## 2 Preliminaries
### Fuzzy Ontology
To enhance ontologies' capacity for representation and provide exact representation of imprecise data, a fuzzy ontology is utilized. An ontology that employs fuzzy logic to create a more intuitive representation of imprecise and vague knowledge for the purpose of facilitating reasoning about it is called a fuzzy ontology [22]. The idea of fuzzy relations is first introduced in fuzzy ontologies. Degrees of relationship between concepts or individuals may be stated in the interval [0, 1], allowing for a variety of possible degrees of relationship. To construct the imprecision connection, a fuzzy membership function may be used in this manner [14]. The modification is entirely incremental; conversion to a fuzzy ontology adds membership values to the currently existing relations and may also add new entries in the ontology. The ontology membership is normalized with respect to each of the terms in the ontology; that is, the sum of the membership values of each term in the ontology is equal to 1. The fuzzification process of the crisp ontology is shown diagrammatically in Figure 2.  
Fuzzy ontology is the hierarchical description of every instance with its fuzzy membership value belonging to a class. A fuzzy domain ontology is a 4-tuple \(O_{F}=(C,P_{F},R_{F},A_{F})\), where (1) A collection of concepts is called \(C\). Each concept has certain properties here that have the values of a fuzzy concept or fuzzy set, which differs from the notion of a crisp ontology. (2) A group of properties is called \(P_{F}\). A property \(p_{F}\in P_{F}\) is described as a 5-tuple of the form \(p_{F}(c,v_{F},q_{F},f,U)\), where \(c\in C\) is an ontology concept, \(v_{F}\) is a representation of property values, \(q_{F}\) is a model of linguistic qualifiers that can control or modify the strength of a property value \(v_{F}\), \(f\) is a representation of restriction facets on \(v_{F}\), and \(U\) is the universe of discourse. Both \(v_{F}\) and \(q_{F}\) are fuzzy concepts at \(U\), but \(q_{F}\) modifies \(v_{F}\)'s degree of fuzziness. (3) \(R_{F}\) is a set of inter-concept relations between concepts. \(r_{F}\in R_{F}\) defined as a 5-tuple of the form \(r_{F}(c_{1},c_{2},t,S_{F},U)\), where \(c_{1},c_{2}\in C\) are ontology concepts, \(t\) represents relation type, \(S_{F}\) models relation strengths and is fuzzy concept at \(U\), which can represent the strength of association between concept-pairs \(\langle c_{1},c_{2}\rangle\). (4) \(A_{F}\) is a set of fuzzy rules. In a fuzzy system the set of fuzzy rules is used as knowledge base. Table 1 shows the difference between fuzzy and classical (crisp) ontology [20]. An extended fuzzy ontology is a 7-tuple \(O_{F}=(c_{a},C_{F},R,F,Q,O,U)\) where \(c_{a},C_{F},R,\)\(F,U\) have same interpretations as defined in fuzzy ontology and \(Q\) is the set of the linguistic qualifiers. A qualifier \(q\in Q\) and a fuzzy concept \(c_{F}\in C_{F}\) compose a composition fuzzy concept that can be the value of \(c_{a}\). \(O\) is the set of fuzzy operators at \(U\), which is isomorphic to \(Q\). Most academics analyze applications that incorporate fuzzy ontologies or use the standard techniques for evaluating crisp ontologies while evaluating fuzzy ontologies. Only one study, [24], in which a unique metric is established as follows, is clear about assessing fuzzy ontologies. \[K=\sum_{(p\
eq 0)}p/n-\sum_{pi=0}0.1/n-\log(n_{1}/n) \tag{2}\]