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The Parameterized Complexity of Learning Monadic Second-Order Logic

Authors Steffen van Bergerem , Martin Grohe , Nina Runde

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ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Fixed parameter tractability
  • Computing methodologies → Logical and relational learning
  • Computing methodologies → Supervised learning
Keywords
  • monadic second-order definable concept learning
  • agnostic probably approximately correct learning
  • parameterized complexity
  • clique-width
  • fixed-parameter tractable
  • Boolean classification
  • supervised learning
  • monadic second-order logic

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Abstract

Within the model-theoretic framework for supervised learning introduced by Grohe and Turán (TOCS 2004), we study the parameterized complexity of learning concepts definable in monadic second-order logic (MSO). We show that the problem of learning an MSO-definable concept from a training sequence of labeled examples is fixed-parameter tractable on graphs of bounded clique-width, and that it is hard for the parameterized complexity class para-NP on general graphs.
It turns out that an important distinction to be made is between 1-dimensional and higher-dimensional concepts, where the instances of a k-dimensional concept are k-tuples of vertices of a graph. For the higher-dimensional case, we give a learning algorithm that is fixed-parameter tractable in the size of the graph, but not in the size of the training sequence, and we give a hardness result showing that this is optimal. By comparison, in the 1-dimensional case, we obtain an algorithm that is fixed-parameter tractable in both.

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Steffen van Bergerem, Martin Grohe, and Nina Runde. The Parameterized Complexity of Learning Monadic Second-Order Logic. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.8

Author Details

Steffen van Bergerem
  • Humboldt-Universität zu Berlin, Germany
Martin Grohe
  • RWTH Aachen University, Germany
Nina Runde
  • RWTH Aachen University, Germany

Funding

  • van Bergerem, Steffen: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 431183758 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 431183758).
  • Grohe, Martin: Funded by the European Union (ERC, SymSim, 101054974). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Runde, Nina: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 453349072 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 453349072).

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