## 3 Problem Definition  
The fundamental concepts of the Subgroup Discovery (SD) technique are provided in this section. Additionally, these concepts are extended and detailed in Appendix A. First, an attribute \(a\) is a unique characteristic of an object, which has an associated value. An example of an attribute is \(a=age\):30. Therefore, the domain of an attribute \(a\) (denoted as \(dom(a)\)) can be defined as the set of all the unique values that said attribute can take. An attribute can be nominal or numeric, depending on its domain. On the other hand, an instance \(i\) is a tuple \(i=(a_{1},\ldots,a_{M})\) of attributes. Given the attributes \(a_{1}=age\):25 and \(a_{2}=sex\):\(woman\), an example of an instance is \(i=(age\):25, _sex:\(woman)\)_. Additionally, a dataset \(d\) is a tuple \(d=(i_{1},\ldots,i_{N})\) of instances. Given the instances \(i_{1}=(age\):30, _sex:\(man)\)_ and \(i_{2}=(age\):25, _sex:\(woman)\)_, an example of a dataset is \(d=((age\):30, _sex:\(man)\)_, (age:25, _sex:\(woman))\). It is necessary to state that all values from a dataset \(d\) can be indexed with two integers, \(x\) and \(y\). We use the notation \(v_{x,y}\) to indicate the value of the x-th instance \(i_{x}\) and of the y-th attribute \(a_{y}\) from a dataset \(d\). Given an attribute \(a_{y}\) from a dataset \(d\), a binary \(operator\in\{=,\
eq,<,>,\leq,\geq\}\) and a value \(w\in dom(a_{y})\), a selector \(e\) is a 3-tuple of the form \((a_{y}\),\(characteristic, operator,w)\). Note that when an attribute \(a_{y}\) is nominal, only the \(=\) and \(\
eq\) operators are permitted. Informally, a selector is a binary relation between an attribute from a dataset and a value in the domain of that attribute. This relation represents a property of a subset of instances from that dataset. It is essential to bear in mind that the first element of a selector refers only to the attribute name, i.e., the characteristic, and not to the complete attribute itself. **Definition 1** (Selector covering). : _Given an instance \(i_{x}\) and an attribute \(a_{y}\) from a dataset \(d\), and a selector \(e=(a_{y}.\textit{characteristic},\textit{operator},w\in\textit{dom}(a_{y}))\), then \(i_{x}\) is covered by \(e\) (denoted as \(i_{x}\sqsubset e\)) if the binary expression \"\(v_{x,y}\) operator \(w\)\" holds true. Otherwise, we say that it is not covered by \(e\) (denoted as \(i_{x}\sqsubset e\))._  
For example, given the instance \(i_{x}=(\textit{age}25,\textit{sex:woman})\) and the selectors \(e_{1}=(\textit{age},\textit{<},20)\) and \(e_{2}=(\textit{sex},=,\textit{woman})\), it will be noted that \(i_{x}\
ot\sqsubset e_{1}\) and \(i_{x}\sqsubset e_{2}\). Subsequently, a pattern \(p\) is a list of selectors \(<e_{1},\ldots,e_{j}>\) in which all attributes of the selectors are different. Moreover, its size (denoted as \(|p|\)) is defined as the number of selectors that it contains. In general, a pattern is interpreted as a list of selectors (i.e., as a conjunction) that represents a list of properties of a subset of instances from a dataset. **Definition 2** (Pattern covering). : _Given an instance \(i_{x}\) from a dataset \(d\) and a pattern \(p\), then \(i_{x}\) is covered by \(p\) (denoted as \(i_{x}\sqsubset p\)) if \(\forall e\in p,i_{x}\sqsubset e\). Otherwise, we say that it is not covered by \(p\) (denoted as \(i_{x}\sqsubset p\))._  
Following these definitions, a subgroup \(s\) is a pair (pattern, selector) in which the pattern is denoted as _s.description_ and the selector is denoted as _s.target_. Given the dataset \(d=((\textit{fever:yes},\textit{sex:man},\textit{flux:yes})\), (\(\textit{fever:yes},\textit{sex:woman},\textit{flux:no}))\), an example of a subgroup is \(s=(<(\textit{fever},=,\textit{yes}),(\textit{sex},=,\textit{woman})>,(\textit{flu },=,\textit{yes}))\). **Definition 3** (Subgroup refinement \(s^{\prime}\)).