P-reducible relative to U0 and U2, respectively. Finally, the third layer tU1(bU2(K)) contains a mixture of rules and ontology axioms, but all the rules are positive and all predicate symbols of all rule body literals belong to U1, so the layer is O-reducible relative to U1.

The layers of the updating hybrid knowledge base K′are shown in Fig. 3. It can be easily verified that they satisfy the same reducibility criteria, so U is indeed an update-enabling sequence for hK, K′i. In order to arrive at a dynamic MKNF model of hK, K′i with respect to U, a dynamic MKNF model of each layer is computed separately and models of previous layers serve to "import" information to the current layer.

In our case, we first need to find the minimal change update model X0 of the first layer of K updated by the first layer of K′. Due to the TBox axioms, this results in c1 no longer being a member of CherryTomato. The HTS code of c1 also changes to '07020010'. Note that the conflict between old and new knowledge is properly resolved by the minimal change update semantics.

Subsequently, the dynamic stable model semantics is used to find the dynamic MKNF
model X1 of the second layer of K updated by the second layer of K′. The rule update results in i2 no longer being an approved importer for c2 . As before, the conflict that arose is resolved by the rule update semantics.

Given the dynamic MKNF models of the first two layers, the model of the third layer of K is now different because i2 is no longer an expeditable importer of c2 . As a consequence, c2 is no longer a member of the concept LowRiskEUCommodity. Also, due to the update of the third layer, c3 is also not a member of LowRiskEUCommodity. The conflicting situation was again resolved by the minimal change update operator and results in the dynamic MKNF model X2 of the third layer.

Finally, due to the changes in all three previous layers, the rules in the fourth layer now imply that CompliantShpmt(s1 ) does not hold and, as a consequence, FullInspection(s1 )
holds. Also, PartialInspection(s2 ) holds because c2 is not a low risk commodity. But even though c3 is also not a low risk commodity, PartialInspection(s3 ) does not hold. This is due to the rule update of the fourth layer according to which the inspection of s3 must be waived because s3 comes from an EU registered producer.

## 4 Discussion

The class of updatable hybrid knowledge bases for which we defined an update semantics in the previous section is closely related to multi-contextsystems (Brewka and Eiter 2007).

Each layer of a hybrid knowledge base relative to a particular update-enabling splitting sequence can be viewed as a context together with all its bridge rules. At the same time, the constraints we impose guarantee that each such context either contains only rules, so the context logic can be the stable model semantics, or it contains only DL axioms so that first-order logic can be used as its logic. On the other hand, different splitting sequences induce different multi-context systems, though their overall semantics stays the same. We believe that a further study of this close relationship may bring about new insights.

Another direction in which the proposed framework can be generalised is by letting the ontology and rule update operators be given as parameters instead of using a fixed