A duality theoretic view on limits of finite structures: Extended version

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 (finitely additive)measures arises -- via Stone-Priestley duality and the notion of types frommodel theory -- by enriching the expressive power of first-order logic withcertain "probabilistic operators". We provide a sound and complete calculus forthis extended 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.Comment: arXiv admin note: text overlap with arXiv:1907.0403