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

A duality theoretic view on limits of finite structures