# Heuristic Approaches to Attribute Reduction for Generalized Decision Preservation
## 2 Preliminaries  
### Rough Approximations in a Decision System  
In this section, we will review some fundamental notions of RST [1], such as a decision system, discernibility relation, and rough approximation. Studies of RST typically start with the concept of a decision system. Assume that \(U=\{x_{1},x_{2},\ldots,x_{n}\}\) is the universe, \(C=\{a_{1},a_{2},...,a_{m}\}\) and \(D=\{d\}\) are a condition attribute set and a decision attribute set, \(V_{a}\) is the domain of attribute values for all \(a\in AT\), and \(I_{a}:U\times\{a\}\to V_{a}\) is an information function such that \(I(x,a)=a(x)\in V_{a}\) for all \(x\in U\) and \(a\in AT\). Then, the 4-tuple \(DS=(U,C\cup D,V_{a},I_{a})\) is referred as to a decision system [1] in RST. For brevity, a decision system can be also denoted by a 2-tuple \(DS=(U,C\cup D)\). For instance, Table 1 shows a decision system \(DS=(U,C\cup D,V_{a},I_{a})\), where the universe is \(U=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\), the condition attribute set is \(C=\{a_{1},a_{2},a_{3},a_{4}\}\), and the decision attribute set is \(D=\{d\}\). **Definition 1**. : _[_1_]_ _Given a decision system \(DS=(U,C\cup D,V_{a},I_{a})\), for \(\forall Q\subseteq C\), if \(x_{i},x_{j}\in U\), \(a(x_{i})\) and \(a(x_{j})\) are the values of objects \(x_{i}\) and \(x_{j}\), with respect to the attribute \(a\), and the indiscernibility relation regarding \(Q\) is defined as_  
\[Ind(Q)=\{(x_{i},x_{j})\in U\times U|\forall a\in Q,a(x_{i})=a(x_{j})\}. \tag{1}\]  
Obviously, \(Ind(Q)\) is symmetric, transitive, and reflexive. Hence, \(Ind(Q)\) is an equivalence relation and can be calculated as \(Ind(Q)=\bigcap_{a\in Q}Ind(\{a\})\). \(Ind(Q)\) divides the universe \(U\) into a collection of indiscernible granules (equivalence classes); that is, \(U/Ind(Q)=U/Q=\{E_{IND(Q)}(x_{i})|x_{i}\in U\}=\{E_{Q}(x_{i})|x_{i}\in U\}\), where \(E_{IND(Q)}(x_{i})=E_{Q}(x_{i})\) is an indiscernible granule containing \(x_{i}\). From classical RST, indiscernible granules can constitute two definable sets, called the lower approximation and the upper approximation. An arbitrary decision class in the universe can be approximated by the lower and upper approximations, which are defined as follows:  
**Definition 2**. : _[_1_]_ _Given a decision system \(DS=(U,C\cup D,V_{a},I_{a})\), \(\forall Q\subseteq C\), \(D_{j}\in U/D\), the lower approximation (LA) and the upper approximation (UA) of \(D_{j}\), with respect to \(Q\), are defined as follows:_  
\[\underline{\mathit{Apr}}_{Q}(D_{j})=\{x_{i}|E_{Q}(x_{i})\subseteq D_{j}\}= \bigcup\{E_{Q}(x_{i})|E_{Q}(x_{i})\subseteq D_{j}\}, \tag{2}\]  
\[\overline{\mathit{Apr}}_{Q}(D_{j})=\{x_{i}|E_{Q}(x_{i})\cap D_{j}\
eq\emptyset \}=\bigcup\{E_{Q}(x_{i})|E_{Q}(x_{i})\cap D_{j}\
eq\emptyset\}. \tag{3}\]  
For \(\forall Q\subseteq C\), \(\underline{\mathit{Apr}}_{Q}(D_{j})\) is the union of all \(E_{Q}(x_{i})\) which are included in \(D_{j}\) totally, whereas \(\overline{\mathit{Apr}}_{Q}(D_{j})\) is the union of all \(E_{Q}(x_{i})\) which are included in \(D_{j}\) partially. \(U-\overline{\mathit{Apr}}_{Q}(D_{j})\) is the union of the \(E_{Q}(x_{i})\) which are not included in \(D_{j}\) completely. The positive region (PR), boundary region (BR), and negative region (NR) of \(D_{j}\) in the universe are defined, respectively, as follows: \(Pos_{Q}(D_{j})=\underline{\mathit{Apr}}_{Q}(D_{j})\), \(\underline{\mathit{Bnd}}_{Q}(D_{j})=\overline{\mathit{Apr}}_{Q}(D_{j})- \underline{\mathit{Apr}}_{Q}(D_{j})\), and \(Neg_{Q}(D_{j})=U-\overline{\mathit{Apr}}_{Q}(D_{j})\). The relationship among the three regions (PR, BR, and NR) is presented as follows:  
\[Pos_{Q}(D_{j})+\mathit{Bnd}_{Q}(D_{j})+Neg_{Q}(D_{j})=U. \tag{4}\]  
**Definition 3**.