Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems)

• Published in: arXiv.org
• Main Authors: Bodirsky, Manuel, Mottet, Antoine, Olšák, Miroslav, Opršal, Jakub, Pinsker, Michael, Ross, Willard
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 17.05.2019
• Subjects: Algebra; Computer Science - Computational Complexity; Computer Science - Logic in Computer Science; Mathematics - Logic; Nesting; Polymorphism; Polynomials
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1901.04237

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable when the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to this conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-trivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of non-trivial height 1 identities differ for \(\omega\)-categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.