6 Search Results for "Sato, Tetsuya"

Document
DOI: 10.4230/LIPIcs.STACS.2026.65
Demystifying Codensity Monads via Duality

Authors: Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Codensity monads provide a universal method to generate complex monads from simple functors. Recently, a wide range of important monads in logic, denotational semantics, and probabilistic computation, such as several incarnations of the ultrafilter monad, the Vietoris monad, and the Giry monad, have been presented as codensity monads, using complex arguments. We propose a unifying categorical approach to codensity presentations of monads, based on the idea of relating the presenting functor to a dense functor via a suitable duality between categories. We prove a general presentation result applying to every such situation and demonstrate that most codensity presentations known in the literature emerge from this strikingly simple duality-based setup, drastically alleviating the complexity of their proofs and in many cases completely reducing them to standard duality results. Additionally, we derive a number of novel codensity presentations using our framework, including the first non-trivial codensity presentations for the filter monads on sets and topological spaces, the lower Vietoris monad on topological spaces, and the expectation monad on sets.
Cite as

Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat. Demystifying Codensity Monads via Duality. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lenke_et_al:LIPIcs.STACS.2026.65,
  author =	{Lenke, Fabian and Wittrock, Nico and Milius, Stefan and Urbat, Henning},
  title =	{{Demystifying Codensity Monads via Duality}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{65:1--65:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.65},
  URN =		{urn:nbn:de:0030-drops-255549},
  doi =		{10.4230/LIPIcs.STACS.2026.65},
  annote =	{Keywords: Codensity, Monad, Duality}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.29
Correspondences Between Codensity and Coupling-Based Liftings, a Practical Approach

Authors: Samuel Humeau, Daniela Petrisan, and Jurriaan Rot

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0,1], and as an infimum, based on probabilistic couplings. Orthogonally, there are categorical generalisations of both presentations proposed in the literature, in the form of codensity liftings and what we refer to as coupling-based liftings. Both lift endofunctors on the category Set of sets and functions to that of pseudometric spaces, and both are parameterised by modalities from coalgebraic modal logic. A generalisation of the Kantorovich-Rubinstein duality has been more nebulous - it is known not to work in some cases. In this paper we propose a compositional approach for obtaining such generalised dualities for a class of functors, which is closed under coproducts and products. Our approach is based on an explicit construction of modalities and also applies to and extends known cases such as that of the powerset functor.
Cite as

Samuel Humeau, Daniela Petrisan, and Jurriaan Rot. Correspondences Between Codensity and Coupling-Based Liftings, a Practical Approach. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{humeau_et_al:LIPIcs.CSL.2025.29,
  author =	{Humeau, Samuel and Petrisan, Daniela and Rot, Jurriaan},
  title =	{{Correspondences Between Codensity and Coupling-Based Liftings, a Practical Approach}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.29},
  URN =		{urn:nbn:de:0030-drops-227861},
  doi =		{10.4230/LIPIcs.CSL.2025.29},
  annote =	{Keywords: Kantorovich distance, behavioural metrics, Kantorovich-Rubinstein duality, functor liftings}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.21
A Formalization of the Lévy-Prokhorov Metric in Isabelle/HOL

Authors: Michikazu Hirata

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
The Lévy-Prokhorov metric is a metric between finite measures on a metric space. The metric was introduced to analyze weak convergence of measures. We formalize the Lévy-Prokhorov metric and prove Prokhorov’s theorem in Isabelle/HOL. Prokhorov’s theorem provides a condition for the relative compactness of sets of finite measures and plays essential roles in proofs of the central limit theorem, Sanov’s theorem in large deviation theory, and the existence of optimal coupling in transportation theory. Our formalization includes important results in mathematics such as the Riesz representation theorem, which is a theorem in functional analysis and used to prove Prokhorov’s theorem. We also apply the Lévy-Prokhorov metric to show that the measurable space of finite measures on a standard Borel space is again a standard Borel space.
Cite as

Michikazu Hirata. A Formalization of the Lévy-Prokhorov Metric in Isabelle/HOL. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hirata:LIPIcs.ITP.2024.21,
  author =	{Hirata, Michikazu},
  title =	{{A Formalization of the L\'{e}vy-Prokhorov Metric in Isabelle/HOL}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{21:1--21:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.21},
  URN =		{urn:nbn:de:0030-drops-207492},
  doi =		{10.4230/LIPIcs.ITP.2024.21},
  annote =	{Keywords: formalization of mathematics, measure theory, metric spaces, topology, L\'{e}vy-Prokhorov metric, Prokhorov’s theorem, Isabelle/HOL}
}
Document
DOI: 10.4230/LIPIcs.ITP.2023.18
Semantic Foundations of Higher-Order Probabilistic Programs in Isabelle/HOL

