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DOI: 10.4230/LIPIcs.CSL.2024.33

Decidable (Ac)counting with Parikh and Muller: Adding Presburger Arithmetic to Monadic Second-Order Logic over Tree-Interpretable Structures

Authors: Luisa Herrmann, Vincent Peth, and Sebastian Rudolph

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract

We propose ωMSO⋈BAPA , an expressive logic for describing countable structures, which subsumes and transcends both Counting Monadic Second-Order Logic (CMSO) and Boolean Algebra with Presburger Arithmetic (BAPA). We show that satisfiability of ωMSO⋈BAPA is decidable over the class of labeled infinite binary trees, whereas it becomes undecidable even for a rather mild relaxations. The decidability result is established by an elaborate multi-step transformation into a particular normal form, followed by the deployment of Parikh-Muller Tree Automata, a novel kind of automaton for infinite labeled binary trees, integrating and generalizing both Muller and Parikh automata while still exhibiting a decidable (in fact PSpace-complete) emptiness problem. By means of MSO-interpretations, we lift the decidability result to all tree-interpretable classes of structures, including the classes of finite/countable structures of bounded treewidth/cliquewidth/partitionwidth. We generalize the result further by showing that decidability is even preserved when coupling width-restricted ωMSO⋈BAPA with width-unrestricted two-variable logic with advanced counting. A final showcase demonstrates how our results can be leveraged to harvest decidability results for expressive μ-calculi extended by global Presburger constraints.

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Luisa Herrmann, Vincent Peth, and Sebastian Rudolph. Decidable (Ac)counting with Parikh and Muller: Adding Presburger Arithmetic to Monadic Second-Order Logic over Tree-Interpretable Structures. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{herrmann_et_al:LIPIcs.CSL.2024.33,
  author =	{Herrmann, Luisa and Peth, Vincent and Rudolph, Sebastian},
  title =	{{Decidable (Ac)counting with Parikh and Muller: Adding Presburger Arithmetic to Monadic Second-Order Logic over Tree-Interpretable Structures}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{33:1--33:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.33},
  URN =		{urn:nbn:de:0030-drops-196768},
  doi =		{10.4230/LIPIcs.CSL.2024.33},
  annote =	{Keywords: MSO, BAPA, Parikh-Muller tree automata, decidability, MSO-interpretations}
}

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Decidable (Ac)counting with Parikh and Muller: Adding Presburger Arithmetic to Monadic Second-Order Logic over Tree-Interpretable Structures

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