A systematic theory of structural limits for finite models has been developed by Nesetril and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises --- via Stone-Priestley duality and the notion of types from model theory --- by enriching the expressive power of first-order logic with certain ``probabilistic operators''. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively.

中文翻译：

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有限结构极限的二元论观点

Nesetril 和 Ossona de Mendez 开发了有限模型结构极限的系统理论。它基于这样的见解，即可以通过称为 Stone 配对的映射将有限结构的集合嵌入到度量空间中，在该空间中可以计算所需的限制。我们展示了一个密切相关但粒度更细的度量空间——通过 Stone-Priestley 对偶性和模型理论中的类型概念——通过使用某些“概率算子”丰富一阶逻辑的表达能力. 我们为这个扩展逻辑提供了一个健全和完整的演算，并揭示了这个构造的函数性质。后果是双重的。一方面，我们确定了结构极限理论的逻辑要点。另一方面，我们的构建表明，斯通配对的对偶理论变体捕获了一层量词的添加，从而与最近关于单词逻辑中半环量词的工作建立了紧密的联系。在此过程中，我们将类型的模型理论概念确定为该链接背后的统一概念。这些结果有助于弥合计算机科学中的逻辑链，这些逻辑分别侧重于语义和更多算法和复杂性相关领域。