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Arity Hierarchies for Quantifiers Closed Under Partial Polymorphisms

Authors Anuj Dawar , Lauri Hella , Benedikt Pago

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LIPIcs.CSL.2026.9.pdf
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ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • finite model theory
  • constraint satisfaction problems
  • generalized quantifiers

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Abstract

We investigate the expressive power of generalized quantifiers closed under partial polymorphism conditions motivated by the study of constraint satisfaction problems. We answer a number of questions arising from the work of Dawar and Hella (CSL 2024) where such quantifiers were introduced. For quantifiers closed under partial near-unanimity polymorphisms, we establish hierarchy results clarifying the interplay between the arity of the polymorphisms and of the quantifiers: The expressive power of (𝓁+1)-ary quantifiers closed under 𝓁-ary partial near-unanimity polymorphisms is strictly between the class of all quantifiers of arity 𝓁-1 and 𝓁. We also establish an infinite hierarchy based on the arity of quantifiers with a fixed arity of partial near-unanimity polymorphisms. Finally, we prove inexpressiveness results for quantifiers with a partial Maltsev polymorphism. The separation results are proved using novel algebraic constructions in the style of Cai-Fürer-Immerman and the quantifier pebble games of Dawar and Hella (2024).

Cite As Get BibTex

Anuj Dawar, Lauri Hella, and Benedikt Pago. Arity Hierarchies for Quantifiers Closed Under Partial Polymorphisms. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CSL.2026.9

Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Lauri Hella
  • Faculty of Information Technology and Communication Sciences, Tampere University, Finland
Benedikt Pago
  • Department of Computer Science and Technology, University of Cambridge, UK

Funding

Research funded in part by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee: grant number EP/X028259/1.

Acknowledgements

We thank Victor Lagerkvist for pointers to the literature on the use of partial polymorphisms in CSP algorithms.

References

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