## 3 Knowledge Representation Model  
Knowledge representation is one of the most important areas in Artificial Intelligence. Although several methods for knowledge representation can be found in the literature, ontologies are the most popular knowledge representation schema. An ontology can be defined as a shared and common understanding of some domain that can be communicated across people and computers [24]. One of the most important types of ontologies is domain ontology, which can be defined as the specification of a conceptualization of domain knowledge [5]. In this study, ontologies are modeled using Multiple Hierarchical Restricted Domains (MHRD), where an MHRD can be understood as a finite set of concepts such that:  
1. The set contains at least two concepts. Formally, Cardinal (MHRD) \(\geq\) 2. 2. Every concept is defined through a finite, non-empty set of attributes. Formally, if AT(c) stands for the set of attributes defining a concept c \(\in\) MHRD, this condition states that Cardinal (AT(c)) \(\geq\) 1. Furthermore, the relationship between a given concept cj of an MHRD and a given attribute ak, written HAS_ATi(cj, ak), is defined as a logical function whose evaluation is true if ak \(\in\) AT(cj). 3. Among a given pair of concepts, there may be either a taxonomic relationship so that (multiple) inheritance of attributes is possible or a partonomic one. Formally, a taxonomic relationship between two concepts, cj and cj of an MHRD, written IS-A (cj, cj), is defined as a logical function whose evaluation is true if cj is a semantic subcategory of cj. A partonomic relationship between two concepts cj and cj of an MHRD, written PART-OF (cj, cj), is defined as a logical function whose evaluation is true if and only if cj is a semantic element/component of cj. 4. Taxonomic relationships are irreflexive, antisymmetric, and transitive, while partonomic relationships are irreflexive and antisymmetric but not transitive [4]. 5. The set of concepts, together with the relationships between those, can be expressed through graphs, where the nodes represent the concepts (including their attributes), and the arrows account for the relationships so that every concept is linked (through a taxonomic or a partonomic relationship) to n concepts, n \(\geq\) 1. Some definitions that consider the two above-referenced types of semantic relationships embracing subsets of concepts in an MHRD are introduced next. **Definition 1**. : _Taxonomic MHRD._  
A taxonomic MHRD, written TMHRD, is an MHRDR subset where all its concepts are related to each other only through taxonomic relationships. Formally, let M and T be two MHRDs such that T \(\subseteq\) M. T is said to be a taxonomic MHRD of M, written TMHRD if \(\forall\)c\({}_{i}\in\) T, \(\exists\) c\({}_{j}\in\) T such that [IS-A (c\({}_{i}\), c\({}_{j}\)) \(\vee\) IS-A (c\({}_{j}\), c\({}_{i}\))] ^ \(\sim\) [PART-OF (c\({}_{i}\), c\({}_{j}\)) \(\vee\) PART-OF (c\({}_{j}\), c\({}_{i}\))]. **Definition 2**. : _Partonomic MHRD._  
A partonomic MHRD, written PMHRD, is an MHRD subset where all its concepts are related to each other only through partononic relationships. Formally, let M and P be two MHRDs such that P \(\subseteq\) M. P is said to be a partonomic MHRD of M, written PMHRD, if \(\forall\)c\({}_{i}\in\) P, \(\exists\) c\({}_{j}\in\) P such that [PART-OF (c\({}_{i}\), c\({}_{j}\)) \(\vee\) PART-OF (c\({}_{j}\), c\({}_{i}\))] ^ \(\sim\) [IS-A (c\({}_{i}\), c\({}_{j}\)) \(\vee\) IS-A (c\({}_{j}\), c\({}_{i}\))]. **Definition 3**. : _Uniform MHRD._  
Let M and U be two MHRDs such that U \(\subseteq\) M. U is said to be a Uniform Multiple Hierarchical Restricted Domain, written UMHRD, of M if U \(\in\) [T, P], where T is a TMHRD of M, and P is a PMHRD of M.  
Based on the previous definitions, an immediate result is stated in the following corollary. **Corollary**. : _Let M be an MHRD and let U be a UMHRD of M. Then, the following inequality holds 2 \(\leq\) Cardinal (U) \(\leq\) Cardinal (M)._  
Proof. : Based on the definition of UMHRD, U is an MHRD.