On extending support for modeling artifact-centric business processes

sync rule that is used to synchronize all abstract transitions together can be defined as follow. For every L-fragment l ,⇒,⇒ ,⇒ ∈ , we have, Chapter 5. Facilitating Inter-Organizational Business Process Collaboration 128 ∃ , ∈ . \ , ⇒ ∗ ⇒⇒ ⟶ ∈ _ , ∧ ∈ _ , . Example 5.. In Figure 5. (a), we can see that AS-region contains two fragments and with sync rules and . When applying _ Π, ), we have abstract lifecycle of and abstract lifecycle of with abstract sync rule . In addition, Figure 5. (b) shows a case of sync abstraction for AS-region which contains multiple entry transitions and multiple exit transitions ASL-fragments (both ASL-fragments and ). Figure 5.: Examples of sync abstraction In the above example, we demonstrate the sync abstraction of two synchronized fragments. However, it is possible that an AS-region contains more than two ASLfragments. For wider understanding of sync abstraction, we illustrate a sync abstraction of more than two SL-fragments in Figure 5. (with artifact extended to the example in Figure 5. (b)). Example 5.. In Figure 5., we can see that AS-region contains synchronized fragments and of artifact and , respectively, and of artifact . As all three fragments can be considered as ASL-fragments in , we can validly apply function _ Π, ) and the abstract lifecycles of artifacts {A , A , A } with abstract sync rules {r , r } are returned. Chapter 5. Facilitating Inter-Organizational Business Process Collaboration 129 Figure 5.: Example of abstraction of more than two ASL-fragments Next, we discuss the case of an AS-region containing SL-fragments with a nested (sub) SL-fragment that synchronizes with other lifecycle. Intuitively, the sub SLfragment and its synchronized lifecycle should be also taken into account when its super fragment has to be abstracted. Therefore, we need to induce the abstraction to its sub fragment together with its counterpart if they both can satisfy the property of AS-region (which is considered as a sub AS-region of the whole). In other words, we can say that the entire AS-region should contain such counterpart in order to have a valid abstraction. We show an example of AS-region consisting of sub SL-fragment in Figure 5.. Figure 5.: Example of abstraction on nested sub-SL-fragment Example 5.. In Figure 5., we can see that L-fragment is a synchronized fragment of L-fragment which is nested under L-fragment . The abstraction Chapter 5. Facilitating Inter-Organizational Business Process Collaboration 130 yields abstract transitions with abstract sync rule that is in the lifecycle of A and in the lifecycle of A . Next, we show that sync abstraction function _ preserves the B-consistency of between two synchronized lifecycles of the input and two outputted abstract lifecycles. THEOREM 5. (SYNCHRONIZED FRAGMENTS B-CONSISTENT ABSTRACTION). Let , be an AS-region in ACP model Π that is to be abstracted in ACP model Π and let be a set of abstract artifact lifecycles resulted from applying sync abstraction function _ Π, . Then the following statement holds. ∀ , ∈ |l ≺ ∧ l ∈ , ∃ , ∈ ,