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On the VC Dimension of First-Order Logic with Counting and Weight Aggregation

Authors Steffen van Bergerem , Nicole Schweikardt

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Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • VC dimension
  • VC density
  • stability
  • nowhere dense graphs
  • first-order logic with weight aggregation
  • first-order logic with counting

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Abstract

We prove optimal upper bounds on the Vapnik-Chervonenkis density of formulas in the extensions of first-order logic with counting (FOC_1) and with weight aggregation (FOWA_1) on nowhere dense classes of (vertex- and edge-)weighted finite graphs. This lifts a result of Pilipczuk, Siebertz, and Toruńczyk [Michał Pilipczuk et al., 2018] from first-order logic on ordinary finite graphs to substantially more expressive logics on weighted finite graphs. Moreover, this proves that every FOC_1 formula and every FOWA_1 formula has bounded Vapnik-Chervonenkis dimension on nowhere dense classes of weighted finite graphs; thereby, it lifts a result of Adler and Adler [Hans Adler and Isolde Adler, 2014] from first-order logic to FOC_1 and FOWA_1.
Generalising another result of Pilipczuk, Siebertz, and Toruńczyk [Michał Pilipczuk et al., 2018], we also provide an explicit upper bound on the ladder index of FOC_1 and FOWA_1 formulas on nowhere dense classes. This shows that nowhere dense classes of weighted finite graphs are FOC_1-stable and FOWA_1-stable.

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Steffen van Bergerem and Nicole Schweikardt. On the VC Dimension of First-Order Logic with Counting and Weight Aggregation. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.15

Author Details

Steffen van Bergerem
  • 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).

Acknowledgements

We thank the anonymous reviewers for their valuable comments that helped to improve the presentation of this paper.

References

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