## 5 Synthesis Example
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(S_{C}\) & \(PC(S_{c})\) & \(S_{T}\) & \(FC(S_{T})\) & \(I2_{h}\) & \(AV_{h}^{1}\) & \(D_{h}^{1}\) & \(h\) \\ \hline \(S_{6}\) & 01 & \(S_{4}\) & 0010 & \(\frac{i_{3}}{i_{3}}\) & \(a_{4}\) & \(D_{3}\) & 1 \\  & & \(S_{5}\) & 0101 & \(\frac{i_{3}}{i_{3}}\) & \(a_{2}\) & \(D_{2}D_{4}\) & 2 \\ \hline \(S_{8}\) & 10 & \(S_{7}\) & 0110 & \(\frac{i_{3}}{i_{3}}\) & \(a_{2}a_{3}\) & – & 3 \\  & & \(S_{4}\) & 0010 & \(\frac{i_{3}}{i_{3}}\) & \(-\) & \(D_{2}\) & 4 \\  & & \(S_{9}\) & 1010 & \(\frac{i_{3}}{i_{7}}\) & \(a_{3}\) & \(D_{2}D_{3}\) & 5 \\ \hline \(S_{9}\) & 11 & \(S_{4}\) & 0010 & \(\frac{i_{5}}{i_{7}}\) & \(-\) & \(-\) & 7 \\  & & \(S_{8}\) & 1001 & \(\frac{i_{5}}{i_{7}}\) & \(a_{3}\) & \(D_{1}D_{4}\) & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Table of _CorePC(S^2)_. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(S_{C}\) & \(PC(S_{c})\) & \(S_{T}\) & \(FC(S_{T})\) & \(I2_{h}\) & \(AV_{h}^{1}\) & \(D_{h}^{1}\) & \(h\) \\ \hline \(S_{2}\) & 01 & \(S_{6}\) & 1000 & \(\frac{i_{3}}{i_{3}}\) & \(a_{4}\) & \(D_{1}\) & 2 \\  & & \(S_{3}\) & 0001 & \(\frac{i_{3}}{i_{3}}\) & \(a_{2}\) & \(D_{3}\) & 3 \\ \hline \(S_{5}\) & 10 & \(S_{1}\) & 0000 & \(\frac{i_{5}}{i_{5}}\) & \(a_{2}a_{4}\) & – & 4 \\  & & \(S_{2}\) & 0100 & \(\frac{i_{5}}{i_{6}}\) & \(a_{3}\) & \(D_{2}\) & 5 \\  & & \(S_{7}\) & 0110 & \(\frac{i_{5}}{i_{6}}\) & \(a_{2}a_{3}\) & \(D_{2}D_{3}\) & 6 \\ \hline \(S_{7}\) & 11 & \(S_{4}\) & 0010 & \(\frac{i_{2}}{i_{2}}\) & \(a_{3}\) & \(D_{3}\) & 7 \\  & & \(S_{8}\) & 1001 & \(\frac{i_{2}}{i_{2}}\) & \(a_{2}a_{4}\) & \(D_{1}D_{4}\) & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Table of _CorePC(S^1)_.  
Figure 9: Partial state codes for Mealy FSM \(A1\). The transparent approach is used to construct SBFs (22) and (23). For example, the functions \(D_{1}^{0}\), \(D_{1}^{1}\), and \(D_{1}^{2}\) are represented as:  
\[ \begin{split} D_{1}^{0}=\overline{T_{1}}\ \overline{T_{2}}T_{4}i_{1};\\ D_{1}^{1}=\overline{v_{1}}v_{2}\overline{i_{2}}i_{5}\lor v_{1}v_{2} \overline{i_{2}};\\ D_{1}^{2}=v_{3}\overline{v_{4}}\ \overline{i_{3}}\ \overline{i_{7}}\lor v_{3}v_{4} \overline{i_{5}}\ \overline{i_{7}}.\end{split}  \tag{31}\]  
In the same way, we can obtain the following SOPs:  
\[ \begin{split} a_{3}^{0}=\overline{T_{1}}\ \overline{T_{2}}\ \overline{T_{3}}\ \overline{T_{4}}i_{1}\lor\overline{T_{1}}\ \overline{T_{2}}\ \overline{T_{3}}\ \overline{T_{4}}\ \overline{i_{1}};\\ a_{3}^{1}=v_{1}\overline{v_{2}}\ \overline{i_{5}}\lor\underline{v_{1}}v_{2} \overline{i_{2}};\\ a_{3}^{2}=v_{3}\overline{v_{4}}i_{3}\lor v_{3}\overline{i_{4}} \ \overline{i_{3}}\ \overline{i_{7}}\lor v_{3}v_{4}\ \overline{i_{5}}\ \overline{i_{7}}.\end{split}  \tag{32}\]  
Step 10. The block _LUTerFA_ is based on Table 5. Table 5 includes the following columns: Function (this is an assembled function produced by _LUTerFA_), _CoreFC_, and _CorePC_. If some function belonging to the set \(D\cup AV\) is generated by a LUT of the block _CoreFC_, then there is a 1 in the intersection of the row containing this function and the column _CoreFC_. The opposite situation is marked by 0. The column _CorePC_ is divided by \(J\) subcolumns corresponding to the classes \(S^{j}(j\in\{1,\ldots,J\})\). The same principle is in play for placing either 1 or 0 in the rows of this part of Table 5.  
We use Table 2 to fill the rows of column _CoreFC_ of Table 5. To fill the rows of subcolumn \(S^{1}\) (\(S^{2}\)), we use Table 3 (Table 4). Table 5 determines the \(R1+R2\) disjunctions of partial Boolean functions. The following disjunctions represent the circuit of the block _LUTerFA_:  
\[ \begin{split} D_{1}=D_{1}^{0}\lor D_{1}^{1}\lor D_{1}^{2};\quad D _{2}=D_{2}^{0}\lor D_{2}^{1}\lor D_{2}^{2};\\ D_{3}=D_{3}^{0}\lor D_{3}^{1}\lor D_{3}^{2};\quad D_{4}=D_{4}^{0} \lor D_{4}^{1}\lor D_{4}^{2};\\ a_{1}=a_{1}^{0};\quad a_{2}=a_{1}^{1}\lor a_{2}^{2};\\ a_{3}=a_{3}^{0}\lor a_{3}^{1}\lor a_{3}^{2};\quad a_{4}=a_{4}^{0} \lor a_{4}^{1}\lor a_{4}^{2}.\end{split}  \tag{33}\]  
Step 11.