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Fig. 2. Example of an OPM graph according to the improved definition.
—DerivedFrom ⊆ Art × Art; and —InformedBy ⊆ Proc × Proc. In extension of our original terminology, the elements of GeneratedBy! ∪ Used! ∪ DerivedFrom! are called precise edges, and the elements of GeneratedBy ∪ Used ∪ DerivedFrom ∪ InformedBy are called imprecise edges; all together, they are called edges. For future use, we introduce the following notation. When the distinction between precise and imprecise derived-from edges is of no consequence, we use the following set to refer to all derived-from edges of an OPM graph: DerivedEdges = DerivedFrom ∪ {(A, B) | (A, r, B) ∈ DerivedFrom!}. Example 3.2. The graph shown in Figure 2 extends the graph of Figure 1 with some new features allowed by the improved definition. All derived-from edges involved in the making of the invoice and e-book have now become precise. Moreover, we have added an imprecise used-edge from “Third-Party Process” to “order” and an imprecise generated-by edge from “toy” to “Take Order.” 3.1. Legality
The OPM reference specification defines a notion of legal graph as a directed graph without cycles in the derived-from edges, in which each artifact is generated by at most one process. It turns out that cycles in the derived-from edges do not strictly need to be forbidden; we will discuss this issue in more detail in Section 5.2. On the other hand, we can refine the notion of legality to take advantage of the new possibility of precise derived-from edges. This leads to the following. Definition 3.3 (Legal OPM graph). An OPM graph is called legal if —for each artifact A, there is at most one process P with a precise generated-by edge ! A → P; and ACM Transactions on the Web, Vol. 9, No. 2, Article 10, Publication date: May 2015.