Towards Measuring Similarity in Description Logics

We review several kinds of previously studied concept similarity measures, and then rephrase them in terms of a simple DL. We discuss the difficulties encountered in trying to generalize these formulations to more complex DLs, and settle on one based on probability/information theory as being the most principled. 1 Motivation and Goals The idea of measuring concept similarity has received considerable attention in several domains, including psychology, cognitive science, and computational linguistics and information retrieval. This work has been applied and extended recently in the field of Information Integration, which often relies on ontologies and hence concepts described in DLs. Most past work has concentrated on the similarity of “atomic” concepts (word senses), rather than composite, defined concepts, which are the stock-in-trade of DLs. This has sometimes been characterized as the difference between considering nouns appearing in a dictionary, e.g. WordNet, and general arbitrary noun phrases. The modest goal of this paper is to consider the problem of generalizing previous efforts to define similarity for primitive concepts to composite ones. To this end, we review three classes of approaches found in the current literature, rephrase them in terms of a very simple description logic, and then try to generalize this to a complex DL. These extensions invite a host of new questions that we leave open for further research. To begin with, we make several observations about possible real-valued function(s) sim(C,D) that would measure concept similarity in general. First, any effort to assess simlarity via absolute numeric values seems ill-advised — in small experiments even human judgments have been shown to correlate only in the . 91 to . 94 range. A less exacting target is to use sim to provide relative orderings over concepts. Some researchers have studied a complementary measure, distance dist(C,D), which can similarly be used for ordering alternatives. To the extent that two functions provide similar orderings (e.g., consistently differing by a constant factor) they are essentially indistinguishable for our purposes. Nonetheless, there are two important general properties that are widely agreed to be desirable for sim(C,D): 1. sim(C,D) is positively correlated with the amount of commonality between C and D. 2. sim(C,D) is negatively correlated with the amount of difference between C and D. The key question then becomes how to measure “commonality”; in turn, this is related to how information about the concept is captured. 2 Three approaches to concept similarity Clearly, in order to compare concepts we need to consider what information we have available about them. We follow Rodriguez [14] in categorizing a variety of approaches, and for each we give a few of the best known functions, which we shall follow. Feature-based Models In this approach, a concept C is characterized by a a set of features ftrs(C). In his pioneering work on similarity, the psychologist Amos Tversky [16] introduced two families of measures: the “contrast model”, where the similarity of C and D is a linear function: contrasttv(C,D) = θf(ftrs(C) ∩ ftrs(D)) − αf(ftrs(C) \ ftrs(D))− βf(ftrs(D) \ ftrs(C)), where \ is set difference, θ, α, and β are non-negative constants, and where f(.) is often taken as the count of features in the set (written |.|). Tversky also proposed a normalized “ratio model”, where similarity is a fraction involving these sets: