Enumerating Justifications using Resolution

If a conclusion follows from a set of axioms, then its justification is a minimal subset of axioms for which the entailment holds. An entailment can have several justifications. Such justifications are commonly used for the purpose of debugging of incorrect entailments in Description Logic ontologies. Recently a number of SAT-based methods have been proposed that can enumerate all justifications for entailments in light-weight ontologies languages, such as \(\mathcal {EL}\). These methods work by encoding \(\mathcal {EL}\) inferences in propositional Horn logic, and finding minimal models that correspond to justifications using SAT solvers. In this paper, we propose a new procedure for enumeration of justifications that uses resolution with answer literals instead of SAT solvers. In comparison to SAT-based methods, our procedure can enumerate justifications in any user-defined order that extends the set inclusion relation. The procedure is easy to implement and, like resolution, can be parametrized with ordering and selection strategies. We have implemented this procedure in PULi—a new Java-based Proof Utility Library, and performed an empirical comparison of (several strategies of) our procedure and SAT-based tools on popular \(\mathcal {EL}\) ontologies. The experiments show that our procedure provides a comparable, and often better performance than those highly optimized tools. For example, using one of the strategies, we were able for the first time to compute all justifications for all entailed concept subsumptions in one of the largest commonly used medical ontology Snomed CT.