## 2 WFT-Net Model
### Data Table  
**Definition 1** (Data table). : _A 3-tuple \(R=(U,V,\tau)\) is a data table where_  
1. \(U\) _is a finite set of attributes, and there is a special identifier attribute (or primary key) id, which holds id_ \(\in U\)_;_
2. \(V\)\(=\) _is a set of domains of elements in_ \(U\)_;_
3. \(\tau\)_:_ \(U\)\(\rightarrow\)\(V\) _is a mapping from attributes to domains._  
A data table \(R=(U,V,\tau)\), where \(U=\{u_{1},\cdots,u_{m}\}\) and \(u_{1}\) is the identifier attribute, can be denoted as \(R(\underline{u_{1}},\cdots,u_{m})\). An element of the Cartesian product of \(\tau(u_{1}),\cdots,\tau(u_{m})\) is a _tuple_ of \(R\), and tuples can be represented by attributes and their values, e.g., \(R[u_{i}(u_{i}^{*})]\) denotes all the tuples where the value of \(u_{i}\) is \(u_{i}^{*}\). Additionally, we use \(R[u_{i}(u_{i}^{*})](u_{i})\) to denote the value of the attribute \(u_{j}\) in \(R[u_{i}(u_{i}^{*})]\). Note that the value of the identifier attribute \(u_{1}\) in each tuple cannot be null or overlap. Take the data table \(R^{\prime}\) in Figure 1 for instance: \(a_{11}\
eq a_{12}\). And here, two tuples of \(R\) are considered equal if \(u_{1}\) has the same value. We use \(R[u_{1}]\) to represent all the tuples in \(R\). Each data table is two-dimensional, consisting of several attributes and their values. The persistent data operations on tables mainly include _select_, _update_, _insert_, and _delete_. For a table \(R(\underline{u_{1}},\cdots,u_{m})\), we have the following:  
* The _select_ operation can retrieve attributes and tuples in \(R\), and there are two cases: one, the attribute \(u_{i}\) can be retrieved with a value \(u_{i}^{*}\), and the tuples containing the value \(u_{i}^{*}\) of \(u_{i}\) are output, denoted as \(sel\langle R[u_{1}](u_{i})\diamond u_{i}^{*}\rangle\), e.g., \(sel\langle R^{\prime}[A_{1}](A_{2})\diamond a_{21}\rangle\) represents \"_select_\(A_{1}\)_from_\(R^{\prime}\)_where_\(A_{2}=a_{21}\)\", and its outcome is \(a_{1}\) (which means \(R^{\prime}[a_{11}]\)); the other one is that the tuples \(R[u_{i}(u_{i}^{*})]\) can be retrieved, and the values of an attribute \(u_{j}\) in them are output, denoted as \(sel\langle R[u_{i}(u_{i}^{*})](u_{j})\rangle\), e.g., \(sel\langle R^{\prime}[A_{1}(a_{11})](A_{2})\rangle\) represents \"_select_\(A_{2}\)_from_\(R^{\prime}\)_where_\(A_{1}=a_{11}\)\", and its outcome is \(a_{21}\);
* The _insert_ operation can add a new tuple \((u_{1}^{*},\cdots,u_{m}^{*})\) to \(R\), denoted as \(ins\langle R[u_{1}]\diamond(u_{1}^{*},\)\(\cdots,\)\(u_{m}^{*})\rangle\), e.g., \(ins\langle R^{\prime}[A_{1}]\diamond(u_{1}^{*},\)\(\cdots,\)\(u_{m}^{*})\rangle\) represents \"_insert_ into_\(R^{\prime}(A_{1},\cdots,\)\(A_{m})\)_values_\((a_{1}^{*},\)\(\cdots,\)\(a_{m}^{*})\)\", and then \(R^{\prime}\) is changed from \(R^{\prime}=\{(a_{11},\)\(\cdots,\)\(a_{m1})\), \((a_{12},\)\(\cdots,\)\(a_{m2})\}\) to \(R^{\prime}=\{(a_{11},\)\(\cdots,\)\(a_{m1})\), \((a_{12},\)\(\cdots,\)\(a_{m2})\), \((a_{1}^{*},\)\(\cdots,\)\(a_{m}^{*})\}\);
* The _update_ operation can change the value of an attribute \(u_{j}\) with a new value \(u_{j}^{*}\) in \(R[u_{i}(u_{i}^{*})]\) of \(R\), denoted as \(upd\langle R[u_{i}(u_{i}^{*})](u_{j})\diamond u_{j}^{*}\rangle\), e.g., \(upd\langle R^{\prime}[A_{1}(a_{11})](A_{2})\diamond a_{2}^{*}\rangle\) represents \"_update_\(R^{\prime}\)_set_\(A_{2}=a_{2}^{*}\) where \(A_{1}=a_{11}\)\", and then \(R^{\prime}\) is changed from \(R^{\prime}=\{(a_{11},\)\(a_{21},\)\(\cdots,\)\(a_{m1})\), \((a_{12},\)\(a_{22},\)\(\cdots,\)\(a_{m2})\}\) to \(R^{\prime}=\{(a_{11},\)\(a_{2}^{*},\)\(\cdots,\)\(a_{m1})\), \((a_{4. \(\mathcal{O}_{D}\): \(T\)\(\rightarrow\)\(\{rd\), \(wt\), \(dt\}\) represents process data operations on \(\mathcal{D}\), in which rd (resp. wt, dt) is a labeling function of reading (resp. writing, deleting) data, and the set of all data elements read (resp. written, deleted) by transition \(t\in T\) is denoted as \(\mathcal{D}_{t}^{rd}\) (resp. \(\mathcal{D}_{t}^{wt}\), \(\mathcal{D}_{t}^{dt}\));
5.