Arity hierarchies for quantifiers closed under partial polymorphisms

Anuj Dawar; Lauri Hella; Benedikt Pago

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arXiv · 2511.11326

Arity hierarchies for quantifiers closed under partial polymorphisms

Anuj Dawar·Lauri Hella·Benedikt Pago

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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 $(\ell+1)$-ary quantifiers closed under $\ell$-ary partial near-unanimity polymorphisms is strictly between the class of all quantifiers of arity $\ell-1$ and $\ell$. 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-F\"urer-Immerman and the quantifier pebble games of Dawar and Hella (2024).

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Anuj Dawar, Lauri Hella, Benedikt Pago. 2025-11-14. Arity hierarchies for quantifiers closed under partial polymorphisms. https://arxiv.org/abs/2511.11326

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