Pinpointing in the Description Logic $\mathcal {EL}^+$

Abstract

Axiom pinpointing has been introduced in description logics (DLs) to help the user understand the reasons why consequences hold by computing minimal subsets of the knowledge base that have the consequence in question. Until now, the pinpointing approach has only been applied to the DL $\mathcal {ALC}$ and some of its extensions. This paper considers axiom pinpointing in the less expressive DL $\mathcal {EL}^+$, for which subsumption can be decided in polynomial time. More precisely, we consider an extension of the pinpointing problem where the knowledge base is divided into a static part, which is always present, and a refutable part, of which subsets are taken. We describe an extension of the subsumption algorithm for $\mathcal {EL}^+$ that can be used to compute all minimal subsets of (the refutable part of) a given TBox that imply a certain subsumption relationship. The worst-case complexity of this algorithm turns out to be exponential. This is not surprising since we can show that a given TBox may have exponentially many such minimal subsets. However, we can also show that the problem is not even output polynomial, i.e., unless P=NP, there cannot be an algorithm computing all such minimal sets that is polynomial in the size of its input and output. In addition, we show that finding out whether there is such a minimal subset within a given cardinality bound is an NP-complete problem. In contrast to these negative results, we also show that one such minimal subset can be computed in polynomial time. Finally, we provide some encouraging experimental results regarding the performance of a practical algorithm that computes one (small, but not necessarily minimal) subset that has a given subsumption relation as consequence.