A Logic for Concepts and Similarity

Categorization 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 categorizing 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 article, we propose a DL-based integration of the two classification methods. Our formalism, called $\mathrm{SL}+\mathrm{ALCQIO}$, extends the expressive DL $\mathrm{ALCQIO}$ by means of the constructors of the similarity logic $\mathrm{SL}$ 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, $\mathrm{SL}+\mathrm{ALCQIO}$ 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 $\mathrm{SL}+\mathrm{ALCQIO}$ can be decomposed into the corresponding problems for its DL-part $\mathrm{ALCQIO}$ and similarity part $\mathrm{SL}$. Then we investigate reasoning in $\mathrm{SL}$ 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 $\mathrm{ALCQIO}$, we prove that reasoning in $\mathrm{SL}+\mathrm{ALCQIO}$ is NExpTime-complete. As the `closer' operator of $\mathrm{SL}$ has the same expressive power as the standard implication > of conditional logic, these results may have interesting consequences for conditional logic as well.