Relating Distances and Abstractions

We establish a formal relation between quantitative and semantic approximations—formalized by pre-metrics and upper closure operators (ucos), respectively—by means of Galois connections. This connection reveals that it is far from trivial for a pre-metric to uniquely identify a uco, highlighting the structural constraints and, more generally, the distinct identity inherent to semantic approximations. Building on this foundation, we introduce a general composition of semantic and quantitative approximations. This allows us to define a new confidentiality property, called Partial Abstract Non-Interference, that measures bounded variations in program behavior over abstract properties of data. We then relate this property to Partial Completeness in abstract interpretation, revealing a deeper connection between static analysis precision and security guarantees.