## 4 Model of the Agent
### Repository of Categories with Prototype  
As can be seen in Figure 2, it is assumed that models of all categories known to the agent are stored in a dedicated repository. The repository is an important part of the internal representation of the so-called embodied ontology, representing how the agent perceives the conceptual order of the external world. Another basic element of such an ontology is the basic mental space introduced above. Each category with a prototype is built in relation to the accessible mental basic space \(O\). In our approach, we try to reflect as many structural and functional features as possible, which have been provided for categories in the so-called standard version of the prototype semantics [9]. In particular, each category is assigned a specialized object \(o_{c}^{\star}\in O\) distinguished from all other objects in this category, called a prototype of the category. It is perceived by the agent as being the most representative and in some sense \"central\" element of the category. An agent's attitude towards a membership of an object to a given category should be directly related to the degree of that object's similarity (or distance) to the prototype. Such an approach translates to a requirement that the agent should be equipped with a cognitive ability allowing for comparison of any object \(o\in O\) with a prototype \(o_{c}^{\star}\) of any category \(c\). Additionally, this ability enables the agent to order objects according to their similarity (or distance) to the prototype. Naturally, the most similar (closest to the prototype) objects are considered as belonging to the category. On the other hand, the least similar (farthest from the prototype) objects are considered as not belonging to the category. This is going to be determined within the agent by a set of ranges assigned to the category by the agent itself during the process of category learning. Consistent with the assumption that the boundary of a category may be fuzzy, our model reflects it by using ranges in order to introduce an area of the mental basic space in which a membership of an object to a category is considered uncertain (understood here as _partial_ rather than as _probable_) by the agent. These ideas lead to the formulation of category's model consisting of the following:  
* a prototype of a category--a mental representation of a prototypical object, considered the very center of the category,
* a core of a category--an area of a basic universe of mental representations naturally assigned by an agent to the category; for natural reasons, the core should surround the prototype,
* a boundary of a category--an area of a basic universe of mental representations where an applicability of the category is questionable,
* an outer of a category--an area of a basic universe of mental representations where the concept is not applicable. Assuming the distance-based approach, perhaps in the simplest but still effective way the category can formally be defined as a triplet:  
\[\langle o_{c}^{\star},\tau_{c}^{+},\tau_{c}^{-}\rangle, \tag{6}\]  
where:  
* \(o_{c}^{\star}\) is a prototype of \(c\),
* \(\tau_{c}^{+}\) is a radius defining _Core_ of \(c\),
* \(\tau_{c}^{-}\) is a radius defining _Boundary_ of \(c\),
* and the following condition holds: \[\tau_{c}^{+}<\tau_{c}^{-}.\] (7)  
Figure 3 graphically presents a deliberately simplified model of a category with a prototype using a visual notion of the cognitive distance (reflected as graphical distance between points on the plane). The category is defined over the basic mental space \(O=\{o_{1},\ldots,o_{8}\}\). The object \(o_{7}\) is indicated as the prototype of the category. Objects \(o_{1}\) and \(o_{7}\) are located in the core of the category and therefore are both perceived as belonging to the category \(c\), although object \(o_{1}\) is not assessed by the agent as fully conforming to the prototype \(o_{7}\).