Automata-based Pinpointing for DLs

Recent years have seen a boom in the creation and development of ontologies. Unfortunately, the maintenance of such ontologies is an error-prone process. On one side, it is in general unrealistic to expect a developer to be simultaneously a domainand an ontology-expert. This leads to problems when a part of the domain is not correctly understood, or when, although correctly understood, is translated wrongly to the ontology language. On the other side, most of the larger ontologies are developed by a group of individuals. The difference in their points of view can produce unexpected consequences. Whenever an error is identified, one would like to be able to detect the portion of the ontology responsible for such it; additionally, it would also be desirable to modify the ontology as little as possible to remove the error. If, for instance, an ontology is expresed by a TBox of an expressive Description Logic (DL), an unwanted consequence could be the unsatisfiability of a certain concept term C. Given that C is indeed unsatisfiable, we can search for a minimal sub-TBox that still leads to unsatisfiability of the concept (explaining the consequence), or for a maximal sub-TBox where C is satisfiable (removing the consequence). Finding these sets by hand in large ontologies is not a viable option. Schlobach and Cornet [14] describe an algorithm for computing the minimal subsets of an unfoldable ALC-terminology that keep the unsatisfiability of a concept. The algorithm extends the known tableau-based satisfiability algorithm for ALC [15], using labels to keep track of the axioms responsible of the generation of an assertion during the execution of the algorithm. A similar approach was actually presented previously in [2], for checking consistency of ALC-ABoxes. The main difference between the algorithms in [14] and [2] is that the latter does not directly compute the minimal subsets that have the consequence, but rather a Boolean formula, called pinpointing formula, whose minimal satisfying valuations correspond to the minimal sub-ABoxes that are inconsistent. The ideas sketched by these algorithms have been applied to other tableau-based decision algorithms for more expressive DLs (see, e.g. [13, 12, 11]), and generalized in [3] where so-called general tableaux are extended into pinpointing algorithms that compute a formula as in [2]. This general approach was then successfuly applied for explaining subsumption relations in EL [4]. The main drawback of the general approach in [3] is that it assumes that the original tableau algorithm stops after a finite number of steps without the