## 3 Related Definition  
### Definition of Semantic Threshold Similarity and Definition of Semantic Threshold Equality  
**Definition 1**.: _Formal description of Web services [9]. A Web service can be formally described as_  
\[WS_{i}(I_{i},O_{i},P_{i},Q_{i})\,,\]  
_where \(I_{i}\) is the input parameter set of Web service \(WS_{i}\), \(O_{i}\) is the output parameter set of Web service \(WS_{i}\), \(P_{i}\) is the precondition set of Web service \(WS_{i}\), and \(Q_{i}\) is the QoS set of Web service \(WS_{i}\)._  
**Definition 2**.: _Semantic threshold similarity for two ontologies and semantic threshold equality for two ontologies. There are two ontology concepts, \(c_{i}\) and \(c_{j}\). By calculating their semantic similarity, we can obtain the value of \(sim(c_{i},c_{j})\). We set a semantic similarity threshold, \(k(0\leq k\leq 1)\). If \(sim(c_{i},c_{j})\geq k\), then the relation of semantic threshold similarity is built between \(c_{i}\) and \(c_{j}\), denoted as \(c_{i}\exists c_{j}\). If \(c_{i}\exists c_{j}\), and \(c_{j}\exists c_{i}\), then the relation of semantic threshold equality is built between \(c_{i}\) and \(c_{j}\), denoted as \(c_{i}\cong c_{j}\)._  
**Definition 3**.: _Semantic threshold similarity for two ontology sets and semantic threshold equality for two ontology sets. There are two concept sets \(C_{i}\{c_{i}\}\) and \(C_{j}\{c_{j}\}\). We set a semantic similarity threshold, \(k(0\leq k\leq 1)\). If any concept \(c_{i}\) in \(C_{i}\) exists corresponding \(c_{j}\) in \(C_{j}\) to satisfy \(c_{i}\exists c_{j}\), then the relation of semantic threshold similarity is built between \(C_{i}\) and \(C_{j}\), denoted as \(C_{i}\exists C_{j}\). If \(C_{i}\exists C_{j}\), and \(C_{j}\exists C_{i}\), then the relation of semantic threshold equality is built between \(C_{i}\) and \(C_{j}\), denoted as \(C_{i}\cong C_{j}\)._  
**Definition 4**.: _Semantic threshold similarity for Web service preconditions and semantic threshold similarity for Web services QoS. There are two Web services, \(WS_{i}(I_{i},O_{i},P_{i},Q_{i})\) and \(WS_{j}(I_{j},O_{j},P_{j},Q_{j})\); if any logical expression \(p_{i}\) in \(P_{i}\) exists corresponding \(p_{j}\) in \(P_{j}\), and \(p_{i}\) is **true**, then \(p_{j}\) is **true**, the relation of semantic threshold similarity is built between \(P_{i}\) and \(P_{j}\), denoted as \(P_{i}\exists P_{j}\). If any logical expression of QoS \(q_{i}\) in \(Q_{i}\) exists corresponding \(q_{j}\) in \(Q_{j}\), and \(q_{i}\) is **true**, then \(q_{j}\) is **true**, the relation of semantic threshold similarity is built between \(Q_{i}\) and \(Q_{j}\), denoted as \(Q_{i}\exists Q_{j}\)._  
**Definition 5**.: _Semantic threshold similarity for two Web services and semantic threshold equality for two Web services. There are two Web services, \(WS_{i}(I_{i},O_{i},P_{i},Q_{i})\) and \(WS_{j}(I_{j},O_{j},P_{j},Q_{j})\). We set a semantic similarity threshold \(k(0\leq k\leq 1)\), if \(I_{i}\exists I_{j},O_{i}\exists O_{j},P_{i}\exists P_{j}\) and \(Q_{i}\exists Q_{j}\), then the relation of semantic threshold similarity is built between \(WS_{i}\) and \(WS_{j}\), denoted as \(WS_{i}\exists WS_{j}\). If \(WS_{i}\exists WS_{j}\) and \(WS_{j}\exists WS_{i}\), then the relation of semantic threshold equality is built between \(WS_{i}\) and \(WS_{j}\), denoted as \(WS_{i}\cong WS_{j}\)._