TY - GEN

T1 - Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad

AU - Dagnino, Francesco

AU - Farjudian, Amin

AU - Moggi, Eugenio

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2023

Y1 - 2023

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.

KW - Enriched category

KW - Monad

KW - Quantale

KW - Robustness

KW - Topology

UR - http://www.scopus.com/inward/record.url?scp=85178574644&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-47963-2_19

DO - 10.1007/978-3-031-47963-2_19

M3 - Conference contribution

AN - SCOPUS:85178574644

SN - 9783031479625

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 313

EP - 331

BT - Theoretical Aspects of Computing – ICTAC 2023 - 20th International Colloquium, Proceedings

A2 - Ábrahám, Erika

A2 - Dubslaff, Clemens

A2 - Tarifa, Silvia Lizeth

PB - Springer Science and Business Media Deutschland GmbH

T2 - 20th International Colloquium on Theoretical Aspects of Computing, ICTAC 2023

Y2 - 4 December 2023 through 8 December 2023

ER -