Concept Dissimilarity with Triangle Inequality

ABSTRACT. Several researchers have developed properties that ensure compatibility of a concept similarity or dissimilarity measure with the formal semantics of Description Logics. While these authors have highlighted the relevance of the triangle inequality, none of their proposed dissimilarity measures satisfies it. In this work we present several dissimilarity measures with this property: first, a simple dissimilarity measure, based on description trees for the lightweight Description Logic EL; secondly, a general framework based on concept relaxations; last, an instantiation of the general framework using dilation operators from mathematical morphology, exploiting the link between Hausdorff distance and dilations using balls of the ground distance as structuring elements. A comparison between these definitions and their properties is provided as well.