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 -