The Parameterized Complexity of Learning Monadic Second-Order Logic

Authors Steffen van Bergerem , Martin Grohe , Nina Runde



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Author Details

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

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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

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.

Subject Classification

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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