# A duality theoretic view on limits of finite structures

## TL;DR

This paper explores the connection between structural limits of finite models and model theory by extending the Stone pairing with probabilistic operators, revealing a duality-based framework that links logic, semantics, and complexity.

## Contribution

It introduces a finer measure space via Stone-Priestley duality, enriched with probabilistic operators, and provides a complete calculus for this extended logic, bridging semantics and algorithmic logic.

## Key findings

- Identifies the logical core of the theory of structural limits.
- Shows that the duality-based Stone pairing captures quantifier addition.
- Links model theoretic types to semiring quantifiers in logic.

## 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 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.