On the Impossibility of Universally Transforming Similarity Metrics into Partial Metrics

Abstract

[eng] Partial metrics generalize traditional metrics by allowing non-zero self-distances. This distinguished property makes them suitable for the development of many applications to computer science, artificial intelligence and applied mathematics. Such distances are interpreted as a dissimilarity measure. However, in cases where the measurement method must quantify the degree of common information between two objects, rather than quantifying the level of di erence between them, it is required to handle the notion of similarity. This paper studies a duality relationship between the so-called similarity metrics and partial metrics. In particular, we focus on the search of a characterization of those functions that transform every similarity metric into a partial metric. While transformations can exist for specific similarity metrics, we prove the non-existence of a function that can universally transform any similarity metric into a partial metric.