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Arity hierarchies for quantifiers closed under partial polymorphisms

Anuj Dawar, Lauri Hella, Benedikt Pago

TL;DR

This work analyzes the expressive power of generalized quantifiers closed under partial polymorphisms, focusing on the interaction between quantifier arity and the arity of the governing polymorphisms. It proves an arity-reduction bound for near-unanimity-closed quantifiers and establishes a strict infinite arity hierarchy, plus a Maltsev-oriented expressiveness result, using Cai–Ffer–Immerman style constructions and quantifier pebble games. The results hinge on FO-definable reductions to transform templates and on carefully designed CFI-like gadgetry to separate logics, revealing nuanced, nontrivial hierarchies beyond the classical arity framework. The work also outlines limitations for Maltsev-closed quantifiers and highlights open questions surrounding equivalence-relations, game-based characterizations, and extending the hierarchy to broader polymorphism families.

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-Fürer-Immerman and the quantifier pebble games of Dawar and Hella (2024).

Arity hierarchies for quantifiers closed under partial polymorphisms

TL;DR

This work analyzes the expressive power of generalized quantifiers closed under partial polymorphisms, focusing on the interaction between quantifier arity and the arity of the governing polymorphisms. It proves an arity-reduction bound for near-unanimity-closed quantifiers and establishes a strict infinite arity hierarchy, plus a Maltsev-oriented expressiveness result, using Cai–Ffer–Immerman style constructions and quantifier pebble games. The results hinge on FO-definable reductions to transform templates and on carefully designed CFI-like gadgetry to separate logics, revealing nuanced, nontrivial hierarchies beyond the classical arity framework. The work also outlines limitations for Maltsev-closed quantifiers and highlights open questions surrounding equivalence-relations, game-based characterizations, and extending the hierarchy to broader polymorphism families.

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 -ary quantifiers closed under -ary partial near-unanimity polymorphisms is strictly between the class of all quantifiers of arity and . 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).

Paper Structure

This paper contains 12 sections, 54 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Arity reductions
  4. Arity hierarchies for partial near-unanimity quantifiers
  5. Construction of polymorphism-closed templates
  6. Constructing instances
  7. (In-)distinguishability of the instances
  8. Limitations of partial-Maltsev-closed quantifiers
  9. Conclusion
  10. Details omitted from Section \ref{['sec:collapses']}
  11. Details omitted from Section \ref{['sec:ArityHierarchy']}
  12. Limitations of partial-Maltsev-closed quantifiers

Key Result

Proposition 2

Let $k \ge r$ be positive integers. Then Duplicator has a winning strategy in $\mathrm{BP}^k_r(\mathbf{A},\mathbf{B})$ if, and only if, $\mathbf{A}\equiv_{\mathcal{L}^k(\mathcal{Q}_r)}\mathbf{B}$.

Theorems & Definitions (63)

  • Definition 1
  • Proposition 2: Hella96
  • Corollary 3
  • Example 4
  • Lemma 5: dawarHella2024
  • Lemma 6
  • Corollary 7
  • Proposition 7
  • Lemma 8
  • Lemma 9
  • ...and 53 more