Collapsing the bounded width hierarchy for infinite-domain CSPs: when symmetries are enough

TL;DR

This paper demonstrates that certain infinite-domain CSPs with specific polymorphisms have low relational width, leading to a collapse of the bounded width hierarchy and extending finite structure results to infinite domains.

Contribution

It introduces a new characterization of bounded width for infinite-domain CSPs using canonical pseudo-WNU polymorphisms, extending finite structure theorems.

Findings

Relational structures with specific polymorphisms have low relational width.

Bounded width hierarchy collapses for many classes of infinite-domain CSPs.

Characterizations of bounded width for first-order reducts and MMSNP sentences are provided.

Abstract

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities $n \geq 3$) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu, ten Cate, Lutz, and Wolter. In particular, the bounded width hierarchy collapses in those cases as well. Our results extend the scope of theorems of Barto and Kozik characterizing bounded width for finite structures, and show the applicability of infinite-domain CSPs to other fields.