Example 2.17 **Consider again the running Example 2.4. We provide the results**
of the computation.

| P1 = {D(b), o(a), o(b)}       | N1 = KA(KG) \ {E(b), D(a)}       |
|-------------------------------|----------------------------------|
| P2 = {D(b), o(a), o(b), p(a)} | N2 = KA(KG) \ {E(b), D(a), p(b)} |
| P3 = P2                       | N3 = N2                          |

Thus we obtain the well-founded MKNF model as MW F = {π(O)} ∪ {D(b), o(a), o(b), p(a)} ∪ {not E(b), not D(a), not p(b)}.

Consequently, E(a**) is undefined.**
To ease some proofs in the following section, we also adapt the notion of unfounded sets [van Gelder et al. 1991] for Hybrid MKNF which relates to the sequence Ni **(see [Knorr et al. 2011]). The essential advantage is that the reasons**
why a certain atom is considered false are better characterized. For that purpose, we first need to define a notion of dependency that captures more **precisely the**
derivations from OBO,S, for some S**, by the operator** DK.

Definition 2.18 Let K = (O,P**) be a ground Hybrid MKNF knowledge base,** H an atom with H ∈ KA(K), and S **a (possibly empty) set of atoms with** S ⊆ KA(K).

We say that H depends on S **if and only if**
(i) OBO,S |= H and
(ii) there is no S
′ **with** S
′ ⊂ S **such that** OBO,S′ |= H.

Intuitively, S is a minimal set that, in combination with O**, allows us to derive** H**. Note that there may exist several such minimal sets. Based on this notion** of dependency, the notion of an unfounded set can be extended to Hybrid MKNF KBs.

Definition 2.19 Let K be a ground Hybrid MKNF knowledge base and (T, F) a pair of sets of atoms with T, F ⊆ KA(K). We say that U ⊆ KA(K) is an **unfounded**
set (of K) with respect to (T, F) if, for each atom H ∈ U**, the following conditions**
are satisfied:
(Ui) for each rule H ← B in P **at least one of the following holds.**
(Uia) Some atom A appears in B **and in** U ∪ F. (Uib) Some negative literal not B appears in B **and in** T .

(Uic) OBO,T |= ¬H
(Uii) for each (possibly empty) S on which H depends, with S ⊆ KA(K**) and** OBO,S
consistent, there is at least one atom A such that OBO,S\{A} 6|= H and A in U ∪F.

The union of all unfounded sets of K w.r.t. (T, F) is called the **greatest unfounded**
set of K w.r.t. (T, F) and denoted UK(**T, F**).

It can be shown that the computation of Ni based on Γ′K **directly corresponds**
to the computation of the greatest unfounded set w.r.t. (Pi−1, Ni−1).

ACM Transactions on Computational Logic, Vol. V, No. N, September 2018.