Fajar Haifani

Abduction in description logics finds extensions of a knowledge base to make it entail an observation. As such, it can be used to explain why the observation does not follow, to repair incomplete knowledge bases, and to provide possible explanations for unexpected observations. We consider TBox abduction in the lightweight description logic $$\mathcal {EL}$$EL , where the observation is a concept inclusion and the background knowledge is a TBox, i.e., a set of concept inclusions. To avoid useless answers, such problems usually come with further restrictions on the solution space and/or minimality criteria that help sort the chaff from the grain. We argue that existing minimality notions are insufficient, and introduce connection minimality. This criterion follows Occam’s razor by rejecting hypotheses that use concept inclusions unrelated to the problem at hand. We show how to compute a special class of connection-minimal hypotheses in a sound and complete way. Our technique is based on a translation to first-order logic, and constructs hypotheses based on prime implicates. We evaluate a prototype implementation of our approach on ontologies from the medical domain.

A clause C is syntactically relevant in some clause set N, if it occurs in every refutation of N. A clause C is syntactically semi-relevant, if it occurs in some refutation of N. While syntactic relevance coincides with satisfiability (if C is syntactically relevant then $$N\setminus \{C\}$$N \ { C } is satisfiable), the semantic counterpart for syntactic semi-relevance was not known so far. Using the new notion of a conflict literal we show that for independent clause sets N a clause C is syntactically semi-relevant in the clause set N if and only if it adds to the number of conflict literals in N. A clause set is independent, if no clause out of the clause set is the consequence of different clauses from the clause set.Furthermore, we relate the notion of relevance to that of a minimally unsatisfiable subset (MUS) of some independent clause set N. In propositional logic, a clause C is relevant if it occurs in all MUSes of some clause set N and semi-relevant if it occurs in some MUS. For first-order logic the characterization needs to be refined with respect to ground instances of N and C.

We prove the SOS strategy for first-order resolution to be refutationally complete on a clause setNand set-of-supportSif and only if there exists a clause inSthat occurs in a resolution refutation from $$N\cup S$$N∪S . This strictly generalizes and sharpens the original completeness result requiringNto be satisfiable. The generalized SOS completeness result supports automated reasoning on a new notion of relevance aiming at capturing the support of a clause in the refutation of a clause set. A clauseCisrelevantfor refuting a clause setNifCoccurs in every refutation ofN. The clauseCissemi-relevant, if it occurs in some refutation, i.e., if there exists an SOS refutation with set-of-support $$S = \{C\}$$S={C} from $$N\setminus \{C\}$$N\{C} . A clause that does not occur in any refutation fromNisirrelevant, i.e., it is not semi-relevant. Our new notion of relevance separates clauses in a proof that are ultimately needed from clauses that may be replaced by different clauses. In this way it provides insights towards proof explanation in refutations beyond existing notions such as that of an unsatisfiable core.