## 5 Synthesis Example
The column \(AV^{0}_{h}\) includes the additional variables \(a_{r}\in AV\) equal to 1 in codes of generated OCs. These variables are also produced by some blocks of _LUTterPC_. The column \(AV1_{h}\) includes the additional variables \(a_{r}\in AV\) generated only by the block _CoreFC_. Obviously, these variables are not produced by any block of _LUTterPC_. Table 2 represents the block _CoreFC_ for the given example. The columns \(AV^{0}_{h}\) and \(AV1_{h}\) are created in the following manner. For example, there is an OC \(O_{4}\) written in line 1 of Table 1. Analysis of Figure 8 gives the code \(K(O_{4})=1011\). This code determines the variables \(a_{1}\), \(a_{3}\), and \(a_{4}\). Therefore, the first line of Table 2 includes the variables \(a_{3}\) and \(a_{4}\) in the column \(AV^{0}_{h}\), as well as the variable \(a_{1}\) in the column \(AV1_{h}\). All other lines of Table 2 are created using a similar approach. Using Table 2, we can obtain SBFs (19)-(21). For example, the function \(a_{1}\in AV\) is represented as the following:  
\[a_{1}=\overline{T_{1}}\,\overline{T_{2}}\,\overline{T_{3}}\vee\overline{T_{1} }\,\overline{T_{2}}T_{3}i_{1}. \tag{30}\]  
The block _CoreFC_ determines the set \(D^{0}=D\). We will show a bit later the SOPs for functions \(D^{0}_{1}\) and \(a^{0}_{3}\). Step 8. The codes for states \(s_{m}\in S^{1}\) use the variables \(v_{1},v_{2}\in SV\). The codes for states \(s_{m}\in S^{2}\) are based on the variables \(v_{3}\), \(v_{4}\in SV\). The code combination \(v_{1}=v_{2}=0\) indicates that a particular state belongs to a class other than \(S^{1}\). The code combination \(v_{3}=v_{4}=0\) indicates that a particular state belongs to a class other than \(S^{2}\). Due to the fulfillment of condition (10), the codes do not affect the number of LUTs in the circuit of _CorePC_. Therefore, the partial state codes can be arbitrary. We have chosen the following approach: the smaller the subscript (m) of a state, the more nulls its partial code contains. The obtained partial state codes are shown in Figure 9. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(S_{C}\) & \(FC(S_{c})\) & \(S_{T}\) & \(FC(S_{T})\) & \(\Pi_{h}\) & \(AV^{0}_{h}\) & \(AV1_{h}\) & \(D^{0}_{h}\) & \(h\) \\ \hline \multirow{2}{*}{\(S_{1}\)} & \multirow{2}{*}{0000} & \(S_{2}\) & 0100 & \(i_{1}\) & \(a_{3}a_{4}\) & \(a_{1}\) & \(D_{2}\) & 1 \\  & & \(S_{3}\) & 0001 & \(\overline{i_{1}}\) & \(-\) & \(a_{1}\) & \(D_{4}\) & 2 \\ \hline \multirow{2}{*}{\(S_{3}\)} & \multirow{2}{*}{0001} & \(S_{6}\) & 1000 & \(i_{1}\) & \(a_{4}\) & \(a_{1}\) & \(D_{1}\) & 3 \\  & & \(S_{1}\) & 0000 & \(\overline{i_{1}}\) & \(a_{3}\) & \(a_{1}\) & \(-\) & 4 \\ \hline \multirow{2}{*}{\(S_{4}\)} & \multirow{2}{*}{0010} & \(S_{1}\) & 0000 & \(i_{1}\) & \(-\) & \(a_{1}\) & \(-\) & 5 \\  & & \(S_{4}\) & 0010 & \(\overline{i_{1}}\) & \(-\) & \(-\) & \(D_{3}\) & 6 \\ \hline \end{tabular}
\end{table} 
Table 2: Table of _CoreFC_ for Mealy FSM \(A1\). Using Figure 9, we can obtain the following partial codes: \(PC(s_{2})=PC(s_{6})=01\), \(PC(s_{5})=PC(s_{8})=10\), and \(PC(s_{7})=PC(s_{9})=11\). Using them allows for creating tables representing _CorePC_. Step 9. The block _CorePC_ includes two blocks of LUTs. The block \(CorePC(S^{1})\) corresponds to the set \(S^{1}\), whereas the block \(CorePC(S^{2})\) corresponds to the set \(S^{2}\). The table of \(CorePC(S^{1})\) (Table 3) is based on lines 3-5, 10-12, and 15-16 (Table 1). Table 4 represents the block \(CorePC(S^{2})\). The table is constructed using the lines 13-14 and 17-22 of the initial STT. In these tables, the current states are represented by their partial codes \(PC(s_{C})\); the states of transition are represented by their full codes \(FC(s_{T})\). The column \(O_{CT}\) of STT is replaced by the columns \(AV_{h}^{1}\) and \(AV_{h}^{2}\), respectively. These columns include additional variables equal to \(1\) in the codes of the OCs.