A duality theoretic view on limits of finite structures
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The study of limits of sequences of finite structures plays a crucial role in finite model theory. It is motivated by an attempt to understand the behaviour of dynamical systems, such as computer networks evolving over time. For this purpose, starting in 2012, Nesetril and Ossona de Mendez have been developing a theory of structural limits of finite models. It is based on the insight that the collection of finite structures can be embedded in a space of measures, where the desired limits can be computed. This embedding they call the Stone pairing. We show that a closely related but finer grained space of measures arises --- via Stone duality --- by enriching the expressive power of the logic with certain "probabilistic operators". 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 our duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to the recent work on semiring quantifiers in logic on words. These results connect two branches of logic in computer science which thus far have employed different techniques and tools: formal languages-and-logic on words and structural limits of finite models.
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