A Duality Theoretic View on Limits of Finite Structures

Abstract

A systematic theory of structural limits for finite models has been developed by Nešetřil 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 firstorder 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 twofold. 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.

Priestley duality for MV-algebras and beyond

“…In this setting, the law (MV6) may be rendered as a transparent first-order condition on the two partial binary operations (see Section 4.2). We expect the ideas laid out in this paper to support many similar applications, and note that other applications of this duality theory are already under way (see, e.g., [13]).…”