## 4 Knowledge Evaluation and Integration Method
### Evaluation of Semantic Relationships  
The proposed method to evaluate and integrate semantic relationships takes into account the overall concepts and semantic relationships included in all the (input) ontologies to be evaluated. Thus, a list of concepts is built as the union set of the concepts in all the input ontologies. Then, a similar procedure is followed regarding the taxonomic and the partonomic relationships in all the input ontologies. Subsequently, for each input ontology, the following is performed.  
For each type of semantic relationship (i.e., for taxonomic and partonomic relationships), an evaluation matrix is generated, having the concepts of all input ontologies as rows and columns. Then, each cell of the matrix associated with an input ontology is marked with the value 1 if there exists a semantic relationship in the input ontology under question between the involved concepts in the corresponding row and column. Otherwise, it is marked with the value 0.  
Formally, the semantic relationships evaluation method can be stated in an algorithmic fashion as follows. Let C be \(\bigcup_{n}\) UMHRD\({}_{i}\), where UMHRD\({}_{i}\) stands for a UMHRD of the MHRD corresponding to the i-th input ontology, i = 1,..., n; let TM\({}_{i}\)(c\({}_{i}\), c\({}_{k}\)) be the value of the cell formed by the concepts c\({}_{j}\) and c\({}_{k}\) in the evaluation matrix of the taxonomic relationships for the i-th input ontology; and let PM\({}_{i}\)(c\({}_{j}\), c\({}_{k}\)) be the value of the cell formed by the concepts c\({}_{j}\) and c\({}_{k}\) in the evaluation matrix of the partonomic relationships for the i-th input ontology. Then, the proposed method can be stated as follows (Algorithm 1):  
```
For i = 1 to n For j = 1 to Cardinal (C) For k = 1 to Cardinal (C) If (cj \(\in\) UMHRD\({}^{i}\) (ck \(\in\) UMHRD\({}^{i}\)) \({}^{\sim}\)[IS-A (cj, ck) \(\lor\) IS-A (ck, cj)] then TM\(i\)(cj, ck) = 1 else TM\(i\)(cj, ck) = 0; If (cj \(\in\) UMHRD\({}^{i}\)) \({}^{\sim}\)(ck \(\in\) UMHRD\({}^{i}\) [PART-OF (cj, ck) \(\lor\) PART-OF (ck, cj)] then PM\(i\) (cj, ck) = 1 else PM\(i\)(cj, ck) = 0;
```  
**Algorithm 1**Method for assessing semantic relations.  
To calculate the support for a taxonomic relationship, the following is carried out. For every input ontology, the support for a certain semantic relationship involving the concepts of that ontology is obtained by summing the values in the cells of the corresponding matrix for that relationship and the counterparts of the rest of the input ontologies when these take the value 1. Formally, the _taxonomic support for the i-th ontology_, written TS\({}^{i}\), is defined using the following equation:  
\[\text{TS}^{i}=\sum_{j,k}^{N}TM_{i}\big{(}c_{j},c_{k}\big{)}+\sum_{\begin{array} []{c}l=1\\ l\
eq i\end{array}}^{n}\sum_{j,\ k}^{N}TM_{i}\big{(}c_{j},c_{k}\big{)}\ TM_{l} \big{(}c_{j},c_{k}\big{)} \tag{1}\]  
where N = Cardinal (C); i = 1, 2,..., n; n = number of input ontologies.  
Proceeding in an analogous manner, the _partonomic support for the i-th ontology_, written PS\({}^{i}\), may be defined using the following equation:  
\[PS^{i}=\sum_{j,k}^{N}PM_{i}\big{(}c_{j},c_{k}\big{)}+\sum_{\begin{array}{c}l =1\\ l\
eq i\end{array}}^{n}\sum_{j,k}^{N}PM_{i}\big{(}c_{j},c_{k}\big{)}\ PM_{l} \big{(}c_{j},c_{k}\big{)} \tag{2}\]Finally, the _relational support for the i-th ontology_, written RS\({}^{\text{i}}\), is defined using Equation (3) below.  
\[\text{RS}^{\text{i}}=\text{TS}^{\text{i}}+\text{PS}^{\text{i}} \tag{3}\]