## 4 Property-Based Ontology Representation
### Property-Based Metrics  
The property-oriented ontology model provides an opportunity to introduce specific property-based quality metrics for ontologies. The main principle is that the property distribution in the taxonomy must be well balanced and consistent. To measure this quality, we propose the following measures: \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) & \(\mathbf{a_{4}}\) & \(\mathbf{a_{5}}\) \\ \hline \#chair1 & 1 & 0 & 0 & 0 & 0 \\ \hline \#peter1 & 0 & 1 & 1 & 0 & 0 \\ \hline \#employee1 & 0 & 0 & 0 & 1 & 1 \\ \hline \#dog1 & 0 & 0 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: T1: Generated context table. * \(m_{A}=\frac{|A|}{|C|}:\) the relative number of the properties. If the value is lower (near or below 1), there are too few properties. If the number is too high, most of the properties are not used in the taxonomy construction. * \(m_{e}=\frac{|C_{\emptyset}|}{|C|}:\) the ratio of concepts with empty local (not inherited) property sets. From the viewpoint of FCA, if this value is greater than 0, then the ontology is invalid. * \(m_{=}=\frac{|C_{\_}{\_}{\infty}|}{|C|}:\) the ratio of concepts having non-unique properties set. From the viewpoint of FCA, if this value is greater than 0, then the ontology is invalid. * \(m_{l}=\frac{\sum_{\_}{c\in C_{\_}{\infty}}|(A(c)|)}{|C|}\). This measures shows the average length of the local (not empty) property sets. A high value means that many attributes are not relevant in the taxonomy construction. * \(m_{c}=\frac{\sum_{\_}{c\in D_{\_}{\infty}^{inf}}d(c,D_{\_}{\infty}^{inf})+ \sum_{\_}{c\in D_{\_}{\infty}^{inf}}d(c,D_{\_}{\infty}^{(dc)})}{2*|A|}\): This value shows the total distance between the declared and inferred domain concepts. The distance function \(d()\) is defined as the length of the shortest path in taxonomy graph between the elements. In the best case, \(m_{c}\) is equal to 0. As an example, we compare two schema descriptions; the first is an ontology oriented model, while the second is example of the UML-OOP model. In this example, we first take a sample ontology found on the internet [41]. For the sake of simplicity, we use only the following concept taxonomy fragment: \begin{tabular}{l} Person \\ Employee \\ Professor \\ AssistantProfessor \\ AssociateProfessor \\ Lecturer \\ Assistant \\ Student \\ UndergraduateStudent \\ GraduateStudent \\ Organization \\ EducationOrganization \\ Department \\ Program \\ Course \\ Seminar \\ \end{tabular}   
The list of related properties in the ontology, where the first argument denotes the declared domain and the second argument, is the range parameter. \begin{tabular}{l} advisor(Student, Professor) \\ affiliateOf(Organization, Person) \\ affiliatedOrganization(Organization, Organization) \\ alumnus(Organization, Person) \\ emailAddress(Person,.STRING) \\ head(Organization, Person) \\ listedcourse(Program, Course) \\ name(Person,.STRING) \\ takesCourse(Student, Course) \\ teacherOf(Department, Course) \\ \end{tabular}   
The ontology schema does not contain any individual declarations; thus, the set \(D_{\_}{a}^{inf}\) contains only the \(\top\) concept for every attribute \(a\). The proposed metrics have the following values for this sample ontology:\(m_{A}=10/16=0.62\)
* \(m_{e}=11/16=0.69\). * \(m_{=}=0\)
* \(m_{l}=2\)
* \(m_{c}=1.4\)  
The second example schema is the UML class diagram presented in Figure 2. If we consider this UML diagram as an ontology model, we have the following metric values:  
* \(m_{A}=12/11=1.10\)
* \(m_{e}=4/11=0.36\). * \(m_{=}=0\)
* \(m_{l}=2\)
* \(m_{c}=16/12=1.33\)  
The third example is the standard pizza ontology [43] presented in Figure 3. The presented reduced ontology contains only fourteen concepts, while the depth of the taxonomy hierarchy is 4. The ontology contains three properties. The first step of ontology processing is to generate the corresponding properties. In this reduced ontology, fifteen attributes were generated from the properties.