In the description logic community, the emphasis has not been on views, but on querying incomplete information with constraints. Our positive query implication problems relate to work in the description logic community on hybrid closed and open world query answering or DBoxes, in which the schema is divided into closed-world and open-world relations. Given a Boolean CQ, we want to find out if it holds in all instances that can add facts to the open-world relations but do not change the closed-world relations. In the non-Boolean case, the generalization is to consider which tuples from the initial instance are in the query answer on all such instances. Thus closed-world and open-world relations match our notion of visible and invisible, and the hybrid closed and open world query answering problem matches our notion of positive query implication, except that we restrict to the case where the open-world/visible relations of the instance are empty. It is easy to see that this restriction is actually without loss of generality: one can reduce the general case to the case we study with a simple linear time reduction, making a closed-world copy R ′ of each open-world relation R, and adding an inclusion dependency from R ′ to R. As with the database community, the main distinction between our study of the Positive Query Implication problem and the prior work in the DL community concerns the classes of constraints considered. Lutz et al. [LSW12] study the complexity of this problem for the constraint languages EL and DL-LITE, giving a dichotomy between co-NP-hard and first-order rewritable sets of constraints. They also show that in all the tractable cases, the problem coincides with the classical open-world query answering problem. Franconi et al. [FIS11] show co-NP-completeness for a disjunction-free description logic. Our results on the **data** complexity of PQI consider the same problem, but for decidable constraint languages that are more expressive, and in particular, can handle relations of arbitrary arity, rather than arity at most 2 as in [LSW12,FIS11].