Authors: Michikazu Hirata, Yasuhiko Minamide, and Tetsuya Sato

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract
Higher-order probabilistic programs are used to describe statistical models and machine-learning mechanisms. The programming languages for them are equipped with three features: higher-order functions, sampling, and conditioning. In this paper, we propose an Isabelle/HOL library for probabilistic programs supporting all of those three features. We extend our previous quasi-Borel theory library in Isabelle/HOL. As a basis of the theory, we formalize s-finite kernels, which is considered as a theoretical foundation of first-order probabilistic programs and a key to support conditioning of probabilistic programs. We also formalize the Borel isomorphism theorem which plays an important role in the quasi-Borel theory. Using them, we develop the s-finite measure monad on quasi-Borel spaces. Our extension enables us to describe higher-order probabilistic programs with conditioning directly as an Isabelle/HOL term whose type is that of morphisms between quasi-Borel spaces. We also implement the qbs prover for checking well-typedness of an Isabelle/HOL term as a morphism between quasi-Borel spaces. We demonstrate several verification examples of higher-order probabilistic programs with conditioning.
Cite as

Michikazu Hirata, Yasuhiko Minamide, and Tetsuya Sato. Semantic Foundations of Higher-Order Probabilistic Programs in Isabelle/HOL. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hirata_et_al:LIPIcs.ITP.2023.18,
  author =	{Hirata, Michikazu and Minamide, Yasuhiko and Sato, Tetsuya},
  title =	{{Semantic Foundations of Higher-Order Probabilistic Programs in Isabelle/HOL}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.18},
  URN =		{urn:nbn:de:0030-drops-183933},
  doi =		{10.4230/LIPIcs.ITP.2023.18},
  annote =	{Keywords: Higher-order probabilistic program, s-finite kernel, Quasi-Borel spaces, Isabelle/HOL}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.102
*-Liftings for Differential Privacy

Authors: Gilles Barthe, Thomas Espitau, Justin Hsu, Tetsuya Sato, and Pierre-Yves Strub

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
Recent developments in formal verification have identified approximate liftings (also known as approximate couplings) as a clean, compositional abstraction for proving differential privacy. There are two styles of definitions for this construction. Earlier definitions require the existence of one or more witness distributions, while a recent definition by Sato uses universal quantification over all sets of samples. These notions have different strengths and weaknesses: the universal version is more general than the existential ones, but the existential versions enjoy more precise composition principles. We propose a novel, existential version of approximate lifting, called *-lifting, and show that it is equivalent to Sato's construction for discrete probability measures. Our work unifies all known notions of approximate lifting, giving cleaner properties, more general constructions, and more precise composition theorems for both styles of lifting, enabling richer proofs of differential privacy. We also clarify the relation between existing definitions of approximate lifting, and generalize our constructions to approximate liftings based on f-divergences.
Cite as

Gilles Barthe, Thomas Espitau, Justin Hsu, Tetsuya Sato, and Pierre-Yves Strub. *-Liftings for Differential Privacy. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 102:1-102:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barthe_et_al:LIPIcs.ICALP.2017.102,
  author =	{Barthe, Gilles and Espitau, Thomas and Hsu, Justin and Sato, Tetsuya and Strub, Pierre-Yves},
  title =	{{*-Liftings for Differential Privacy}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{102:1--102:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.102},
  URN =		{urn:nbn:de:0030-drops-74358},
  doi =		{10.4230/LIPIcs.ICALP.2017.102},
  annote =	{Keywords: Differential Privacy, Probabilistic Couplings, Formal Verification}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2015.156
Codensity Liftings of Monads

Authors: Shin-ya Katsumata and Tetsuya Sato

Published in: LIPIcs, Volume 35, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)

Abstract
We introduce a method to lift monads on the base category of a fibration to its total category using codensity monads. This method, called codensity lifting, is applicable to various fibrations which were not supported by the categorical >>-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preorders, topological spaces and extended psuedometric spaces to the category of sets, and also the fibration from the category of binary relations between measurable spaces. We next study the liftings of algebraic operations to the codensity-lifted monads. We also give a characterisation of the class of liftings (along posetal fibrations with fibred small limits) as a limit of a certain large diagram.
Cite as

Shin-ya Katsumata and Tetsuya Sato. Codensity Liftings of Monads. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 156-170, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{katsumata_et_al:LIPIcs.CALCO.2015.156,
  author =	{Katsumata, Shin-ya and Sato, Tetsuya},
  title =	{{Codensity Liftings of Monads}},
  booktitle =	{6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
  pages =	{156--170},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-84-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{35},
  editor =	{Moss, Lawrence S. and Sobocinski, Pawel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.156},
  URN =		{urn:nbn:de:0030-drops-55329},
  doi =		{10.4230/LIPIcs.CALCO.2015.156},
  annote =	{Keywords: Monads, Lifting, Fibration, Giry Monad}
}
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