# Quality-Aware Resource Model Discovery
## 3 Quality-Aware Resource Model Discovery
### Resource Model Generation
#### 3.2.1 Transition Systems  
Transition systems are composed of states, events, and transitions, and they can be produced from event logs through a series of steps [12]. States represent the status where an event in a process is performed, while transitions refer to the relationship that changes a state as an event is performed in a particular state. In addition, a transition system has one or more initial states and final states. In a nutshell, a transition system is built by \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**CaseID** & **Event** & **Activity** & **Originator** & **Timestamp** & **Attribute (Quality)** \\ \hline C1 & E1 & A & M1 & 1 January 2021 10:30 & 0.95 \\ C1 & E2 & B & M2 & 1 January 2021 12:00 & 0.95 \\ C1 & E3 & C & M4 & 1 January 2021 17:00 & 0.95 \\ C1 & E4 & D & M6 & 2 January 2021 09:00 & 0.95 \\ C2 & E5 & A & M1 & 4 January 2021 09:00 & 0.85 \\ C2 & E6 & B & M3 & 4 January 2021 12:00 & 0.85 \\ C2 & E7 & C & M5 & 4 January 2021 15:00 & 0.85 \\ C2 & E8 & D & M6 & 5 January 2021 13:00 & 0.85 \\ C3 & E9 & A & M1 & 6 January 2021 09:00 & 0.9 \\ C3 & E10 & B & M2 & 6 January 2021 14:00 & 0.9 \\ C3 & E11 & C & M5 & 7 January 2021 13:00 & 0.9 \\ C3 & E12 & D & M6 & 7 January 2021 18:00 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A snippet of event logs. connecting a chain of transitions from initial states to final states. Definition 2 describes the formal definition of building transition systems from event logs. **Definition 2**. : _(State representation function, Event representation function, Transition system). Let \(L\) be an event log. Let \(E^{*}\) and \(E\) be a set of possible traces and events. A state representation function \(\mathit{rep}^{s}\in E^{*}\to R^{s}\) is a function that maps traces to state representations, where \(R^{s}\) is the set of possible state representations. An event representation function \(\mathit{rep}^{s}\in E\to R^{e}\) is a function that maps events to event representations, where \(R^{e}\) is the set of possible event representations. A transition system TS is defined as a triplet (S, E, T), where \(S=\left\{\mathit{rep}^{s}\left(\mathit{hd}^{k}(\sigma)\right)\middle|c\in L \wedge\sigma=\beta_{t}(c)\wedge 0\leq k\leq\left|\sigma\right|\right\}\), \(E=\left\{\mathit{rep}^{s}(\sigma(k))\middle|c\in L\wedge\sigma=\beta_{t}(c) \wedge 1\leq k\leq\left|\sigma\right|\right\}\), and \(T\subseteq S\times E\times S\) with \(T=\left\{\mathit{rep}^{s}\left(\mathit{hd}^{k}(\sigma)\right)\right.\), \(\mathit{rep}^{s}(\sigma(k+1))\), \(\mathit{rep}^{s}\left(\mathit{hd}^{k+1}(\sigma)\right)\middle|c\in L\wedge \sigma=\beta_{t}(c)\wedge 0\leq k\leq\left|\sigma\right|\right\}\) is the set of states, the set of events, and the set of transitions._  
Transition systems can be produced in different forms from the same event log \(L\) based on event and state representation functions. First, the event representation function \(\mathit{rep}^{s}(e)\) is associated with what types (i.e., event attributes) of transition systems are derived. If we want to build a transition system relevant to the activity, the event representation function should be established such that \(\mathit{rep}^{e}(e)=\beta_{a}(e)\). Alternatively, if interested in the originator, it is necessary to determine the event representation function such that \(\mathit{rep}^{e}(e)=\beta_{o}(e)\). We can also apply both the activity and the originator at the same time such that \(\mathit{rep}^{e}(e)=\left[\beta_{a}(e)\times\beta_{o}(e)\right]\). As far as states are concerned, the state representation function \(\mathit{rep}^{s}(\sigma)\) is utilized with different types of abstractions, where \(\sigma\) is a trace. In addition, we can consider the horizon, which means how many events are included from the prefix to determine the states with \(\mathit{hd}^{k}(\sigma)\). Here, \(\mathit{hd}^{k}(\sigma)\) signifies to the head of the sequence \(\sigma\) with the first k elements from the trace \(\sigma\).