Decidability of SHI with Transitive Closure of Roles

This paper investigates a Description Logic, namely $\mathcal{SHI}_+$, which extends $\mathcal{SHI}$ by adding transitive closure of roles. The resulting logic $\mathcal{SHI}_+$ allows transitive closure of roles to occur not only in concept inclusion axioms but also in role inclusion axioms. We show that $\mathcal{SHI}_+$ is decidable by devising a terminating, sound and complete algorithm for deciding satisfiability of concepts in $\mathcal{SHI}_+$ with respect to a set of concept and role inclusion axioms.