Forgetting and Uniform Interpolation in Extensions of the Description Logic EL

Terminologies used in life sciences, health-care, and other knowledge intensive areas are often very large and comprehensive. Examples are the medical terminology SNOMED CT (Systematised Nomenclature of Medicine Clinical Terms) [17] containing about 380 000 concept definitions and the National Cancer Institute ontology (NCI) [16] containing more than 60 000 axioms. For terminologies T of this size, it is often of interest to forget a subvocabularyΣ of the vocabulary of T ; i.e., to transform T into a new terminology TΣ (called aΣ-interpolant of T ) that contains no symbols from Σ and that is indistinguishable from T regarding its consequences that do not use Σ. In AI, this problem has been studied under a variety of names such as forgetting and variable elimination [14, 4, 10]. In mathematical logic, this problem has been investigated as the uniform interpolation problem [18]. Computing Σ-interpolants of terminologies has a number of potential applications, e.g., Re-use of ontologies: when using terminologies such as SNOMED CT in an application, often only a very small fraction of its vocabulary is of interest. In this case, one could use a Σ-interpolant instead of the whole terminology, where Σ is the vocabulary not of interest for the application. Predicate hiding: a terminology developer might not want to publish a terminology completely because a certain part of its vocabulary is not intended for public use. Again, publishing Σ-interpolants, where Σ is the vocabulary to be hidden, appears to be a solution to this problem. Exhibiting hidden relations between terms: large terminologies are difficult to maintain as small changes to its axioms can have drastic and damaging effects. To analyze possibly unwanted consequences over a certain part Γ of the vocabulary, an ontology developer can automatically generate a complete axiomatization of the relations between terms over Γ by computing a Σ-interpolant, where Σ is the complement of Γ . Ontology versioning: to check whether two versions of a terminology have the same consequences over their common vocabulary (or a subset thereof), one can first compute their interpolants by forgetting the vocabulary not shared by the two versions and then check whether the two interpolants are logically equivalent (i.e., have the same models). In the description of Σ-interpolants given above, we have neither specified a language in which they are axiomatized nor did we specify the language wrt. which Σinterpolants should be indistinguishable from the original termininology. The choice of