## 5 Synthesis Example
There are \(M=9\) elements in the set \(S\). Therefore, using (1) gives \(R1=4\). This value determines the sets \(SV=\{T_{1},...,T_{4}\}\) and \(D=\{D_{1},...,D_{4}\}\). As shown in [17], it is necessary to cover the states from the same class using the minimum possible number of generalized cubes of \(R1\)-dimensional Boolean space. Such an outcome decreases the number of literals in functions (19)-(21). One of the possible outcomes is shown in Figure 6. To encode the states by MBCs, we used the algorithm JEDI [40]. As we can see from the analysis of the resulting Karnaugh map (Figure 7), the states \(s_{m}\in S^{0}\) are covered by the generalized cube 00 xx. The states \(s_{m}\in S^{1}\) are represented by the generalized cube x100. The cube 1x00 covers the states \(s_{m}\in S^{2}\). Therefore, for our example, each class is placed into a single generalized cube. \begin{table}
\begin{tabular}{c c c c c c} \hline \(S_{C}\) & \(S_{T}\) & \(I_{CT}\) & \(O_{CT}  
Step 6. The analysis of Table 1 gives \(Q=10\) output collections. They are the following: \(O_{1}=\overline{O}\), \(O_{2}=\{o_{1},o_{7}\}\), \(O_{3}=\{o_{4}\}\), \(O_{4}=\{o_{3}\}\), \(O_{5}=\{o_{2},o_{6}\}\), \(O_{6}=\{o_{1},o_{4}\}\), \(O_{7}=\{o_{5}\}\), \(O_{8}=\{o_{2}\}\), \(O_{9}=\{o_{1},o_{5},o_{7}\}\), and \(O_{10}=\{o_{4},o_{5},o_{6}\}\). Using (5) gives \(R2=4\) and the set \(AV=\{a_{1},...,a_{4}\}\). Each literal in the sum-of-product (SOP) of a Boolean function corresponds to an interconnection between the input source and a corresponding LUT. To reduce the number of interconnections, the number of literals in SOPs should be decreased. To encode the output collections, we used the methods presented in classical work [17]. Using the approach from [17] gives the codes shown in Figure 8.  
We encoded the OCs in a way where the variable \(a_{1}\in AV\) is generated only by one LUT of _CoreFC_. To do this, we have analyzed Table 1. The analysis of Table 1 shows that the following OCs are generated during the transitions from states \(s_{m}\in S^{0}\): \(O_{4}\), \(O_{5}\), \(O_{8}\), and \(O_{10}\). Therefore, we have divided the Karnaugh map (Figure 8) into two parts. The first part corresponds to \(a_{1}=0\), and the second part corresponds to \(a_{1}=1\). We have placed the OCs \(O_{4}\), \(O_{5}\), \(O_{8}\), and \(O_{10}\) into the second part. Now, we can obtain the following system of functions:  
\[ \begin{gathered} o_{1}=O_{2}\lor O_{6}\lor O_{9}=a_{2};\\ o_{2}=O_{5}\lor O_{8}=a_{1}\overline{a_{3}};\\ o_{3}=O_{4}\lor O_{10}=a_{1}a_{3};\\ o_{4}=O_{3}\lor O_{6}=\overline{a_{1}}a_{4};\\ o_{5}=O_{7}\lor O_{9}\lor O_{10}=a_{3}\overline{a_{4}};\\ o_{6}=O_{5}\lor O_{10}=a_{1}\overline{a_{4}};\\ o_{7}=O_{2}\lor O_{9}=a_{2}\overline{a_{4}}.\end{gathered}  \tag{29}\]  
The SBF (29) determines the circuit of _LUTerOF_. The function \(o_{1}\) is represented by a corresponding output of _LUTerFA_. Therefore, the circuit of _LUTerOF_ consists of 6 LUTs. Figure 8: Codes of output collections for Mealy FSM \(A1\). Figure 7: Maximum binary state codes for FSM \(A1\). Analysis of system (29) shows that there are 12 literals in the SOPs of the implemented functions. This determines 12 interconnections between _LUTterOF_ and other circuit blocks. Using the results of [19] gives the maximum number of interconnections. In our case, it is equal to \(R2*Q=28\). Thus, due to using the proposed approach, the number of interconnects is reduced by 2.33 times. Step 7. To construct the table of _CoreFC_, it is necessary to select the lines of STT with transitions from states \(s_{m}\in S^{0}\). In the discussed case, we should select lines 1-2 and 6-9 of STT (Table 1). The table of _CoreFC_ includes 5 additional columns (compared to the baseline STT). These columns are: \(FC(s_{C})\), \(FC(s_{T})\), \(D^{0}_{h}\), \(AV^{0}_{h}\), and \(AV1_{h}\). There is a self-explanatory meaning of columns \(FC(s_{C})\) and \(FC(s_{T})\). The column \(D^{0}_{h}\) includes IMFs creating the code \(FC(s_{T})\) (to load it into the code register).