On O’Neill Partial Metrics, Similarity Metrics, Transformations and Topology

Partial metrics have shown to be useful dissimilarity measures when incomplete information, partial states, or inherent uncertainty is involved. The main characteristic of this kind of distances is allowing non-zero self-distances. This distinctive property makes them particularly appropriate for applications to computer science, artificial intelligence, pattern recognition and bioinformatics. Nevertheless, in these fields it is often more relevant to quantify the amount of shared information between objects rather than their dissimilarity. In this context, similarity metrics have proven to be a valuable tool. The literature has suggested the existence of a duality relationship between partial metrics and similarity metrics. In this paper we investigate such a relationship. Specifically, we focus on identifying the properties of functions that induce a similarity metric from a partial metric in the sense of O’Neill. We provide a characterization of these functions, showing that they coincide with the class of strictly decreasing and convex functions on the set of non-negative real numbers. We also show that these functions preserve the topology and the partial order, that is, the partial order and topology generated by the induced similarity metric and by the original partial metric are the same. Besides, we characterize the class of functions capable of generating an O’Neill partial metric from a similarity metric showing that such a class is formed by strictly decreasing and concave functions on the set of real numbers. In this case we also show that the partial orders and the topologies generated by the induced partial metric and by the original similarity metric coincide. The results are supported and clarified by appropriate examples.