kim, and Noleen Köhler. CMSO-Transducing Tree-Like Graph Decompositions. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{campbell_et_al:LIPIcs.STACS.2025.22,
  author =	{Campbell, Rutger and Guillon, Bruno and Kant\'{e}, Mamadou Moustapha and Kim, Eun Jung and K\"{o}hler, Noleen},
  title =	{{CMSO-Transducing Tree-Like Graph Decompositions}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.22},
  URN =		{urn:nbn:de:0030-drops-228474},
  doi =		{10.4230/LIPIcs.STACS.2025.22},
  annote =	{Keywords: MSO-transduction, MSO-definability, graph decomposisions}
}

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Documents authored by Campbell, Rutger

The Role of Counting Quantifiers in Laminar Set Systems

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CMSO-Transducing Tree-Like Graph Decompositions

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