Extending ALCQIO with reachability

We introduce a description logic ALCQIO_{b,Re} which adds reachability assertions to ALCQIO, a sub-logic of the two-variable fragment of first order logic with counting quantifiers. ALCQIO_{b,Re} is well-suited for applications in software verification and shape analysis. Shape analysis requires expressive logics which can express reachability and have good computational properties. We show that ALCQIO_{b,Re} can describe complex data structures with a high degree of sharing and allows compositions such as list of trees. We show that the finite satisfiability and implication problems of ALCQIO_{b,Re}-formulae are polynomial-time reducible to finite satisfiability of ALCQIO-formulae. As a consequence, we get that finite satisfiability and finite implication in ALCQIO_{b,Re} are NEXPTIME-complete. Description logics with transitive closure constructors have been studied before, but ALCQIO_{b,Re} is the first description logic that remains decidable on finite structures while allowing at the same time nominals, inverse roles, counting quantifiers and reachability assertions,