## 5 The Strategy of Initial Learning of Categories with Prototypes
### Outline of the Category Learning Steps  
This section builds upon a strategy for learning categories with prototypes, which was in a brief manner presented in [64] to show how categories can be computationally and relatively consistently with intuition extracted from collected learning experience. The agent starts with the learning experience related to a particular category \(c\) aggregated into the above introduced multiset \(Exp_{c}\) (see Formula (9)). It intends to derive a model of a category with a prototype \(c\), that is, the category's prototype, core, and boundary. Following the model of category with prototype introduced in Section 4.3, the algorithm goals are, in particular, to find a prototype \(o_{c}^{\star}\) and to determine two related radii \(\tau_{c}^{+}\) and \(\tau_{c}^{-}\). The basic ideas behind the algorithm are as follows: at first, to use an adopted basic mental space enriched with a specific distance or similarity function \(f\) (see Definition 4); at second, to determine candidates for prototype; at third, to analyze all the candidates according to some criteria; and at the end, to choose the prototype, if it exists. It is generally expected that the prototype should be surrounded by a core of the category with the currently largest possible radius, and contain a substantially large part of the supporting (positive) examples of the learning data. In a case where no candidates fulfill such requirements, the procedure should fail which means that the current experience does not support building a model of category \(c\) as a category with a prototype. The algorithm iterates over objects \(o\in O\) occurring (mentioned) in the learning experience. Therefore, to make further presentation more convenient, some additional notions (subsets) are derived from \(Exp_{c}\):  
* \(E_{c}^{+}(Exp_{c})=\{o:o\in O\wedge k_{o,c}^{+}>0\}\) denotes a set of objects \(o\in O\) supported by at least one positive example of learning experience given in \(Exp_{c}\),
* \(E_{c}^{-}(Exp_{c})=\{o:o\in O\wedge k_{o,c}^{-}>0\}\) denotes a set of objects \(o\in O\) supported by at least one negative example of learning experience given in \(Exp_{c}\),
* \(E_{c}^{+}(Exp_{c})=\{k_{o,c}^{+}(o,\text{is-}c)\in Exp_{c}\}\) denotes a multiset aggregating positive part (labeled by _is-c_) of learning experience given in \(Exp_{c}\). An idea behind the method for category formation [64] is to start with a set of suitable candidates for a prototype of a category for which the agent is trying to build an internal model. Omitting the whole philosophical discussion related to whether a prototype should be a reflection of a particular existing object the agent has ever observed in the past, or whether hypothetical objects from the whole mental space are also allowed, we skip to the conclusion by saying that the prototype should be, in a way, a central element of a multiset containing learning experience positively supporting a particular category, namely a multiset \(\hat{E}_{c}^{+}(Exp_{c})\). An algorithm presented in the next section assumes a particular set of candidates for prototype as being extracted using a function \(extractCandidates\). In a majority of practical contexts, candidates are chosen based on certain optimization criteria. When the basic mental space is equipped with a distance function \(f\), usual cases of candidates \(o_{c}\) for prototype \(o_{c}^{\star}\) are given as follows:  
* the most common element of \(\hat{E}_{c}^{+}(Exp_{c})\)--an honorable mention rather than a serious option--while it is the easiest one to calculate, it ignores distance-based relations between elements of \(\hat{E}_{c}^{+}(Exp_{c})\).