Corollary 71 Let U be a splitting set for a hybrid knowledge base K. An MKNF formula φ is MKNF entailed by K if and only if, for every solution h**X, Y** i of K with respect to U, X ∩Y |= φ.

Proof Follows from Theorem 7.

## Appendix C Proof Of Splitting Sequence Theorem For Hybrid Mknf Knowledge Bases

Many parts of the proofs in this section are adapted from (Turner 1996).

## Remark 72

It is easy to see that solutions to K with respect to a splitting sequence hU, Pi are the same as the solutions to K with respect to the splitting set U.

Let U = hUαiα<µ be a splitting sequence for a hybrid knowledge base K, and let hXαiα<µ be a sequence of MKNF interpretations. Then:

$\mathbf{u}=\mathbf{u}^{\dagger}$
pr(bU0(K)) ⊆ U0

$$\operatorname*{pr}\left(e_{U_{\alpha}}\left(b_{U_{\alpha+1}}(\mathcal{K}),\bigcap_{\eta\leq\alpha}X_{\eta}\right)\right)\subseteq U_{\alpha+1}\setminus U_{\alpha}{\mathrm{~whenever~}}\alpha+1<\mu$$

Furthermore, when X is a solution to K with respect to U, then X0 is saturated relative to U0 and for every α such that α + 1 < µ, Xα+1 is saturated relative to Uα+1 \ Uα.

Also note that for any limit ordinal α, Xα = M, so Xα is saturated relative to any set of predicate symbols. Lemma 73 Let hUαiα<µ be a sequence of sets of atoms and hXαiα<µ be a sequence of members of M such that for all **α < µ**, Xα is saturated relative to Uα. Then Tα<µ Xα is saturated relative to Sα<µ Uα.

Proof Let U =Sα<µ Uα and X =Tα<µ Xα and suppose I
[U] belongs to X[U]. Then there is some J ∈ X such that I
[U] = J
[U]. This means that for every ground atom p with pr(p) ⊆ U,
p ∈ I ⇐⇒ p ∈ J .

We need to show that I belongs to X. Take some **β < µ** and any atom p such that pr(p) ⊆
Uβ. Since Uβ is a subset of U, we immediately obtain p ∈ I ⇐⇒ p ∈ J .

Furthermore, since J ∈ X ⊆ Xβ, it follows that I
[Uβ] belongs to X
[Uβ]
β. Moreover, Xβ is saturated relative to Uβ, so we conclude that I belongs to Xβ. Since the choice of β was arbitrary, I belongs to Xβ for all **β < µ**. Thus, I belongs to X as well.