Splitting and Updating Hybrid Knowledge Bases (Extended Version) 47 Corollary 83 Let M ∈ M. Then M ⊕ M = M.

Proof Follows from Lemma 82. Lemma 84 Let M, N ∈ M. If M is empty or N is empty, then M ⊕ N is empty.

Proof Follows from Lemma 82.

## D.2 Splitting And Updating A Sequence Of First-Order Theories

We will be using the following, inductive definition of a minimal change update model of a sequence of first-order theories. It is equivalent to the definition from Sect. 2, just more precise. Definition 85 (*Minimal Change Update Model*)
Let U = hUiii<n be a finite sequence of first-order theories. We define:
mod(U) = Mhmod(Ui)ii<n , where

$$\bigoplus\left\langle\,\right\rangle={\mathcal{I}}\;\;{\mathrm{and}}$$

MhMiii<k+1 =
MhMiii<k⊕ Mk for any k ≥ 0.

If mod(U) is nonempty, we say it is the *minimal change update model of* U.

Remark 86 Note that the above definition is compatible with the definition of a minimal change update of **T ⊕U** in the following sense: M is a minimal change update model of **T ⊕U** if and only if M is a minimal change update model of hT , U i. This follows from the above definition and from the fact that I ⊕ X = X for any X ∈ M (by Lemma 82).

Lemma 87 Let T be a first-order theory. Then either T has no MKNF model and mod(T ) = ∅, or T
has a unique MKNF model that coincides with mod(T ). Moreover, mod(T ) = \
φ∈T
mod(φ)
Additionally, the S5 and MKNF models of T coincide with S5 and MKNF models of
{ K φ | φ **∈ T }**.

Proof Follows from the definition of an S5 and MKNF models. Proposition 88 Let T be a first-order theory. Then M is the MKNF model of T if and only if M is the minimal change update model of **hT i**.

Proof