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Quantifiers on languages and codensity monads

Published online by Cambridge University Press:  23 June 2021

Mai Gehrke ,
Daniela Petrişan  and
Luca Reggio Open the ORCID record for Luca Reggio [Opens in a new window]
Mai Gehrke
Affiliation:
Laboratoire J. A. Dieudonné, CNRS and Université Côte d’Azur, Nice, France
Daniela Petrişan
Affiliation:
IRIF, CNRS and Université Paris Diderot, Paris, France
Luca Reggio*
Affiliation:
Department of Computer Science, University of Oxford, Oxford OX1 2JD, UK
*
*Corresponding author. Email: luca.reggio@cs.ox.ac.uk
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Abstract

This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.

Keywords

Formal languageslogic on wordssemiring quantifiersstone dualitycodensity monadssemiring-valued measures
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 10 , November 2020 , pp. 1054 - 1088
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

*

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.670624). This work was carried out while the third author was a PhD student supported by Sorbonne Paris Cité (PhD agreement USPC IDEX – REGGI15RDXMTSPC1GEHRKE).

References

Adámek, J., Chen, L.-T., Milius, S. and Urbat, H. (2016a). Profinite monads, profinite equations and Reiterman’s theorem. Preprint available at https://arxiv.org/abs/1511.02147.Google Scholar
Adámek, J., Chen, L.-T., Milius, S. and Urbat, H. (2016b). Profinite monads, profinite equations, and Reiterman’s theorem. In: FoSSaCS, Lecture Notes in Computer Science, vol. 9634, Springer, 531–547.Google Scholar
Adámek, J., Milius, S., Myers, R. and Urbat, H. (2015). Varieties of languages in a category. In: LICS, IEEE Computer Society, 414–425.CrossRefGoogle Scholar
Almeida, J. and Weil, P. (1998). Profinite categories and semidirect products. Journal of Pure and Applied Algebra 123 (1) 1–50.CrossRefGoogle Scholar
Bojańczyk, M. (2015). Recognisable languages over monads. In: DLT, Lecture Notes in Computer Science, vol. 9168, Springer, 1–13.CrossRefGoogle Scholar
Borceux, F. (1994). Handbook of Categorical Algebra 2, Categories and Structures, Encyclopedia of Mathematics and its Applications, Cambridge, New York, Cambridge University Press.Google Scholar
Chen, L.-T. and Urbat, H. (2016). Schützenberger products in a category. In: DLT, Lecture Notes in Computer Science, vol. 9840, Springer, 89–101.CrossRefGoogle Scholar
Engelking, R. (1989). General Topology, 2nd edn., Sigma Series in Pure Mathematics, vol. 6, Berlin, Heldermann Verlag.Google Scholar
Gehrke, M., Grigorieff, S. and Pin, J.-É. (2008). Duality and equational theory of regular languages. In: ICALP (2), Lecture Notes in Computer Science, vol. 5126, Springer, 246–257.CrossRefGoogle Scholar
Gehrke, M., Grigorieff, S. and Pin, J.-É. (2010). A topological approach to recognition. In: ICALP (2), Lecture Notes in Computer Science, vol. 6199, Springer, 151–162.CrossRefGoogle Scholar
Gehrke, M., Petrişan, D. and Reggio, L. (2016). The Schützenberger product for syntactic spaces. In: ICALP, LIPIcs, vol. 55, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 112:1–112:14.Google Scholar
Gehrke, M., Petrişan, D. and Reggio, L. (2017). Quantifiers on languages and codensity monads. In: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, 1–12.CrossRefGoogle Scholar
Kennison, J. F. and Gildenhuys, D. (1971). Equational completion, model induced triples and pro-objects. Journal of Pure and Applied Algebra 1 (4) 317–346.CrossRefGoogle Scholar
Kock, A. (1970). Monads on symmetric monoidal closed categories. Journal of The Australian Mathematical Society 21 1–10.Google Scholar
Kuratowski, K. (1966). Topology, Vol. I. New edition, revised and augmented. Translated from the French by J. Jaworowski. New York-London, Academic Press; Warsaw, Państwowe Wydawnictwo Naukowe.Google Scholar
Leinster, T. (2013). Codensity and the ultrafilter monad. Theory and Applications of Categories 28 (13) 332–370.Google Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician, 2nd edn., Graduate Texts in Mathematics, vol. 5, New York, Springer-Verlag.Google Scholar
McNaughton, R. and Papert, S. (1971). Counter-Free Automata. Cambridge, MA-London, MIT Press. With an appendix by William Henneman, MIT Research Monograph, No. 65.Google Scholar
Michael, E. (1951). Topologies on spaces of subsets. Transactions of the American Mathematical Society 71 152–182.CrossRefGoogle Scholar
Pin, J.-É. and Sakarovitch, J. (1982). Some operations and transductions that preserve rationality. In: Proceedings of the 6th GI-Conference on Theoretical Computer Science, Springer-Verlag, 277–288.CrossRefGoogle Scholar
Pippenger, N. (1997). Regular languages and Stone duality. Theory of Computing Systems 30 (2) 121–134.CrossRefGoogle Scholar
Reggio, L. (2020). Codensity, profiniteness and algebras of semiring-valued measures. Journal of Pure and Applied Algebra 224 (1) 181–205.CrossRefGoogle Scholar
Reutenauer, C. (1979). Sur les variétés de langages et de monodes. In: Theoretical Computer Science (Fourth GI Conference, Aachen), Lecture Notes in Computer Science, vol. 67, Berlin-New York, Springer, 260–265.Google Scholar
Sakarovitch, J. (2009). Elements of Automata Theory, New York, NY, USA, Cambridge University Press.CrossRefGoogle Scholar
Stone, M. H. (1936). The theory of representations for Boolean algebras. Transactions of the American Mathematical Society 40 (1) 37–111.Google Scholar
Straubing, H. (1994). Finite Automata, Formal Logic, and Circuit Complexity, Progress in Theoretical Computer Science, Boston, MA, Birkhäuser Boston, Inc.CrossRefGoogle Scholar
Straubing, H. and Thérien, D. (2008). Modular quantifiers. In: Logic and Automata, Texts in Logic and Games, vol. 2, Amsterdam University Press, 613–628.Google Scholar
Straubing, H., Thérien, D. and Thomas, W. (1995). Regular languages defined with generalized quantifiers. Information and Computation 118 (2) 289–301.CrossRefGoogle Scholar
Street, R. (1972). The formal theory of monads. Journal of Pure and Applied Algebra 2 (2) 149–168.CrossRefGoogle Scholar
Tesson, P. and Thérien, D. (2007). Logic meets algebra: the case of regular languages. Logical Methods in Computer Science 3 (1–4) 1–37.CrossRefGoogle Scholar
Vietoris, L. (1922). Bereiche zweiter Ordnung. Monatshefte für Mathematik und Physik 32 (1) 258–280.CrossRefGoogle Scholar