Robustness, Scott Continuity, and Computability

TL;DR

This paper explores the relationship between robustness, Scott continuity, and computability in system analyses, especially when dealing with non-compact metric spaces by using compact Hausdorff space embeddings.

Contribution

It establishes a novel connection between robustness and Scott continuity through an adjunction involving compact Hausdorff spaces, extending the theory to non-compact metric spaces.

Findings

Robust analyses correspond to Scott continuous maps in compact metric spaces.

The main result relates robustness and Scott continuity via an adjunction with compact Hausdorff spaces.

Applications include examples involving Banach spaces.

Abstract

Robustness is a property of system analyses, namely monotonic maps from the complete lattice of subsets of a (system's state) space to the two-point lattice. The definition of robustness requires the space to be a metric space. Robust analyses cannot discriminate between a subset of the metric space and its closure, therefore one can restrict to the complete lattice of closed subsets. When the metric space is compact, the complete lattice of closed subsets ordered by reverse inclusion is w-continuous and robust analyses are exactly the Scott continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity, when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the w-continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.