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A Duality Theoretic View on Limits of Finite Structures

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12077)

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 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.

Keywords

  • Stone duality
  • finitely additive measures
  • structural limits
  • finite model theory
  • formal languages
  • logic on words

This project has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 670624). Luca Reggio has received an individual support under the grants GA17-04630S of the Czech Science Foundation, and No. 184693 of the Swiss National Science Foundation.

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Correspondence to Luca Reggio .

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Gehrke, M., Jakl, T., Reggio, L. (2020). A Duality Theoretic View on Limits of Finite Structures. In: Goubault-Larrecq, J., König, B. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2020. Lecture Notes in Computer Science(), vol 12077. Springer, Cham. https://doi.org/10.1007/978-3-030-45231-5_16

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