## 3 Inconsistent Data  
Inconsistent data usually stem from incorrect data access, including improper process and/or persistent data operations. Before formalizing such error, some auxiliary concepts should be introduced, including two _activity relations_ (i.e., _causality_ and _concurrency_) for two transitions, _data logs_, and _shared attribute_. **Definition 6** (Activity relations). : _Let \(N=(P,\,T,\,F,\,\mathcal{D},\,\mathcal{R},\,O_{\mathcal{D}},O_{\mathcal{R}},\,G)\) be a WFT-net, \(t_{1}\) and \(t_{2}\) be two transitions of \(N\), and \(c\) be a configuration of \(N\):_  
1. \(t_{1}\) _and_ \(t_{2}\) _are causal at_ \(c\)_, denoted by_ \(t_{1}{\prec}t_{2}\)_, if_ \(c[t_{1}]\wedge\
eg c[t_{2}]\wedge t_{1}^{\bullet}\cap{}^{\bullet}t_{2}\
eq \emptyset\wedge\exists p\in{}^{\bullet}_{1}\cap{}^{\bullet}t_{2}{:}m(p)=0\wedge \forall p\in{}^{\bullet}_{2}\backslash{}^{\bullet}_{1}{:}m(p){\geq}1;\)__
2. \(t_{1}\) _and_ \(t_{2}\) _are concurrent at_ \(c\)_, denoted by_ \(t_{1}{\parallel}c_{2}\)_, if_ \(c[t_{1}]\wedge c[t_{2}]\wedge({}^{\bullet}t_{1}\cap{}^{\bullet}t_{2}{:}m(p){ \geq}2).\)__  
In a WFT-net, two enabled transitions \(t_{1}\) and \(t_{2}\) are concurrent at \(c\) if they share some resources. For example, assuming a configuration \(c^{\prime}=(m^{\prime},\,\theta^{\prime},\,\omega^{\prime},\,\delta^{\prime})\) can be generated after firing \(t_{4}\) in Figure 2 and \(t_{5}\) and \(t_{6}\) are enabled at \(c^{\prime}\) (i.e., \(c^{\prime}[t_{5}\rangle\wedge c^{\prime}[t_{6}])\), \(t_{5}\) and \(t_{6}\) are concurrent at \(c^{\prime}\) because \({}^{\bullet}t_{5}\cap{}^{\bullet}t_{6}=\emptyset\). And two transitions \(t_{1}\) and \(t_{2}\) of \(N\) are causal at \(c\) if only one of them is enabled at \(c\), and its firing can enable the other disabled one. The causality satisfies transitivity, i.e., if \(\exists t_{3}\in T\), \(c^{*}\in C_{c_{0}}\), such that \(t_{1}{\prec}t_{2}\wedge t_{2}{\prec}t_{3}\), then \(t_{1}\) and \(t_{3}\) are also in causality, denoted as \(t_{1}{\prec}t_{3}\). For example, considering the transitions \(t_{1}\), \(t_{2}\), and \(t_{3}\) (or \(t_{4}\)) in Figure 2 with the above configurations \(c^{\prime}\) and \(c^{\prime\prime}\), if there are \(t_{1}{\prec}t_{2}\) and \(t_{2}{\prec}t_{^{\prime\prime}}t_{3}\) (or \(t_{2}{\prec}t_{4}\)), then it holds that \(t_{1}{\prec}t_{3}\) (or \(t_{1}{\prec}t_{4}\)). In addition, each complete business record can be represented as a sequence of tuples from each table \(R_{i}\in\mathcal{R}\) (\(1{\leq}i{\leq}n\)), in which each foreign-key attribute in different tuples has the same value, corresponding to a data log of \(N\). **Definition 7** (Data log). : _Let \(N=(P,\,T,\,F,\,\mathcal{D},\,\mathcal{R},\,O_{\mathcal{D}},O_{\mathcal{R}},\,G)\) be a WFT-net, in which \(\mathcal{R}=\{R_{1},\,\cdots,\,R_{n}\}\), \(R_{i}=(\underline{u_{i1}},\,\cdots,\,u_{i_{m}})\) (\(1{\leq}i{\leq}n\)), and \(R_{1}\), \(\cdots\), \(R_{n}\) are connected with one another by foreign-key attributes \(u^{1},\,\cdots,\,u^{k}{\in}\{u_{i1},\,\cdots,\,u_{n1}\}\). \(\ell=R_{1}[u_{11}(u_{11}^{*})]\&\cdots\&\For example, the tuple sequence \((aid_{1}\), \(ac_{1})\) and \((bid_{1}\), \(aid_{1}\), \(ac_{1})\) is a data log of the WFT-net in Figure 2 since the value of the unique foreign-key attribute \(Aid\) in it only is \(aid_{1}\). This is not so for the tuple sequence \((aid_{1}\), \(ac_{1})\) and \((bid_{1}\), \(aid_{2}\), \(ac_{1})\). Note that in addition to foreign-key attributes, there may be some other non-identifying attributes that belong to more than two tables, called shared attributes, such as the attribute \(Ac\) in the WFT-net in Figure 2. Formally, for two tables \(R_{i}\) and \(R_{j}\), let \(R_{i}\cap R_{j}\) be the set of attributes that belong to \(R_{i}\) and \(R_{j}\). Then, we define a shared attribute as follows. **Definition 8** (Shared attribute).