Axiom Pinpointing is Hard

We investigate the complexity of several decision, enumeration and counting problems in axiom pinpointing in Description Logics. We prove hardness results that already hold for the propositional Horn fragment. We show that for this fragment, unless p = np, all minimal subsets of a given TBox that have a given consequence, i.e. MinAs, cannot be enumerated in a specified lexicographic order with polynomial delay. Moreover, we show that recognizing the set of all MinAs is at least as hard as recognizing the set of all minimal transversals of a given hypergraph, however whether this problem is intractable remains open. We also show that checking the existence of a MinA that does not contain any of the given sets of axioms, as well as checking the existence of a MinA that contains a specified axiom are both np-hard. In addition we show that counting all MinAs and counting the MinAs that contain a certain axiom are both #p-hard.