How Many Legs Do I Have? Non-Simple Roles in Number Restrictions Revisited

The Description Logics underpinning OWL impose a well-known syntactic restriction in order to preserve decidability: they do not allow to use nonsimple roles—that is, transitive roles or their super-roles—in number restrictions. When modeling composite objects, for example in bio-medical ontologies, this restriction can pose problems. Therefore, we take a closer look at the problem of counting over non-simple roles. On the one hand, we sharpen the known undecidability results and demonstrate that: (i) for DLs with inverse roles, counting over non-simple roles leads to undecidability even when there is only one role in the language; (ii) for DLs without inverses, two transitive and an arbitrary role are sufficient for undecidability. On the other hand, we demonstrate that counting over non-simple roles does not compromise decidability in the absence of inverse roles provided that certain restrictions on role inclusion axioms are satisfied.