# Compact Fuzzy Systems Based on Boolean Relations
## 3 Compact Fuzzy Systems Based on Boolean Relations
### Activation Function with Total Dependence  
In this case, activation is performed by all input sets. Table 12 shows the extent of the table from Boole to Kleene where \(X\) can be \(\{1,u,0\}\) and \(X_{u,1}\) the elements \(\{u,1\}\).  
In this table, if any of the variables \(A_{m}\) is 0 then \(Y_{T}=0\), if all variables \(A_{m}\) are 1, then, \(Y_{T}=1\); finally, if any of the variables \(A_{m}\) is \(u\) and the other ones \(u\) or 1, then \(Y_{T}=u\). Thus, the cases where there is a transition from \(Y_{T}\) between 0 and 1 they are assigned \(u\).  
Using the rules to determine the disjunctive normal form, it is observed that the term \(A_{1}\wedge A_{m}\wedge\ldots\wedge A_{n}\) appears for the expressions when \(Y_{T}\) is 1 or \(u\). In general \(Y_{T}\) can be written as:  
\[Y_{T}=(A_{1}\wedge A_{m}\wedge\ldots\wedge A_{n})\vee(A_{1}\wedge A_{m}\wedge \ldots\wedge A_{n}\wedge F_{E}) \tag{35}\]  
\[Y_{T}=A_{1}\wedge A_{m}\wedge\ldots\wedge A_{n}\]  
where \(F_{E}\) corresponds to the disjunction of all the conjunctions obtained in the truth table for which \(Y_{T}\) is \(u\) eliminating from these the conjunction \(A_{1}\wedge A_{m}\wedge\ldots\wedge A_{n}\).  
It is important to note the term \(A_{1}\wedge A_{m}\wedge\ldots\wedge A_{n}\) is presented in the terms calculated for \(Y_{T}\) when is \(u\), hence when the absorption process is carried out, it is obtained the expression for \(Y_{T}\).  
Example for a Table of Three Variables  
In order to show the case when having an activation function with total dependency, it is used the coding in Table 13.  
The respective expression using Boolean algebra is:  
\[f_{T}=A_{1}\wedge A_{2}\wedge A_{3} \tag{36}\]  
Table 14 displays the configuration considering the third element \(u\) to build the Kleene table.  
Alternatively, the relationship between variables is described by the configuration present in Figure 6.  
\begin{table}
\begin{tabular}{c c c c} \hline \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(f_{T}\) \\ \hline
0 & \(X\) & \(X\) & 0 \\ \(X\) & 0 & \(X\) & 0 \\ \(X\) & \(X\) & 0 & 0 \\
1 & 1 & 1 & 1 \\ \hline \end{tabular}
\end{table}
Table 13: Boolean relations for three variables.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(A_{1}\) & \(A_{2}\) & \(\cdots\) & \(A_{n-1}\) & \(A_{n}\) & \(Y_{T}\) \\ \hline
1 & 1 & \(\cdots\) & 1 & 1 & 1 \\ \(u\) & \(X_{u,1}\) & \(\cdots\) & \(X_{u,1}\  
Applying Kleene algebra for \(f_{T}\) is obtained:  
\[\begin{array}{rcl}f_{T}&=&(A_{1}\wedge A_{2}\wedge A_{3})\vee(A_{1}\wedge A_{2} \wedge A_{3}\wedge\bar{A}_{3})\vee(A_{1}\wedge A_{2}\wedge\bar{A}_{2}\wedge A_ {3})\vee(A_{1}\wedge A_{2}\wedge\bar{A}_{2}\wedge A_{3}\wedge\bar{A}_{3})\vee \\ &&(A_{1}\wedge\bar{A}_{1}\wedge A_{2}\wedge A_{3})\vee(A_{1}\wedge\bar{A}_{1} \wedge A_{2}\wedge A_{3}\wedge\bar{A}_{3})\vee(A_{1}\wedge\bar{A}_{1}\wedge A _{2}\wedge\bar{A}_{2}\wedge A_{3})\vee\\ &&(A_{1}\wedge\bar{A}_{1}\wedge A_{2}\wedge\bar{A}_{2}\wedge A_{3}\wedge\bar{A }_{3})\\ f_{T}&=&(A_{1}\wedge A_{2}\wedge A_{3})\vee[(A_{1}\wedge A_{2}\wedge A_{3}) \wedge[\bar{A}_{3}\vee\bar{A}_{2}\vee(\bar{A}_{2}\wedge\bar{A}_{3})\vee\\ &&\bar{A}_{1}\vee(\bar{A}_{1}\wedge\bar{A}_{3})\vee(\bar{A}_{1}\wedge\bar{A}_ {2})\vee(\bar{A}_{1}\wedge\bar{A}_{2}\wedge\bar{A}_{3})]]\end{array} \tag{37}\]  
In this case, the equation is written as:  
\[f_{T}=(A_{1}\wedge A_{2}\wedge A_{3})\vee(A_{1}\wedge A_{2}\wedge A_{3}\wedge F _{E}) \tag{38}\]  
therefore:  
\[f_{T}=A_{1}\wedge A_{2}\wedge A_{3} \tag{39}\]