Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad

Francesco Dagnino; Amin Farjudian; Eugenio Moggi

doi:10.1007/978-3-031-47963-2_19

Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad

Francesco Dagnino, Amin Farjudian, Eugenio Moggi

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining various notions of behavioral metrics.

We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad P_S, called the Hausdorff-Smyth monad, and when Q is a continuous quantale and X is a Q-metric space, we relate the topology induced by the metric on P_S(X) with the robust topology on the powerset P(X) defined in terms of the metric on X.

Original language	English
Title of host publication	Theoretical Aspects of Computing – ICTAC 2023
Subtitle of host publication	20th International Colloquium, Lima, Peru, December 4–8, 2023, Proceedings
Publisher	Springer
Pages	313–331
Volume	14446
Edition	1
ISBN (Electronic)	9783031479632
ISBN (Print)	9783031479625
DOIs	https://doi.org/10.1007/978-3-031-47963-2_19
Publication status	Published - 23 Nov 2023
Event	20th International Colloquium on Theoretical Aspects of Computing - Lima, Peru Duration: 4 Dec 2023 → 8 Dec 2023

Publication series

Name	Lecture Notes in Computer Science
Publisher	Springer
Volume	14446
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	20th International Colloquium on Theoretical Aspects of Computing
Abbreviated title	ICTAC 2023
Country/Territory	Peru
City	Lima
Period	4/12/23 → 8/12/23

Keywords

Quantale
Robustness
Monad
Topology
Enriched Category

Access to Document

10.1007/978-3-031-47963-2_19Licence: None: All rights reserved

DagninoF2023Robustness
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Cite this

Dagnino, F., Farjudian, A., & Moggi, E. (2023). Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad. In Theoretical Aspects of Computing – ICTAC 2023: 20th International Colloquium, Lima, Peru, December 4–8, 2023, Proceedings (1 ed., Vol. 14446, pp. 313–331). (Lecture Notes in Computer Science; Vol. 14446). Springer. https://doi.org/10.1007/978-3-031-47963-2_19

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Dagnino, F, Farjudian, A & Moggi, E 2023, Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad. in Theoretical Aspects of Computing – ICTAC 2023: 20th International Colloquium, Lima, Peru, December 4–8, 2023, Proceedings. 1 edn, vol. 14446, Lecture Notes in Computer Science, vol. 14446, Springer, pp. 313–331, 20th International Colloquium on Theoretical Aspects of Computing
, Lima, Peru, 4/12/23. https://doi.org/10.1007/978-3-031-47963-2_19

Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad. / Dagnino, Francesco; Farjudian, Amin; Moggi, Eugenio.
Theoretical Aspects of Computing – ICTAC 2023: 20th International Colloquium, Lima, Peru, December 4–8, 2023, Proceedings. Vol. 14446 1. ed. Springer, 2023. p. 313–331 (Lecture Notes in Computer Science; Vol. 14446).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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N2 - Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining various notions of behavioral metrics.We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad PS, called the Hausdorff-Smyth monad, and when Q is a continuous quantale and X is a Q-metric space, we relate the topology induced by the metric on PS(X) with the robust topology on the powerset P(X) defined in terms of the metric on X.

AB - Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining various notions of behavioral metrics.We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad PS, called the Hausdorff-Smyth monad, and when Q is a continuous quantale and X is a Q-metric space, we relate the topology induced by the metric on PS(X) with the robust topology on the powerset P(X) defined in terms of the metric on X.

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BT - Theoretical Aspects of Computing – ICTAC 2023

PB - Springer

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Y2 - 4 December 2023 through 8 December 2023

ER -

Dagnino F, Farjudian A, Moggi E. Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad. In Theoretical Aspects of Computing – ICTAC 2023: 20th International Colloquium, Lima, Peru, December 4–8, 2023, Proceedings. 1 ed. Vol. 14446. Springer. 2023. p. 313–331. (Lecture Notes in Computer Science). doi: 10.1007/978-3-031-47963-2_19