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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).

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