Arity hierarchies for quantifiers closed under partial polymorphisms

TL;DR

This paper explores the expressive power of generalized quantifiers closed under partial polymorphisms, establishing hierarchies and inexpressiveness results relevant to constraint satisfaction problems and algebraic structures.

Contribution

It introduces hierarchy results for quantifiers closed under partial near-unanimity polymorphisms and proves inexpressiveness for those with partial Maltsev polymorphisms.

Findings

Hierarchy between quantifiers of different arities established

Infinite hierarchy based on quantifier arity demonstrated

Inexpressiveness results for quantifiers with partial Maltsev polymorphisms

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).