## 3 A Review of Conceptual Clustering Algorithms  
Conceptual clustering algorithms (conceptual algorithms), in addition to the list of objects belonging to the clusters, provide for each cluster one or several concepts as an explanation of the clusters. Formally, the problem of conceptual clustering is defined as follows [31]. Let \(O=\{o_{1},o_{2},\ldots,o_{n}\}\) be a set of objects, each of which is described by a set of attributes \(A=\{a_{1},a_{2},\ldots,a_{m}\}\), which can be quantitative or qualitative, and take the values \(V=\{V_{11},\ldots,V_{mk}\}\). The problem of conceptual clustering is to organize a set \(O\) into a set of clusters \(C=\{c_{1},c_{2},\ldots,c_{k}\}\), such that the following three conditions are satisfied:  
* Each cluster \(c_{i},I=1,\ldots,k\) has associated an extensional and an intentional description; the extensional description contains the objects that belong to the cluster, whereas the intentional one contains the concepts (statistical or logical properties) that describe the objects belonging to the cluster in terms of the attribute set A and using a given language. Depending on the complexity of the language, the concepts will be difficult or easy to understand by a final user;
* \(\forall o_{i}\in O\), \(i=1,\ldots,n,\exists c_{j}\in c,j=1,\ldots,k\) such that \(o_{i}\in c_{j}\). This property ensures that every object in the collection belongs to at least one cluster;
* \(\
exists c_{j}\in c,j=1,\ldots,k\) such that \(c_{j}=\emptyset\). This property ensures that each cluster in the solution contains at least one object of the collection. In this section, we present an overview of the most common algorithms in the field of conceptual clustering, highlighting their limitations or shortcomings. Refs. [31; 32] presents conceptual optimization-based algorithms that, starting from an initial set of clusters, build a clustering through a continuous process of optimizing this initial set, typically using an objective function. These algorithms require tuning of some parameters (for example, the number of clusters), which can be a difficult problem. In addition, conceptual optimization-based algorithms have high computational complexity. Many authors have proposed conceptual algorithms that use a classification tree or hierarchical structure to organize objects. Each node in this structure represents a class and is labeled with a concept that describes the objects classified under the node. COBWEB [33] is a well-known incremental and hierarchical conceptual algorithm that describes the clusters using pairs of attribute-values, which summarize the probability distribution of objects classified under the sub-tree determined by that cluster. [PERSON] and [PERSON] propose in [34] a stepwise and hierarchical fuzzy conceptual algorithm. This algorithm can place an object in more than one node and describes clusters using a fuzzy center computed based on the objects stored in the cluster. There are also conceptual algorithms that use graph theory to build clusters. As a rule, these algorithms represent a set of objects in the form of a graph in which the vertices are objects, and the edges express the similarity relationship that exists between the objects and build a clustering through covering this graph with a special kind of subgraph. Examples of algorithms based on this approach are presented in [35]. These algorithms have a high computational complexity and can also create clusters described by a large number of terms, which can also be difficult to understand in real problems [36]. [PERSON] presents an agglomerative conceptual algorithm that use a bottom-up approach for clustering [37]. Agglomerative algorithms create clusters, the concepts of which can be difficult to understand in real-world problems for large datasets, and they require tuning of the parameter values used to build clusters, which, as mentioned earlier, can be a difficult problem [35]. Unlike agglomerative algorithms, the literature presents algorithms that use top-down approaches to build conceptual clustering.