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.
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.
Chapter PDF
Similar content being viewed by others
Keywords
References
Abramsky, S.: Domain theory in logical form. Ann. Pure Appl. Logic 51, 1–77 (1991)
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)
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)
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)
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)
Gehrke, M., Petrişan, D., Reggio, L.: Quantifiers on languages and codensity monads (2019), extended version. Submitted. Preprint available at https://arxiv.org/abs/1702.08841
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)
Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Studia Logica 86(1), 31–68 (2007)
Goldblatt, R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44(3), 173–242 (1989)
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)
Johnstone, P.T.: Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press (1986), reprint of the 1982 edition
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)
Kolaitis, P.G., Pichler, R., Sallinger, E., Savenkov, V.: Limits of schema mappings. Theory of Computing Systems 62(4), 899–940 (2018)
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)
Nešetřil, J., Ossona de Mendez, P.: A model theory approach to structural limits. Commentationes Mathematicae Universitatis Carolinae 53(4), 581–603 (2012)
Nešetřil, J., Ossona de Mendez, P.: First-order limits, an analytical perspective. European Journal of Combinatorics 52, 368–388 (2016)
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
Pin, J.-É.: Profinite methods in automata theory. In: 26th Symposium on Theoretical Aspects of Computer Science, STACS. pp. 31–50 (2009)
Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2020 The Author(s)
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-45231-5_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-45230-8
Online ISBN: 978-3-030-45231-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
