Complexity Classification Transfer for CSPs via Algebraic Products

TL;DR

This paper develops a method to transfer complexity classifications of CSPs across structures using algebraic products, leading to a complete classification for certain infinite-domain structures and solving longstanding open problems.

Contribution

It introduces a transfer technique via algebraic products for CSP complexity classification and applies it to key infinite-domain structures, advancing the understanding of tractability.

Findings

Complete classification of CSPs for algebraic powers of $( ext{Q};<)$.

New complexity results for Allen's Interval Algebra and related formalisms.

Resolution of longstanding open problems in AI related to binary-structure CSPs.

Abstract

We study the complexity of infinite-domain constraint satisfaction problems: our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure $\mathfrak A$ can be transferred to a classification of the CSPs of first-order expansions of another structure $\mathfrak B$. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the $n$-fold algebraic power of $(\mathbb{Q};<)$. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of $(\mathbb{Q};<)$ and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen's Interval Algebra, the $n$-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyse with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the AI literature.