Complexity Classification Transfer for CSPs via Algebraic Products
Manuel Bodirsky, Peter Jonsson, Barnaby Martin, Antoine Mottet, Žaneta Semanišinová
TL;DR
The paper addresses the problem of classifying the computational complexity of constraint satisfaction problems over infinite-domain templates, focusing on first-order expansions of the rational order $(\mathbb{Q};<)$ and its algebraic powers. Its main approach is to develop and leverage the algebraic product, which aligns polymorphism clones and enables transfer of finite-domain tractability phenomena to infinite-domain settings, complemented by syntactic normal forms based on i-determined clauses. The authors prove a comprehensive dichotomy for CSPs of first-order expansions of $(\mathbb{Q};<)^{(n)}$: tractable (in P) when the relevant polymorphism clones admit certain pseudo-weak-near-unanimity or ll-like operations, and NP-hard when a uniformly continuous minor-preserving map to $K_3$ exists, with both outcomes captured via pp-interpretations and transfer theorems. They then apply this framework to key AI formalisms — Cardinal Direction Calculus, Allen's Interval Algebra, and the Block Algebra — obtaining strong classification results, including binary-signature refinements via Ord-Horn definability, thereby validating the infinite-domain tractability conjecture for these challenging, nontrivial settings and resolving several longstanding questions in AI literature.
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.