## 3 Fuzzy Information System with Tagaki-Sugeno Reasoning  
### Type-1 Fuzzy Information System and Fuzzy Rule Induction  
This research is based on our interpretation of a fuzzy information system, introduced in [22]. We assume that the values of the system information function are linguistic variables which corresponds to fuzzy sets. This is a granulation information proposal. A classic information system [16] is defined by the elements: (U, A, V, f), where U is a universe, A is a set of attributes, V represents attributes domains: V \(=\)\({}_{\text{df}}\)\(\bigcup\)\({}_{\text{a}}\)V\({}_{\text{a}}\), with nonempty domain V\({}_{\text{a}}\) of the a-th attribute (a \(\in\) A), and f is the information function f: U\(\times\)A \(\rightarrow\) V, V\(\times\)\(\in\) U, a \(\in\) A f(u, a)\(\in\)V\({}_{\text{a}}\). The indiscernibility binary relation (IND), defined over U (IND \(\subseteq\) U\({}^{2}\)), plays in theory an important role. It is an equivalence relation, defined in Equation (7):  
\[\text{IND(B)}=\)\({}_{\text{df}}\)\(\{\text{(u}_{\text{i}},\text{u}_{\text{j}})\in\) U \(\text{`U:`V$a$\in$B f(u${}_{\text{i}},\text{a})=f(u${}_{\text{j}},\text{a})}\}\), u\({}  
attribute)]. The membership function of the medium fuzzy set assumes Gaussian distribution of the attribute data for any \(\mathrm{a}\in\mathrm{A}\). Therefore, the membership functions for the considered fuzzy sets can be easily defined, see Equation (8):  
\[\mu_{\mathrm{medium}}(\mathrm{a})=_{\mathrm{df}}\ \mathrm{e}^{\frac{-( \mathrm{a}-\mathrm{a}_{0})^{2}}{2\sigma^{2}}},\ \mu_{\mathrm{low}}(\mathrm{a})=_{\mathrm{df}}\left\{ \begin{array}{c}1- \mathrm{e}^{\frac{-(\mathrm{a}-\mathrm{a}_{0})^{2}}{2\sigma^{2}}}\ :\ \mathrm{a}\ <\mathrm{a}_{0}\right\},\\ 0:\mathrm{a}\geq\ \mathrm{a}_{0}\end{array} \right\}, \tag{8}\] \[\mu_{\mathrm{high}}(\mathrm{a})=_{\mathrm{df}}\left\{ \begin{array}{c} 0:\mathrm{a}<\ \mathrm{a}_{0}\\ 1-\mathrm{e}^{\frac{-(\mathrm{a}-\mathrm{a}_{0})^{2}}{2\sigma^{2}}}:\mathrm{a} \geq\mathrm{a}_{0}\end{array} \right\},\ \mathrm{a}\ \in\ \mathrm{V}_{\mathrm{a}}\mathrm{with\ expected\ value}(\mathrm{a}_{0})\] and standard deviation \[(\delta)\]  
The above attribute fuzzification introduces a very simple initialization of the information function as well as values' generalization. For example, let us consider the below matrix (Table 3) which represents a fuzzy information system: U = {object1, object2, object3, object4}, A = {attribute1, attribute2, attribute3}, V = \(\mathrm{V}_{\mathrm{attribute1}}\cup\mathrm{V}_{\mathrm{attribute2}}\cup\mathrm{V}_{ \mathrm{attribute3}}\) and we assume for each set \(\mathrm{V}_{\mathrm{attribute1}}\) a Gaussian distribution in order to obtain the fuzzy sets {lowi, mediumi, highi}. So, for example, f(object2, attribute2) is introduced as _high_ because the following inequality is satisfied: \(\mu_{\mathrm{high}}\)(f(object2, attribute2)) \(\geq\) max{ \(\mu_{\mathrm{low}}\)(f(object2, attribute2), \(\mu_{\mathrm{medium}}\)(f(object2, attribute2)) or f(object1, attribute3) is introduced as _low_, as \(\mu_{\mathrm{low}}\)(f(object1, attribute3)) \(\geq\) max{ \(\mu_{\mathrm{medium}}\)(f(object1, attribute3), \(\mu_{\mathrm{high}}\)(f(object1, attribute3))}\). The corresponding partition (P) with respect to IND has two equivalence classes:  
\[P/\mathrm{IND}\mathrm{[attribute1, attribute2, attribute3]}=\{\mathrm{[ object1, object4],\ \mathrm{[object2, object3]}}\}\]  
Next, if a decision attribute (A*) is added to the set of attributes: A = A \(\cup\) A* an information system can be represented by a decision table. Assuming such an extension, the Skowron and Suraj rule induction method can be applied [38,39] and next, transformation of the rules induced for each decision into fuzzy rules is introduced.