## 4 Discovering Inconsistent Data Based on WFT-Nets
### Extended Reachability Graph  
During inconsistent data detection, once the error of missing data occurs, it would lead to some non-terminating configurations in which no transition can be in \(N\) that is enabled. Taking \(t_{5}\|_{c}t_{6}\) as an example, when \(t_{5}\) is fired after \(t_{6}\), \(t_{5}\) will be disabled. This is because the data \(ac\) read by \(t_{5}\) are undefined. In fact, on the premise that the control flow is sound, the error of missing data can be considered as enabling only the control flow but not the data flow. In other words, such missing data can be expressed in general terms as follows: \(\exists t\in T\), \(\exists c=(m,\,\theta,\,\omega,\,\delta)\in C_{q_{0}}\colon c[_{c}t]\,\wedge\
eg c [_{d}t]\). And here, the special state is introduced as a _pseudo may-reachable configuration_.  
**Definition 10** (Pseudo may-reachable configuration).: _Let \(N=(P,\,T,\,F,\,\mathcal{D},\,\mathcal{R},\,O_{\mathcal{D}},\,O_{\mathcal{R}},\,G)\) be a WFT-net, \(c=(m,\,\theta,\,\omega,\,\delta)\in C_{c_{0}}\), \(t\in T\). If \(c[_{c}t)\) (i.e., \(m[t]m^{\prime})\wedge\
eg c[_{d}t]\), then \(c^{\prime}\) is called a pseudo may-reachable configuration from \(c\) via \(t\), where \(c^{\prime}=(m^{\prime},\,\theta,\,\omega,\,\delta)\), denoted as \(c[_{c}t]c^{\prime}\)._  
To detect inconsistent data based on graph traversal, the pseudo may-reachable configurations and the data element/table operations should be extended into the reachability graph of WFT-net, namely, extended reachability graph (ERG).  
**Definition 11** (Extended reachability graph).: _Let \(N=(P,\,T,\,F,\,\mathcal{D},\,\mathcal{R},\,O_{\mathcal{D}},\,O_{\mathcal{R}},\,G)\) be a PT-net. \(ERG(N)=(C_{c_{0}}\cup C^{\prime}_{c_{0}^{\prime}},\,E\cup E^{\prime},\,L\cup L ^{\prime})\) is the extended reachability graph of \(N\), where_  
1. \((C_{c_{0}},\,E,\,L)\) _is the RG of_ \(N\)_;_
2. \(C^{\prime}_{c_{0}}\) _is the pseudo may-reachable configuration set of_ \(N\)_;_
3. \(E^{\prime}=\{(c_{i},\,c_{j})|c_{i}\in C_{c_{0}}\!,c_{j}\in C^{\prime}_{c_{0}}\!,\exists t\in T\colon c_{i}[_{c}t]c_{j}\}\)_;_
4. \(L^{\prime}\colon E^{\prime}\to T\times 2^{\mathcal{D}}\times 2^{\mathcal{D}} \times 2^{\mathcal{D}}\times 2^{\mathcal{D}}\times 2^{\mathcal{D}}\times 2^{ \mathcal{D}}\times 2^{\mathcal{D}}\times 2^{\mathcal{D}}\times 2^{\mathcal{D}} \times(2^{\mathcal{R}}\!\cdot\!\!2^{R})\times(2^{\mathcal{R}}\!\cdot\!\!2^{R}) \times(2^{\mathcal{R}}\!\cdot\!\!2^{R})\times(2^{\mathcal{R}}\!\cdot\!\!2^{R})\) _such that_ \(L^{\prime}(c_{i},\,c_{j})=\langle t;\,\mathcal{D}^{\mathit{rd}}_{t},\,\mathcal{D }^{\mathit{st}}_{t},\,\mathcal{D}^{\mathit{st}}_{t},\,\mathcal{D}^{\mathit{st }}_{t},\,\mathcal{D}^{\mathit{st}}_{t},\,\mathcal{D}^{\mathit{st}}_{t},\, \mathcal{D}^{\mathit{st}}_{t},\,\mathcal{D}^{\mathit{std}}_{t};\,\mathcal{D} ^{\mathit{std}}_{t};\,\mathcal{D}^{\mathit{std}}_{t},\,\mathcal{D}^{\mathit{std }}_{t};\,\mathcal{D}^{\mathit{std}}_{t},\,\mathcal{D}^{\mathit{std}}_{t}, \,\mathcal{D}^{\mathit{std}}_{t};\,\mathcal{D}^{\mathit{std}}_{t},\,\mathcal{D} ^{\mathit{std}}_{t};\,\mathcal{D}^{\mathit{std}}_{t},\,\mathcal{D}^{\mathit{std }}_{t};\,\mathcal{D}^{\mathit{std}}_{t}\rangle\) _if_ \((c_{i},c_{j})\in E^{\prime}\) _and_ \(c_{i}[_{c}t]c_{j}\)_._  
Since the WFT-net is a concept model, generating its extended reachability graph should concretize the abstract values of its data elements. As with our previous work [34], the concept of data refinement is employed to concretize such abstract values. And the given data refining strategy aims to reflect the differences in the values of data elements/records between different business cases. However, it is not enough to display inconsistent data as discussed in this paper. The main reason is that the refined values of each overwritten data element are the same. This makes it difficult to reflect the error of inconsistent data from the value perspective. Therefore, an appropriate data refining strategy for discovering inconsistent data is given as follows.