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

TL;DR

This paper develops a duality theoretic framework for structural limits of finite models, linking logic, measure theory, and model theory to deepen understanding of limits and their logical foundations.

Contribution

It introduces a finer measure space via Stone-Priestley duality and probabilistic operators, providing a complete calculus and connecting logical types with structural limits.

Findings

Identifies the logical core of the theory of structural limits.

Shows that the duality-based measure captures quantifier addition.

Bridges semantics-focused and complexity-focused logic in computer science.

Abstract

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 (finitely additive) 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.