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

TL;DR

This paper explores robustness and imprecision in quantale-valued metric spaces, establishing a connection between the robust topology and the Hausdorff-Smyth hemi-metric through a new monad, with implications for quantitative reasoning in systems.

Contribution

It introduces the Hausdorff-Smyth monad for quantale-valued metric spaces and relates the induced topology to the robust topology, advancing the understanding of robustness in generalized metric spaces.

Findings

Established a relation between the robust topology and the Hausdorff-Smyth hemi-metric.

Defined a preorder-enriched monad, $ extsf{P}_S$, for quantale-valued metric spaces.

Connected the topology on $ extsf{P}_S(X)$ with robustness measures in systems.

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 $\mathsf{P}_S$, 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 $\mathsf{P}_S(X)$ with the robust topology on the powerset $\mathsf{P}(X)$ defined in terms of the metric on $X$.