## 3 Related Definition
### Production Rule of FRCPN
#### 3.3.5 The Fifth Production Rule  
**If \(WS_{i}\) then \(WS_{j1}\) or \(WS_{j2}\)\(\cdots\)or \(WS_{jn}\)** (The semantic similarity threshold \(k\))  
This fifth production rule represents a dependence relation \"\(WS_{i}\to WS_{j1}\)**or**\(WS_{j2}\)\(\cdots\)**or**\(WS_{jn}\)\" (the semantic similarity threshold \(k\)). The FRCPN based on semantic similarity for Web service compositions is shown in Figure 5.  
Figure 4: The FRCPN expression of the fourth production rule. Figure 5: The FRCPN expression of the fifth production rule. \(s_{1}\), \(s_{2}\), \(\cdots\), \(s_{n}\) are the semantic similarity between the outputs of \(WS_{i}\) and the inputs of \"\(WS_{j1},WS_{j2},\cdots,WS_{jn}\),\" respectively. The above rule needs to satisfy some conditions, including \(s_{1}\geq k,s_{2}\geq k,\cdots,s_{n}\geq k\), (\(O_{i}\overline{\rule[0.0 pt]{0.0 pt}{2.5 pt}}\Pi_{j1}\lor(O_{i}\overline{\rule[0.0 pt]{0.0 pt}{2.5 pt}}\Pi_{j2})\lor\cdots(O_{i}\overline{\rule[0.0 pt]{0.0 pt}{2.5 pt}}\Pi_{jn}\)), and all of the conditions in \(P_{i}\), \(Q_{i}\). Therefore,  
\[ \begin{split} s_{1}&=m_{11}\times\text{sim}(O_{i}, I_{j1})+m_{21},\ m_{11}+m_{21}=1\\ s_{2}&=m_{12}\times\text{sim}(O_{i},I_{j2})+m_{22},\ m _{12}+m_{22}=1\\ &\cdots\\ s_{n}&=m_{1n}\times\text{sim}(O_{i},I_{jn})+m_{2n},\ m_{1n }+m_{2n}=1.\end{split}  \tag{13}\]  
In Equation (13), \(\text{sim}(O_{i},I_{j1})\) represents the semantic similarity between the outputs of \(WS_{i}\) and the inputs of \(WS_{j1}\); \(\text{sim}(O_{i},I_{j2})\) represents the semantic similarity between the outputs of \(WS_{i}\) and the inputs of \(WS_{j2}\), and so on. \(m_{11}\) is the weight that represents the attention degree of matching between the outputs of \(WS_{i}\) and the inputs of \(WS_{j1}\); \(m_{12}\) is the weight that represents the attention degree of matching between the outputs of \(WS_{i}\) and the inputs of \(WS_{j2}\), and so on. \(m_{21}\) is the weight that represents the attention degree of satisfying all the conditions in \(P_{i}\), \(Q_{i}\) in \(s_{1}\), \(m_{22}\) is the weight that represents the attention degree of satisfying all the conditions in \(P_{i}\), \(Q_{i}\) in \(s_{2}\), and so on. According to the above definition, we define the reliability values \"\(\Gamma_{1},\Gamma_{2},\cdots,\Gamma_{n}\)\" of the transitions \"\(\Gamma_{1},\Gamma_{2},\cdots,\Gamma_{n}\)\" to equal the semantic similarities between the interfaces of \(WS_{i}\) and the interfaces of \"\(WS_{j1},WS_{j2},\cdots,WS_{jn}\)\", respectively, as follows:  
\[\Gamma_{1}=s_{1},\Gamma_{2}=s_{2},\cdots,\Gamma_{n}=s_{n}. \tag{14}\]  
We can further define the reliability values of \(WS_{j1},WS_{j2},\cdots,WS_{jn}\). This production rule needs only one Web service \(WS_{i}\), we can trigger \"\(WS_{j1},WS_{j2},\cdots,WS_{jn}\)\", so,  
\[ \begin{split}\Gamma_{j1}&=\Gamma_{i}\times\Gamma _{1}=\Gamma_{i}\times s_{1}\\ \Gamma_{j2}&=\Gamma_{i}\times\Gamma_{2}=\Gamma_{i} \times s_{2}\\ &\cdots\\ \Gamma_{jn}&=\Gamma_{i}\times\Gamma_{n}=\Gamma_{i} \times s_{n}.\end{split}  \tag{15}\]  
The five kinds of production rule, above have some common points; the points can be summed up with some general rules:  
1. In the first, fourth, and fifth production rules, the reliability value of the postpositional Web service of the transition equals the product of the reliability of the prepositional Web service of the transition and the reliability value of the transition. 2. In the second production rule, the reliability value of the postpositional Web service of the transition equals the minimum reliability value of all the prepositional Web services multiplied by the reliability value of the transition. 3. In the third production rule, the reliability value of the postpositional Web service of the transitions equals the maximum value of the reliability values of the respective prepositional Web services multiplied by the reliability values of the transitions.