###### Abstract  
Business processes without inconsistent data errors resulting from inappropriate data operations are regarded as a fundamental and important subject in major application areas. At present, such errors are always explored by only focusing on process data or persistent data; studies have seldom considered both data layers. In fact, each business process contains the data of such two layers, and its execution depends on the interaction between them. This paper employs WFT-nets (workflow nets with tables) to model business processes, considering process and persistent data. It extends the WFT-net's reachability graph to capture process data and persistent data interactions, proposes an automated inconsistency detection method, and introduces a data refining strategy to enhance detection accuracy and workflow repair efficiency. Based on WFT-nets, inappropriate data operations that can lead to the error of inconsistent data are given in formula form. The extended reachability graph displaying the data operations of the WFT-net is constructed under the corresponding data refinement. Then, inconsistent data are detected by traversing such reachability graph. Some theoretical arguments are given; a set of comparative experiments with WFD-nets were conducted to illustrate the effectiveness of our inconsistent data detection approach. WFT-nets; data operation formula; inconsistent data; data refinement; extended reachability graph +
Footnote †: journal: Applied sciences  
0000-0001-5000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-00000-0000-00000-000  
consumed during the execution of process activities), if the deleted operation is performed first for them, the value to be read may be missing, preventing the process from running [7]. Similarly, if the two threads are writing the same data, the values accessed by both threads are different. And then, if the two values are stored in different record locations, it can also lead to inconsistent data in the underlying database [8]. To improve the quality of business process design, it is essential to identify such errors of data inconsistency. In business processes, data information is always integrated subsequently into the control flow [9]. In general, the data that are directly manipulated (including _read_, _written_, and _deleted_) by business activities are usually called _process data_. And many previous studies have focused on inconsistent data resulting from inappropriate data read/write/delete operations. By modeling business processes by Petri nets with data operations attached to activities, inconsistent data can be detected based on an activity dependency [10], a graph traversal [11], a DataRecord [12], data anti-patterns [13], and a data-flow Boolean table [14]. For instance, [PERSON] et al. [15] detected such errors in workflow processes using the Maude language and model checking based on workflow nets with data (WFDnets). [PERSON] et al. [11] formalized inconsistent data with weak behaviors and sequential relationships based on WFD-nets, and they detected such errors in concurrent programs by traversing their reachability graph. [PERSON] et al. [16] tackled such errors in workflows arising from improper concurrent operations by heuristically traversing the WFD-net reachability graph. [PERSON] et al. [17] introduced the Petri net with data (PD-net) and its reachability graph to detect such errors from improper data operations in concurrent systems. To mitigate state space explosion, they later proposed PD-net unfolding, leveraging activity structures for such error detection [18]. [PERSON] et al. [19] identified such errors in workflow models resulting from improper concurrent artifact operations using sequential and parallel trees. [PERSON] et al. [20] identified such errors in BPMN models caused by improper operations on data objects, using anti-patterns derived from the unfolded Petri net.