On the Decidability of Expressive Description Logics with Transitive Closure and Regular Role Expressions

We consider fragments of the description logic SHOIF extended with regular expressions on roles. Our main result is that satisfiability and finite satisfiability are decidable in two fragments SHOIF^1 and SHOIF^2, NExpTime-complete for the former and in 2NExpTime for the more expressive latter fragment. Both fragments impose restrictions on regular role expressions of the form r*. SHOIF^1 encompasses the extension of SHOIF with transitive closure of roles (when functional roles have no subroles) and the modal logic of linear orders and successor, with converse. Consequently, these logics are also decidable and NExpTime-complete.