Pinpointing in the Description Logic EL

For a developer or user of a DL-based ontology, it is often quite hard to understand why a certain consequence holds, and even harder to decide how to change the ontology in case the consequence is unwanted. For example, in the current version of the medical ontology SNOMED [16], the concept Amputationof-Finger is classified as a subconcept ofAmputation-of-Arm. Finding the axioms that are responsible for this among the more than 350,000 terminological axioms of SNOMED without support by an automated reasoning tool is not easy. As a first step towards providing such support, Schlobach and Cornet [14] describe an algorithm for computing all the minimal subsets of a given knowledge base that have a given consequence. In the following, we call such a set a minimal axiom set (MinA). It helps the user to comprehend why a certain consequence holds. The knowledge bases considered in [14] are so-called unfoldable ALC-terminologies, and the unwanted consequences are the unsatisfiability of concepts. The algorithm is an extension of the known tableau-based satisfiability algorithm for ALC [15], where labels keep track of which axioms are responsible for an assertion to be generated during the run of the algorithm. The authors also coin the name “axiom pinpointing” for the task of computing these minimal subsets. The problem of computing MinAs of a DL knowledge base was actually considered earlier in the context of extending DLs by default rules. In [2], Baader and Hollunder solve this problem by introducing a labeled extension of the tableaubased consistency algorithm for ALC-ABoxes [9], which is very similar to the one described later in [14]. The main difference is that the algorithm described in [2] does not directly compute minimal subsets that have a consequence, but rather a monotone Boolean formula whose variables correspond to the axioms of the knowledge bases and whose minimal satisfying valuations correspond to the MinAs. The approach of Schlobach and Cornet [14] was extended by Parsia et al. [12] to more expressive DLs, and the one of Baader and Hollunder [2] was extended by Meyer et al. [11] to the case of ALC-terminologies with general concept inclusions (GCIs), which are no longer unfoldable. Axiom pinpointing has also been considered in other research areas, though usually not under this name.