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.

翻译：我们研究无限域约束满足问题（CSP）的复杂性：基本设定是，结构$\mathfrak A$的一阶扩张的CSP复杂性分类可传递至另一结构$\mathfrak B$的一阶扩张的CSP分类。通过利用对应各自多态克隆积的结构积（代数积），我们首次给出了$(\mathbb{Q};<)$的$n$重代数幂的一阶扩张的CSP完整复杂性分类。该证明综合运用了代数与逻辑方法，结合对$(\mathbb{Q};<)$的可处理一阶扩张多态性的认知，以及通过句法受限一阶公式对可表达关系的显式描述。通过将分类结果与通用分类传递技术相结合，我们在艾伦区间代数、$n$维块代数、基数方向演算等高相关形式化体系中获得了显著强化的新分类结果，甚至允许高阶关系存在。我们的研究结果证实了针对那些传统方法难以分析的结构类，无限域可处理性猜想成立。对于二元签名结构的特殊情况，结论可获得实质性强化并与Ord-Horn公式建立紧密关联，这解决了人工智能文献中多个长期悬而未决的开放性问题。