## 3 Preliminaries
### The DUCG Model  
The DUCG is a probabilistic graphical model originally proposed by [PERSON] [45] for KRR, which intuitively represents uncertain causalities between variables and performs probabilistic reasoning. A typical DUCG is shown in Figure 1 composed of three sorts of variables [7,46]:  
1. The root variable \(B\), which can be drawn as a square, represents the root cause;
2. The intermediate variable \(X\), which can be drawn as a circle, represents the consequence and can also be a cause variable;
3. The variable \(F\), which can be written as \(F_{n;i}\) in text, represents a connection event variable and is represented by a certain probability between 0 and 1.  
In Figure 1, a one-way arrow from variable \(B_{1}\) to variable \(X_{2}\) represents that \(B_{1}\) is the cause of \(X_{2}\). Two-way arrows between variables \(X_{2}\) and \(X_{3}\) represent mutual relationships between them. \(F_{2;1}\) indicates the possibility of variable \(B_{1}\) causing variable \(X_{2}\).  
**Definition 9**[16].: _Given a variable \(V_{i}\), \(V_{i}\in\{B_{i},\ X_{i}\}\), the DUCG is characterized as:_  
\[X_{n}=\underset{i}{\overset{}{\sum}}F_{n;i}V_{i} \tag{6}\]  
\[F_{n;i}=(r_{n;i}/r_{n})A_{n;i} \tag{7}\]  
_where \(r_{n;i}\) is the causal relationship between child event \(X_{n}\) and parent event \(V_{i}\), with \(r_{n;i}\geq 0\) and \(r_{n}\equiv\underset{i}{\overset{}{\sum}}r_{n;i}\); \(A_{n;i}\) is the virtual event representing the causality between the child event \(X_{n}\) and the parent event \(V_{i}\). The connection variable \(F_{n;i}\) can be decomposed into the weighted part \((r_{n;i}/r_{n})\) and the virtual event \(A_{n;i}\)._  
Suppose \(B_{i}=(B_{i1},B_{i2},\cdots,B_{ik})\), \(X_{i}=(X_{i1},X_{i2},\cdots,X_{ik})\) and \(B_{i},X_{i}\in R^{k\times 1}\), then the DUCG can be further described as:  
\[X_{nk}=\underset{i}{\overset{}{\sum}}\underset{j}{\overset{}{\sum}}F_{n;i }V_{ij}=\underset{i}{\overset{}{\sum}}(r_{n;i}/r_{n})\underset{j}{\overset{ }{\sum}}A_{n;k;ij}V_{ij} \tag{8}\]  
where \(V_{ij}\) represents a parent event (state \(j\) of parent variable \(i\)) of child event \(X_{nk}\) (state \(k\) of child variable \(n\)). \(F_{n;k;j}\), the element of the connection matrix \(F_{n;j}\), represents the causality between \(X_{nk}\) and \(V_{ij}\); \(A_{n;k;j}\), the element of the virtual event \(A_{n;i}\), denotes the virtual event  
Figure 1: A typical DUCG model.  
representing the causality between \(V_{ij}\) and \(X_{nk}\); \(r_{n;i}\) denotes the causal relationship intensity between \(X_{nk}\) and \(V_{ij}\).  
**Definition 10**[16].: _Suppose that \(a_{nk;ij}=\Pr\left\{A_{nk;ij}\right\}\) and \(x_{nk}=\Pr\{X_{nk}\}\), in which \(Pr\) means the probability of the event, and \(a_{nk;ij}\) represents the occurrence probability between \(V_{ij}\) and \(X_{nk}\), which satisfies \(\sum\limits_{k}a_{nk;ij}=1\), then_  
\[\begin{array}{l}x_{nk}=\Pr\{X_{nk}\}=\sum\limits_{i}(r_{n;i}/r_{n})\sum \limits_{j}a_{nk;ij}\Pr\{V_{ij}\}\\ \qquad=\sum\limits_{i}(r_{n;i}/r_{n})\sum\limits_{j}a_{nk;ij}\gamma_{ij}\\ \qquad=\sum\limits_{i}(r_{n;i}/r_{n})\;a_{nk;ji}\gamma_{ij}\end{array} \tag{9}\]  
_where the knowledge parameters \(a_{nk;ij}\) and \(r_{n;i}\) are usually obtained from domain specialists._  
Note that any condition of a variable cannot bring about any condition of the same variable [16]. As indicated by the standard of logical inconsistency, an event itself cannot be both root and consequence at the same time; therefore, it breaks the coordinated cyclic cases in DUCG.