Reasoning and Query Answering Using Concept Similarity Measures and Graded Membership Functions

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Description Logics (DLs) are a well-investigated family of logic-based knowledge representation languages, which are, e.g., frequently used to formalize ontologies for application domains such as biology and medicine. To define the important notions of such an application domain as formal concepts, DLs state necessary and sufficient conditions for an individual to belong to a concept. Once the relevant concepts of an application domain are formalized this way, they can be used in queries in order to retrieve new information from data. Since traditional DLs are based on classical first-order logic, their semantics is strict in the sense that all the stated properties need to be satisfied for an individual to belong to a concept, and the same is true for answers to queries. In applications where exact definitions are hard to come by, it would be useful to relax this strict requirement and allow for approximate definitions of concepts, where most, but not all, of the stated properties are required to hold. Similarly, if a query has no exact answer, approximate answers that satisfy most of the features the query is looking for could be useful.In order to allow for approximate definitions of concepts, we have introduced the notion of a graded membership function, which instead of a Boolean membership value 0 or 1 yields a membership degree from the interval [0,1]. Threshold concepts can then, for example, require that an individual belongs to a concept C with degree at least 0.8. A different approach, which is based on the use of similarity measures on concepts, was used to relax instance queries (i.e., queries that consist of a single concept). Given a query concept C, we are looking for answers to queries D whose similarity to C is higher than a certain threshold. While these two approaches were developed independently by the two applicants together with doctoral students, it has turned out that there are close connections. Similarity measures can be used to define graded membership functions, and threshold concepts w.r.t. these functions provide a more natural semantics for relaxed instance queries.The goals of this project are to1. explore the connection between the two approaches in detail, and develop and investigate a combined approach; 2. extend the applicability of this combined approach by considering a) more expressive DLs than the DL EL for which the approaches were initially developed; b) more general terminological formalisms; c) more general queries such as conjunctive queries; d) more general inference problems such as determining appropriate threshold values rather than requiring the user to provide them.