A Logic for Concepts and Similarity
Mikhail Sheremet, Dmitry Tishkovsky, Frank Wolter, Michael Zakharyaschev · Journal of Logic and Computation · 2007
Categorisation of objects into classes is currently supported by (at least) two ‘orthogonal’ methods. In logic-based approaches, classifications are defined through ontologies or knowledge bases which describe the existing relationships among terms. Description logic (DL) has become one of the most successful formalisms for representing such knowledge bases, in particular because theoretically well-founded and efficient reasoning tools have been readily available. In numerical approaches, classifications are obtained by first computing similarity (or proximity) measures between objects and then categorising them into classes by means of Voronoi tessellations, clustering algorithms, nearest neighbour computations, etc. In many areas such as bioinformatics, computational linguistics or medical informatics, these two methods have been used independently of each other: although both of them are often applied to the same domain (and even by the same researcher), up to now no formal interaction mechanism has been developed. In this paper, we propose a DL-based integration of the two classification methods. Our formalism, called SL + ALCQIO, extends the expressive DL ALCQIO by means of the constructors of the similarity logic SL which allow definitions of concepts in terms of both comparative and absolute similarity. In the combined knowledge base the user should declare the similarity spaces where the new operators are interpreted. Of course, SL + ALCQIO can only be useful if classifications with this logic are supported by automated reasoning tools. We lay theoretical foundations for the development of such tools by showing that reasoning problems for SL + ALCQIO can be decomposed into the corresponding problems for its DL-part ALCQIO and similarity part SL. Then we investigate reasoning in SL and prove that consistency and many other reasoning problems are ExpTime-complete for this logic. Using this result and a recent complexity result of Pratt-Hartmann for ALCQIO, we prove that reasoning in SL + ALCQIO is