Collapsing the bounded width hierarchy for infinite-domain CSPs: when symmetries are enough
Antoine Mottet, Tomáš Nagy, Michael Pinsker, Michał Wrona
TL;DR
This work addresses when CSPs with infinite templates admit efficient local-consistency algorithms by examining relational width and width-collapse phenomena through the lens of Aut$(\mathbb{B})$-canonical polymorphisms. It develops a general reduction to finite-template CSPs and proves that having pseudo-WNU (or pseudo-totally symmetric) polymorphisms of all arities implies concrete relational-width bounds, thereby collapsing the bounded width hierarchy for broad infinite-domain classes. A new loop lemma for smooth approximations is established, enabling construction of high-arity polymorphisms when algebraic niceties fail, and the results are applied to unary structures and MMSNP to yield explicit width bounds and Datalog-rewritability characterizations. The findings extend finite-domain width-collapse results to significant infinite-domain settings, with implications for tractability and logic (including ontology-mediated querying) and provide a unified algebraic framework connecting model theory, Ramsey theory, and constraint satisfaction.
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.