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Complexity of Clique-Guarded First-Order Logic with Counting

Authors Steffen van Bergerem , Johannes Friedrich Lange , Nicole Schweikardt

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Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Complexity theory and logic
  • Computing methodologies → Supervised learning
Keywords
  • First-order logic with counting
  • VC dimension
  • graph dimension
  • algorithmic metatheorems
  • enumeration
  • nowhere dense
  • locally bounded expansion
  • PAC learning

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Abstract

We introduce clique-guarded first-order logic with counting (cgFOC), a fragment of the first-order logic with counting FOC [Kuske and Schweikardt, LICS 2017], and we study the complexity of this fragment. In particular, we prove computable upper bounds on the Vapnik-Chervonenkis (VC) dimension of cgFOC formulas and on the graph dimension of cgFOC counting terms on nowhere dense classes of relational structures. Furthermore, we show algorithmic metatheorems for cgFOC for query answering, enumeration, and probably approximately correct (PAC) learning for Boolean and multiclass classification problems on classes of locally bounded expansion. On the other hand, we show that a slight extension of cgFOC is already intractable on trees.

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Steffen van Bergerem, Johannes Friedrich Lange, and Nicole Schweikardt. Complexity of Clique-Guarded First-Order Logic with Counting. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.20

Author Details

Steffen van Bergerem
  • Humboldt-Universität zu Berlin, Germany
Johannes Friedrich Lange
  • Humboldt-Universität zu Berlin, Germany
Nicole Schweikardt
  • Humboldt-Universität zu Berlin, Germany

Funding

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 541000908 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 541000908).

References

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