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.
Highlights
While topology plays an important role, via Stone duality, in many parts of semantics, topological methods in more algorithmic and complexity oriented areas of theoretical computer science are not so common
We show that the ambient space of measures for the structural limits of Nesetril and Ossona de Mendez can be obtained via “adding a layer of quantifiers” in a suitable enrichment of first-order logic
Γ-valued measures and [0, 1]-valued measures are tightly related by a retraction-section pair which allows the transfer of properties. These results identify the logical gist of the theory of structural limits and provide a new interesting connection between logic on words and the theory of structural limits in finite model theory
Summary
While topology plays an important role, via Stone duality, in many parts of semantics, topological methods in more algorithmic and complexity oriented areas of theoretical computer science are not so common. Nesetril and Ossona de Mendez view the map A → -, A as an embedding of the finite σ-structures into the space of probability measures over the Stone space dual to the Lindenbaum–Tarski algebra of all first-order formulas in the signature σ. The measures involved are not classical but only finitely additive and they take values in finite semirings rather than in the unit interval This appearance of measures as duals of quantifiers begs the further question whether the spaces of measures in the theory of structural limits may be obtained via Stone duality from a semantic addition of certain quantifiers to classical first-order logic. This general theory is applied in Appendix B to explain how the space Γ and its algebra-like structure are derived
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