## 2 Related Work
### Process Tree  
Playing an important role in organizations, a process model can be used to represent a workflow task execution in a certain process [8]. The use of Petri nets as modeling notation is common in both the discrete event systems and process mining literature [8]. In our framework, we use the different, but still related, process tree notation to represent a process, similar to other approaches in the configurable process models literature [20]. A process tree [20,21] is a tree-structured process model, where the leaf nodes represent the activities, and the non-leaf nodes represent control-flow operators, e.g., sequence (\(\rightarrow\)), exclusive choice (\(\times\)), inclusive choice (\(\vee\)), parallelism (\(\wedge\)) and loop (\(\odot\)). A silent activity is denoted by \(\tau\) and cannot be observed; it is used to model processes where an activity can be skipped under some specific circumstances. The process tree notation ensures soundness, and it is used by a wide range of process mining techniques, such as ETM [22], Inductive Miner (IM) [23] and Inductive Miner-Infrequent (IMI) [24]. Its formal definition is as follows:  
**Definition 3** (Process tree).: _Let \(A\) be a finite set of activities, with \(\tau\
ot\in A\) representing a silent activity. \(\oplus=\{\rightarrow,\times,\vee,\wedge,\Diamond\}\) is the set of process tree operators. \(A\) process tree is recursively defined as follows [8]:_  
* _if_ \(a\in A\cup\{\tau\}\)_, then_ \(Q=a\) _is a process tree,_
* _if_ \(Q_{1},Q_{2},\ldots,Q_{n}\) _are process trees where_ \(n\geq 1\)_, and_ \(\oplus\in\{\rightarrow,\times,\vee,\wedge\}\)_, then_ \(Q=\oplus(Q_{1},Q_{2},\ldots,Q_{n})\) _is a process tree and_
* _if_ \(Q_{1},Q_{2},\ldots,Q_{n}\) _are process trees where_ \(n\geq 2\)_, then_ \(Q=\odot\ (Q_{1},Q_{2},\ldots,Q_{n})\) _is a process tree._  
_The nodes of a process tree, both operator and activity nodes, are denoted as \(N(Q)\)._  
Notice that both Petri net and process tree modeling notations are closely related and that conclusions obtained in one model can be easily extrapolated to the other. Moreover, a mapping between process tree and Petri net-based workflow nets is described in [8], and it can be easily adapted for other representations such as BPMN, YAWL and EPCs, among others [8]. However, process trees also preserve interesting properties for analysis and verification, such as soundness by construction [8] and the block-structure [8]. An example of a process tree model \(Q_{1}\) that contains seven activities and five operators is shown in Figure 3. This process tree contains a sequence operator (\(\rightarrow\)) as a root node, i.e., its branches will be executed from left to right. Hence, the first activity to be executed will be \(a\). Then, there is a loop operator (\(\odot\)). Its leftmost branch represent the \"do\" part of the loop; it will be executed at least once, and the loop execution always starts and ends with it. In this case, thereis an exclusive choice operator (\(\times\)) in the leftmost branch, indicating that either the activity \(b\) or the parallel (\(\wedge\)) activities \(c\) and \(d\) will be executed. The rightmost branch of the loop operator (\(\odot\)) represents the \"redo\" part of the loop, which in this case contains only the activity \(e\). The process ends with an exclusive choice operator (\(\times\)) that indicates that the final activity will be \(f\) or \(g\).