We present a Protégé plugin for explaining missing entailments from OWL ontologies. The importance of explaining description logic reasoning to end-users has long been understood, and has been studied in many forms over the past decades. Indeed, explainability is one of the main advantages of logic-based knowledge representations over sub-symbolic methods. The first approaches to explain why a consequence follows from a Description Logic (DL) ontology were based on step-by-step proofs [8, 18], but soon research focused on justifications [7, 11, 20] that are easier to compute, but still very useful for pointing out the axioms responsible for an entailment. Consequently, the ontology editor Protégé supports black-box methods for computing justifications for arbitrary OWL DL ontologies [12]. More recently, a series of papers investigated different methods of computing good proofs for entailments in DLs ranging from EL to ALCOI [13, 1, 2, 3], and the Protégé plug-ins proof-explanation [13] and Evee [4], as well as the web-based application Evonne [19], were developed to make these algorithms available to ontology engineers. While reasoning can sometimes reveal unexpected entailments that need explaining, very often the problem is not what is entailed, but what is not entailed. In order to explain such missing entailments, and offer suggestions on how to repair them, both counterexamples and abduction have been suggested in the literature.

Logic-based approaches to AI have the advantage that their behavior can in principle be explained to a user. If, for instance, a Description Logic reasoner derives a consequence that triggers some action of the overall system, then one can explain such an entailment by presenting a proof of the consequence in an appropriate calculus. How comprehensible such a proof is depends not only on the employed calculus, but also on the properties of the particular proof, such as its overall size, its depth, the complexity of the employed sentences and proof steps, etc. For this reason, we want to determine the complexity of generating proofs that are below a certain threshold w.r.t. a given measure of proof quality. Rather than investigating this problem for a fixed proof calculus and a fixed measure, we aim for general results that hold for wide classes of calculi and measures. In previous work, we first restricted the attention to a setting where proof size is used to measure the quality of a proof. We then extended the approach to a more general setting, but important measures such as proof depth were not covered. In the present paper, we provide results for a class of measures called recursive, which yields lower complexities and also encompasses proof depth. In addition, we close some gaps left open in our previous work, thus providing a comprehensive picture of the complexity landscape.