(i2) C1 ⊓C2 ⊑⊥: the statement expresses disjointness of C1 and C2**, i.e.,** ¬(C1(x)∧
C2(x)) for all x which is equivalent to C1(x) → ¬C2(x) and C2(x) → ¬C1(x); using the correspondences ¬C1(x) → NC1(x**) and** ¬C2(x) → NC2(x).

(i3) ∃R.C ⊑⊥ **follows the same argument as (i2).**

## This Finishes The Proof.

For the MKNF-inconsistent case, we point out that one result of the transformation into rules is that we obtain a somewhat paraconsistent approach: while an inconsistent ontology allows us to derive anything from it, the process of doubling the rules enables us to derive those consequences that do not depend on inconsistent information contained in the KB as presented in Example 5.6. We leave further details of this paraconsistency to future studies.

Finally, we have to show that the process of translating the ontology into rules and reasoning over the combined set of rules with SLG(O) **also preserves the intended**
polynomial data complexity.

Theorem 6.8 Let K = (O,P) **be a Hybrid MKNF knowledge base with** O in EL+.

An SLG(O) evaluation of a query in KEL+ = (∅,(P
d ∪ PdO)) **is decidable with data**
complexity in P.

Proof. The EL+ **oracle is in fact a transformation of the evaluation of the** EL+
ontology into a set of rules so that evaluation of the Hybrid MKNF KB is **made** w.r.t. a combined set of rules. Note that the polynomial subsumption **algorithm**
for EL+ and the linear transformations to obtain KEL+ **together are in P.**
We consider the data complexity to be the number of answers returned for a given atomic query w.r.t. the number ground facts in the rules, and the number of assertions in the ABox. Note that the transformation of the EL+ **axioms introduces**
a number of facts in the rules at most linear in the size of the ABox (cases a1 and a2 **of Definition 6.6). As a result the number of facts in the transformed system** will be linear in the size of the rule facts plus the ABox of the original system.

Finally, note that Theorem 5.7 ensures polynomial data complexity of **query**
evaluation in Hybrid MKNF. The transformed KB KEL+ **can be considered a Hybrid**
MKNF KB with empty O**, so that by Theorem 5.7 the transformed KB has a data** complexity in P. Since the transformed KB increases the size of the rule facts linearly this proves the statement.

## 7. Discussion And Conclusions 7.1 Related Work

Three other semantics define well-founded models for a combination **of rules and** ontologies, namely the works in [Eiter et al. 2011], [Lukasiewicz 2010], and [Drabent and Ma luszy´nski 2007]. The approach of [Eiter et al. 2011] combines ontologies and rules in a modular way, keeping separate the semantics of both, and **has identical** data complexity to the well-founded MKNF semantics for a tractable **DL. As such,** it has similarities with SLG(O) **in terms of reasoning, in the sense that both treat**
reasoning in the DL separately. However, the approach of [Eiter et **al. 2011], implemented using the dlv hex system [Eiter et al. 2006], has a looser integration, limiting**
ACM Transactions on Computational Logic, Vol. V, No. N, September 2018.