Splitting and Updating Hybrid Knowledge Bases (Extended Version) 63 To sum up, define the sequence of interpretations Z = hZβiβ<ν by Zβ = σ(**M, T**β). We know the following:
- Z0 = σ(**M, T**0) is a dynamic MKNF model of K′0 = bV0(K);
- for any ordinal β such that β + 1 < ν, Zβ+1 = σ(**M, T**β+1) is a dynamic MKNF
model of K′β+1 = eVβ
(bVβ+1 (K),Tη≤β σ(**M, T**η)) = eVβ
(bVβ+1 (K),Tη≤β Zη);
- for any limit ordinal **β < ν**, put Zβ = σ(**M, T**β) = σ(M, ∅) = I.

Thus, Z is a solution to K with respect to V . Moreover, since M is sequence-saturated relative to T , it follows by Proposition 47 that

$$M=\bigcap_{\beta<\nu}\sigma(M,T_{\beta})=\bigcap_{\beta<\nu}Z_{\beta}\;\;.$$

So M is a dynamic MKNF model of K with respect to V .

Proof of the converse implication is symmetric.

Corollary 18 (Compatibility with Def. 12)
Let K be a basic dynamic hybrid knowledge base and U *be a splitting sequence for* K. Then M is a dynamic MKNF model of K if and only if M *is a dynamic MKNF model of* K
with respect to U.

Proof of Corollary 18 Since K is basic, hPi is an update-enabling sequence for K. Also, by the definition, M is a dynamic MKNF model of K if and only if K is a dynamic MKNF model of K with respect to hPi. Finally, by Proposition 17 this holds if and only if M is a dynamic MKNF model of K with respect to U.

Theorem 20 (Generalisation of MKNF Models)
Let K be an updatable hybrid knowledge base and M *be an MKNF interpretation. Then* M *is a dynamic MKNF model of* hKi if and only if M *is an MKNF model of* K.

Proof of Theorem 20 This follows by Theorem 10 and Propositions 105 and 88. Theorem 21 (Generalisation of Minimal Change Update Semantics)
Let hKiii<n, where Ki = hOi,Pii*, be a dynamic hybrid knowledge base such that* Piis empty for all **i < n**. Then M *is a dynamic MKNF model of* hKiii<n if and only if M is the minimal change update model of hζ (Oi)ii<n.

Proof of Theorem 21 This follows by Corollary 18 and Lemma 119.

Theorem 22 (Generalisation of Dynamic Stable Model Semantics)
Let K = hKiii<n, where Ki = hOi,Pii*, be a dynamic hybrid knowledge base such that* Oi*is empty for all* **i < n**. Then M is a dynamic MKNF model of K *if and only if* M =
{ J **∈ I |** I ⊆ J } *for some dynamic stable model* I of hPiii<n.

Proof of Theorem 22