the way the ontology can call back program atoms. In [Eiter et al. 2011], the set of atoms occurring in rules and DLs are disjoint, and links must be established using specific interface atoms in the rules, which can only temporarily add information to the DL part. To the contrary, our semantics does not require any **such restriction**
so that the flow of information between rules and DLs is not limited. The wellfounded semantics for normal dl-programs [Lukasiewicz 2010] does not require any of these limitations either, but it requires that the ontology is decomposable into a positive and a negative part. This severely restricts the applicability **to arbitrary**
DLs although it is shown in [Lukasiewicz 2010] that the approach is applicable to the DL-Lite family. This contrasts with SLG(O)**, which can be applied to any**
decidable DL. Hybrid programs of [Drabent and Ma luszy´nski 2007] are even more restrictive than [Eiter et al. 2011] in the combination: in fact it only allows to transfer information from the ontology to the rules and not the other way around. Moreover, the semantics of this approach differs from MKNF [Motik and Rosati 2010; Knorr et al. 2011] and also [Eiter et al. 2011; Lukasiewicz 2010] in that if an ontology expresses B1 ∨ B2**, then the semantics in [Drabent and Ma luszy´nski 2007]** derives p from rules p ← B1 and p ← B2, p **while MKNF and [Eiter et al. 2011;**
Lukasiewicz 2010] do not. More generally, several well-founded models may exist, contrary to the more common definitions of well-founded models.

In Section 6, we also presented a concrete oracle for SLG(O) **that allows the**
combination of non-monotonic rules with the DL EL+. Using this oracle, **SLG(**O)
remains tractable w.r.t. data complexity, and permits the discovery **of possible**
inconsistencies between the rules and the ontology. These results **contribute to the**
work related to conjunctive query answering with respect to EL+**. Conjunctive**
query answering has been studied, e.g., for acyclic EL+ **in [Mei et al. 2009] as an**
extension to [Lutz et al. 2009], where the limitation to acyclic TBoxes avoids general undecidability (see [Rosati 2007]). In contrast, our work limits the queries to be DL-safe but adds rules as an additional expressive means. As an additional point of comparison, since the concrete oracle operates in a kind of abductive way - by finding the set of atoms which together with the ontology prove the **query - our** work also bears some relation to [Bienvenu 2008] where general complexity results on abduction for EL++ are established. Another concrete oracle for SLG(O) was very recently presented in [Knorr and Alferes 2011] providing a top-down procedure for DL-LiteR**, the DL underlying the tractable OWL 2 profile, OWL 2 QL. As does**
the EL+ oracle, the DL-LiteR **oracle maintains the data complexity of the bottomup approach.**

## 7.2 Conclusions

Together with the alternate computation method of Section 3, SLG(O) **provides a** sound and complete querying method for Hybrid MKNF knowledge bases. Further, SLG(O) **maintains the favorable computational complexity of the well-founded** MKNF model and freely allows bidirectional calls between the ontology **and the** rules, unlike other approaches (as discussed in Section 7.1). As such it presents a significant step towards making Hybrid MKNF knowledge bases practically usable for the Semantic Web.

Future work with regard to concrete oracles includes the (non-trivial) extension of the EL+ oracle to EL++**. A second potentially fruitful extension is the construction**
ACM Transactions on Computational Logic, Vol. V, No. N, September 2018.