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

Authors Luisa Herrmann , Vincent Peth , Sebastian Rudolph



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Luisa Herrmann
  • Computational Logic Group, TU Dresden, Germany
  • Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, Germany
Vincent Peth
  • Département d’informatique de l’ÉNS, École normale supérieure, CNRS, PSL University, Paris, France
Sebastian Rudolph
  • Computational Logic Group, TU Dresden, Germany
  • Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, Germany

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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)
https://doi.org/10.4230/LIPIcs.CSL.2024.33

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Tree languages
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Automated reasoning
Keywords
  • MSO
  • BAPA
  • Parikh-Muller tree automata
  • decidability
  • MSO-interpretations

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