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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)


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.


  • 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.


  1. Abramsky, S.: Domain theory in logical form. Ann. Pure Appl. Logic 51, 1–77 (1991)

    Google Scholar 

  2. Abramsky, S., Dawar, A., Wang, P.: The pebbling comonad in finite model theory. In: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS. pp. 1–12 (2017)

    Google Scholar 

  3. Abramsky, S., Shah, N.: Relating Structure and Power: Comonadic semantics for computational resources. In: 27th EACSL Annual Conference on Computer Science Logic, CSL. pp. 2:1–2:17 (2018)

    Google Scholar 

  4. Gehrke, M., Grigorieff, S., Pin, J.-É.: Duality and equational theory of regular languages. In: Automata, languages and programming II, LNCS, vol. 5126, pp. 246–257. Springer, Berlin (2008)

    Google Scholar 

  5. Gehrke, M., Petrişan, D., Reggio, L.: Quantifiers on languages and codensity monads. In: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS. pp. 1–12 (2017)

    Google Scholar 

  6. Gehrke, M., Petrişan, D., Reggio, L.: Quantifiers on languages and codensity monads (2019), extended version. Submitted. Preprint available at

  7. Gehrke, M., Priestley, H.A.: Canonical extensions of double quasioperator algebras: an algebraic perspective on duality for certain algebras with binary operations. J. Pure Appl. Algebra 209(1), 269–290 (2007)

    Google Scholar 

  8. Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Studia Logica 86(1), 31–68 (2007)

    Google Scholar 

  9. Goldblatt, R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44(3), 173–242 (1989)

    Google Scholar 

  10. van Gool, S.J., Steinberg, B.: Pro-aperiodic monoids via saturated models. In: 34th Symposium on Theoretical Aspects of Computer Science, STACS. pp. 39:1–39:14 (2017)

    Google Scholar 

  11. Johnstone, P.T.: Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press (1986), reprint of the 1982 edition

    Google Scholar 

  12. Jung, A.: Continuous domain theory in logical form. In: Coecke, B., Ong, L., Panangaden, P. (eds.) Computation, Logic, Games, and Quantum Foundations, Lecture Notes in Computer Science, vol. 7860, pp. 166–177. Springer Verlag (2013)

    Google Scholar 

  13. Kolaitis, P.G., Pichler, R., Sallinger, E., Savenkov, V.: Limits of schema mappings. Theory of Computing Systems 62(4), 899–940 (2018)

    Google Scholar 

  14. Matz, O., Schweikardt, N.: Expressive power of monadic logics on words, trees, pictures, and graphs. In: Logic and Automata: History and Perspectives. pp. 531–552 (2008)

    Google Scholar 

  15. Nešetřil, J., Ossona de Mendez, P.: A model theory approach to structural limits. Commentationes Mathematicae Universitatis Carolinae 53(4), 581–603 (2012)

    Google Scholar 

  16. Nešetřil, J., Ossona de Mendez, P.: First-order limits, an analytical perspective. European Journal of Combinatorics 52, 368–388 (2016)

    Google Scholar 

  17. Nešetřil, J., Ossona de Mendez, P.: A unified approach to structural limits and limits of graphs with bounded tree-depth (2020), to appear in Memoirs of the American Mathematical Society

    Google Scholar 

  18. Pin, J.-É.: Profinite methods in automata theory. In: 26th Symposium on Theoretical Aspects of Computer Science, STACS. pp. 31–50 (2009)

    Google Scholar 

  19. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)

    Google Scholar 

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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.

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