Computing Least Common Subsumers in Description Logics

Description logics are a popular formalism for knowledge representation and reasoning. This paper introduces a new operation for description logics: computing the “least common subsumer” of a pair of descriptions. This operation computes the largest set of commonalities between two descriptions. After arguing for the usefulness of this operation, we analyze it by relating computation of the least common subsumer to the well-understood problem of testing subsumption; a close connection is shown in the restricted case of “structural subsumption”. We also present a method for computing the least common subsumer of “attribute chain equalities”, and analyze the tractability of computing the least common subsumer of a set of descriptions -an important operation in inductive learning. Introduction and Motivation Description logics (DLs) or terminological logics are a family of knowledge representation and reasoning systems that have found applications in several diverse areas, ranging from database interfaces [Beck et al., 19891, to software information bases [Devanbu et al., 19911 to financial management [Mays et al., 19871. They have also received considerable attention from the research community (e.g., [Woods and Schmolze, 1.9921.) DLs are to used to reason about descriptions, which describe sets of atomic elements called individuals. Individuals can be organized into primitive classes, which denote sets of individuals, and are related through binary relations called roles (or attributes when the relation is functional). For example, the individuals Springsteen and BorntoRun might be related by the sings role, and Springsteen might be an instance of the primitive class PERSON. Descriptions are composite terms that denote sets of individuals, and are built from primitive classes (such a.s PERSON), and restrictions on the properties an individual may have, such as the kinds or number *On sabbatical leave. of role fillers (e.g., “persons that sing at least 5 things”). For example, the statement “all songs sung by Springsteen (and there are at least 6) are set in New Jersey” could be expressed by attaching to the individual Springsteen the description (AND (AT-LEAST 5 sings) (ALL sings (FILLS setting NJ))). Knowledge base management systems (KBMS) based on DLs perform a number of basic operations on descriptions: for example, checking if a description is incoherent, or determining if two descriptions are disjoint. An especially important operation on descriptions is testing subsumption: DI subsumes D2 iff it is more general than D2. Efficient implementation of such operations allows a KBMS to organize knowledge, maintain its consistency, answer queries, and recognize conditions that trigger rule firings. This paper introduces a new operation for description logics: computing the least common subsumer (LCS) of a pair of concept (i.e., finding the most specific description in the infinite space of possible descriptions that subsumes a pair of concepts.‘) This operation can also be thought of as constructing a concept that describes the largest set of commonalities between two other concepts. In logic programming, similar operations called “least general generalization” and “relative least i! eneral generalization” have been extensively studied Frisch and Page, 1990; Plotkin, 1969; Buntine, 19SS]; in the context of DLs, there are a number of circumstances in which the LCS operation is useful. Learning from examples. Finding the least general concept that generalizes a set of examples is a common operation in inductive learning. For example, [Valiant, 19841 proved that L-CNF (a class of Boolean functions) can be probabilistically learned by computing the LCS of a set of positive examples; also, many experimental learning systems make use ‘This should n ot be confused with the “most specific subsume?’ operation, which searches the (finite) space of raarraed concepts to find the most specific named concept(s) that subsumes a single concept [Woods and Schmolze, 19921. 754 Representat ion and Reasoning: Terminological From: AAAI-92 Proceedings. Copyright ©1992, AAAI (www.aaai.org). All rights reserved.