# Compact Fuzzy Systems Based on Boolean Relations
## 2 Framework
### Fuzzy Inference Systems Based on Boolean Relations  
The fuzzy inference systems based on Boolean relations (FIS-BBR) use coding and design tools of Boolean algebra. The structure of the inference process is given by the truth table associated with the system design. In order to implement the FIS-BBR the two-element Boolean encoding extends to a three-element Kleene encoding.  
In the operation of a FIS-BBR, actions that contribute to carry out a total action on a system are considered. The actions are fully accomplished by weighting an actuator \(v_{m}\) (virtual actuator) using a function \(Y_{m}\) that operates on it (activation function), which is a fuzzy function with values between zero and one. In this way, the output in a FIS-BBR is calculated as the sum of the partial results \(y_{m}=v_{m}Y_{m}\) (virtual outputs), as shown in Equation (20).  
Table 3 shows the coding for the inputs and outputs of a FIS-BBR. The rows of left side represent the rules that provide an activation output. The columns on the right side correspond to the activation functions. In this way two parts are identified in Table 3, one corresponding to an encoding for the sets of each entry; the other corresponds to the activation outputs associated with virtual outputs.  
In Table 3, variables \(a_{k,j}\) and \(f_{k,m}\) represent relationships between \(A_{j}\) and \(Y_{m}\). Value of variables in the Boolean case are \(\{0,1\}\); in Kleene's case the values are \(\{0,u,1\}\). Table 3 can be a Boolean or Kleene truth table [56]. It is noteworthy that \(A_{j}\) can be a Boolean or fuzzy set associated with the input, and \(Y_{m}\) is an activation function associated with the output.  
The combination of sets \(A_{j}\) when using a disjunctive normal form can be expressed as:  
\[\bigwedge_{j=1}^{p}\widehat{A}(a_{k,j},f_{k,m}) \tag{18}\]  
where \(\widehat{A}\) is a function that can be 0, 1, \(A_{j}\), \(\bar{A}_{j}\) or \(A_{j}\wedge\bar{A}_{j}\), it depends on \(a_{k,j}\) and \(f_{k,m}\) respectively [22; 45; 56]. Table 4 shows the values of \(\widehat{A}\) for the case of Kleene or Boole.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Sets Associated to the Input**} & \multicolumn{4}{c}{**Output Activation Functions**} \\ \hline \(A_{1}\) & \(A_{2}\) & \(\cdots\) & \(A_{j}\) & \(\cdots\) & \(A_{p}\) & \(Y_{1}\) & \(Y_{2}\) & \(\cdots\) & \(Y_{m}\) & \(\cdots\) & \(Y_{  
The \(m\)-th activation output \(Y_{m}\), using the DNF can be represented as:  
\[Y_{m}=\bigvee_{k=1}^{q}\bigwedge_{j=1}^{p}\widehat{A}(a_{k,j},f_{k,m}) \tag{19}\]  
Considering the activation outputs of Table 3, the total system output is calculated as:  
\[y=\sum_{m=1}^{w}Y_{m}v_{m} \tag{20}\]  
where \(v_{m}\) corresponds to \(m\)-th virtual actuator; then, it is obtained the \(m\)-th virtual output:  
\[y_{m}=Y_{m}v_{m} \tag{21}\]  
In Equation (21) \(Y_{m}\in[0,1]\) is the activation function and \(v_{m}\in\mathbb{R}\) is the value of the virtual actuator. In control systems, the actuator is used to apply the control action [22]; in some applications (with Boolean logic) it is possible to use several actuators for the control action (several batteries, valves, switches, etc.). However, when there is only one action element, this can be considered as the sum of several actuators called \"virtual\".  
In control applications, the value of \(v_{m}\) is associated with the value (range) of the actuator utilized, for example, in the application of the hydraulic system described in [50], \(v_{m}\) corresponds to the valve's maximum flow for filling the tank. In general, for neuro-fuzzy applications the value of \(v_{m}\) can be an adaptation parameter when the optimization process of the fuzzy system is carried out.  
It is noticeable that when using Kleene algebra in some cases it is possible to have a simplified activation \(Y_{m}\) with a smaller number of literals and connectors. This is the principle applied to have compact fuzzy systems based on Boolean relations.