## 3 Analysis of Related Work
States \(s_{m}\), \(s_{j}\in S\) are compatible if their inclusion in the same class of the partition \(\pi_{S}\) does not lead to the following phenomenon: the required number of LUT inputs exceeds the maximum number of inputs of LUT \(S_{L}\). Why such a phenomenon is possible will be clear from the further text of the article. Three sets characterize any class \(S^{k}\in\pi_{S}\). These sets consist of: 1. inputs determining transitions from states \(s_{m}\in S^{k}\) (a set \(l^{k}\subseteq I\) including \(L_{k}\) elements); 2. outputs produced during the transitions from these states (a set \(O^{k}\subseteq O\)); and 3. IMFs determining MBCs of transition states (a set \(D^{k}\subseteq D\)). If the encoding of OCs is used, then the set \(O^{k}\subseteq O\) is replaced by set \(AV^{k}\subseteq AV\). The set \(AV^{k}\) includes additional variables equal to 1 in the codes of OCs generated during the transitions from states \(s_{m}\in S^{k}\). Each class \(S^{k}\in\pi_{S}\) includes \(M_{k}\) compatible states \(s_{m}\in S\). Inside each class, the states are encoded by partial codes \(PC(s_{m})\). These codes have \(R_{k}\) bits:  
\[R_{k}=\lceil\log_{2}(M_{k}+1)\rceil. \tag{8}\]  
To create the partial codes, a set ASV of additional state variables is created. The states \(s_{m}\in S^{k}\) are encoded using the variables \(v_{r}\in\,ASV^{k}\). The sets \(ASV^{k}\) create the set ASV, which includes R3 elements:  
\[R3=R_{1}+...+R_{K}. \tag{9}\]  
If a state \(s_{m}\in S\) is compatible with states \(s_{p}\in S^{k}\), then including this state into \(S^{k}\) satisfies the condition:  
\[R_{k}+L_{k}\leq S_{L}\quad(k\in\{1,\ldots,K\}). \tag{10}\]  
This approach leads to a \(P_{T}Y\) Mealy FSM. In \(P_{T}Y\) Mealy FSMs, each state \(s_{m}\in S\) has two codes. One of them is a maximum binary full state code \(FC(s_{m})\), and the second is a partial state code \(PC(a_{m})\). The second code determines a particular state as an element of a particular class. Figure 2: Architecture of \(PY\) Mealy FSM. Each class \(S^{k}\in\pi_{S}\) determines the following two systems of PBFs:  
\[D^{k}=D^{k}(ASV^{k},I^{k}); \tag{11}\]  
\[AV^{k}=AV^{k}(ASV^{k},I^{k}). \tag{12}\]  
To obtain the final values of additional variables and IMFs, the following SBFs should be created:  
\[D_{r}=\bigvee_{k=1}^{K}D_{r}^{k}\quad(r\in\{1,\ldots,R1\}); \tag{13}\]  
\[AV=AV(SV,I); \tag{6}\]0  
Next, the codes of the OCs should be transformed into FSM outputs. The outputs are represented by SBF (7). Additionally, the full state codes should be transformed into the corresponding partial codes. The transformation is represented by the following SBF:  
\[AV=AV(SV,I); \tag{6}\]1  
SBFs (11) and (12) define the first level of a \(P_{T}Y\) Mealy FSM circuit. SBFs (13) and (14) determine its second level. Finally, SBFs (7) and (15) represent the third circuit level. The architecture of a \(P_{T}Y\) Mealy FSM is shown in \[AV=AV(SV,I); \tag{6}\]2  
In this architecture, the block _LUTerK_ generates PBFs (11) and (12). The block _LUTerPF_ implements the system of disjunctions (13) and (14). This block includes the distributed RG controlled by the pulses _Clk_ and _Res_. The block _LUTerOF_ implements the outputs represented by SBF (7). The block LUTerASV implements SBF (15). Therefore, it executes the transformation of state codes. Our previous research [20] shows that the LUT-based circuits of \(P_{T}Y\) FSMs have better characteristics than the circuits of equivalent \(PY\) FSMs. If the conditions  
\[AV=AV(SV,I); \tag{6}\]3  
\[AV=AV(SV,I); \tag{6}\]4 Architecture of \(P_{T}Y\) Mealy FSM. \[AV=AV(SV,I); \tag{6}\]5  
hold, then the circuits of \(P_{T}Y\) FSMs are three-level and are faster than the equivalent \(PY\) Mealy FSMs. Let us represent the circuit (Figure 3) as a combination of a core of partial functions (_CorePF_) and a functional transformer. The core includes blocks _LUTer1-LUTerK_. The functional transformer includes all other blocks shown in \[AV=AV(SV,I); \tag{6}\]6 This leads to the generalized diagram of a \(P_{T}Y\) FSM (Figure 4).