A a sequence of ABoxes and M an MKNF interpretation. Then M **is a minimal**
change dynamic stable model of P ⊕T A if and only if M **is a minimal change**
update model of SP ⊕T A where SP = { p | K p **∈ P }**.

Proof of Proposition 4.5 Since P contains only facts, we can see that P = PM, so M **is a minimal change** dynamic stable model of P ⊕T A **if and only if** M = mod(P ⊕T A**) which by**
definition holds if and only if

$$M=\bigcap_{n\geq0}T_{{\mathcal{P}}_{\oplus}\tau_{\mathcal{A}}}^{n}({\mathcal{I}})$$

Further, we know that T
0 P⊕T A(I) = I
T
1 P⊕T A(I) = mod({ H∗(r) | r ∈ P ∧ I |= B(r) } ⊕T A)
= mod({ H∗(r) | r ∈ P } ⊕T A) = mod(SP ⊕
T A)
T
n P⊕T A(I) = T
1 P⊕T A(I) for all n > 1 Hence, we have
\
n≥0 T
n P⊕T A(I) = mod(SP ⊕
T A) .

So M is a minimal change dynamic stable model of P ⊕ U **if and only if** M =
mod(SP ⊕T A) which is by definition equivalent to M **being a minimal change** update model of SP ⊕T A.

Lemma D.1 Let T be a TBox, A = (A1, A2, . . . , An) a sequence of ABoxes (where n ≥ **1) and**

$$i_{\oplus}\tau_{\mathcal{A}}(L)=$$

A
′ = (A1, A2, . . . , Ai−1, Ai, ∅, Ai+1**, . . . ,** An)
for some i ∈ { 0, 1, 2, . . ., n }**. Then for any** M ⊆ mod(T **) it holds that**
incorporateT(A, M) = incorporateT(A
′, M)
Proof We will prove by induction on n:
1
◦If n = 1, then i ∈ { 0, 1 }**, so we need to prove that**
incorporateT(A1, M) = incorporateT(A1, incorporateT(∅, M))
and that incorporateT(A1, M) = incorporateT(∅, incorporateT(A1, M)) .

This follows easily from the fact that incorporateT(∅, N) = N **for any** N ⊆ mod(T ).

2
◦ We assume the claim holds for n − 1 and prove it for n. First let i **= 0. Then**
incorporateT(A
′, M) = incorporateT(A, incorporateT(∅, M))
and the claim again follows from the fact that incorporate(∅, N) = N **for any** N ⊆ mod(T ).