# Complexity Classification Transfer for CSPs via Algebraic Products

## Abstract.

*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 \(({\mathbb Q};\lt )\) . This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of \(({\mathbb Q};\lt )\) 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 analyze 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 artificial intelligence (AI) literature.

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**Submitted**: 21 November 2022

**Accepted**: 17 May 2024

**Published online**: 12 September 2024

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**Funding:**The work of the first and fifth authors was supported by European Research Council grant 101071674, POCOCOP, and by Deutsche Forschungsgemeinschaft grant 467967530. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. The work of the second author was partially supported by the Swedish Research Council (VR) under grants 2017-04112 and 2021-04371. The third author was supported by EPSRC grant EP/X03190X/1.

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- Smooth approximations: An algebraic approach to CSPs over finitely bounded homogeneous structuresJournal of the ACM, Vol. 71, No. 5 | 5 October 2024

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