# Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks
## 3 Finite-Time Set Reachability
### State Transfer Graph  
STG reconstruction is the primary approach used in finite-time set reachability analysis. For any subset \(\mathcal{M}\subset\Delta_{2^{KN}}\), the indicator matrix of \(\mathcal{M}\) is expressed as \(\mathbf{D}_{\mathcal{M}}\),  
\[Col_{j}(\mathbf{D}_{\mathcal{M}}):=\left\{ \begin{array}{ll}\delta^{j}_{2^{KN}},& \text{if }\delta^{j}_{2^{KN}}\in\mathcal{M},\\ \delta^{0}_{2^{KN}},&\text{otherwise},\end{array} \right.\]  
where \(\mathbf{D}_{\mathcal{M}}\in\mathscr{B}_{2^{KN}\times 2^{KN}}\). Different from the STG in [42], we define \(\hat{\mathbf{D}}_{\mathcal{M}}\) as follows:  
\[\hat{\mathbf{D}}_{\mathcal{M}}=\left[ \begin{array}{cccc}\mathbf{D}_{\mathcal{M}}&0& \cdots&0\\ 0&\mathbf{D}_{\mathcal{M}}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\mathbf{D}_{\mathcal{M}}^{c}\end{array} \right]_{2^{(m+KN)}\times 2^{(m+KN)}}\]  
Then, \(\hat{\mathbf{D}}_{\mathcal{M}^{c}_{d}}\) and \(\hat{\mathbf{D}}_{\widetilde{\mathcal{M}^{c}_{d}}}\) can be described as follows:  
\[\hat{\mathbf{D}}_{\mathcal{M}^{c}_{2}}:=\left[ \begin{array}{cccc}\mathbf{D}_{ \mathcal{M}^{c}_{d}}&0&\cdots&0\\ 0&\mathbf{D}_{\mathcal{M}^{c}_{d}}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\mathbf{D}_{\mathcal{M}^{c}_{d}}\end{array} \right]_{2^{(m+KN)}\times 2 ^{(m+KN)}}\]  
\[\hat{\mathbf{D}}_{\widetilde{\mathcal{M}^{c}_{d}}}:=\left[ \begin{array}{cccc}\bm {D}_{\widetilde{\mathcal{M}^{c}_{d}}}&0&\cdots&0\\ 0&\mathbf{D}_{\widetilde{\mathcal{M}^{c}_{d}}}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\mathbf{D}_{\widetilde{\mathcal{M}^{c}_{d}}}\end{array} \right]_{2^{(m+KN )}\times 2^{(m+KN)}}\]  
It should be noted that the dimensions of \(\mathcal{M}^{c}_{d}\) and \(\widetilde{M}^{c}_{d}\) are inconsistent. \(\mathcal{M}^{c}_{d}\) is the supplementary set of the overall target subset \(\mathcal{M}_{d}\) in \(\Delta_{2^{KN}}\), and \(\widetilde{\mathcal{M}^{c}_{d}}\) is the supplementary set of the global target subset \(\widetilde{\mathcal{M}}_{d}\) in \(\Delta_{2^{n}}\). Here, the state spaces \(\Delta_{2^{KN}}\) and \(\Delta_{2^{n}}\) could be extended to \(\Delta^{0}_{2^{KN}}\) and \(\Delta^{0}_{2^{n}}\), where \(\Delta^{0}_{2^{KN}}:=\Delta_{2^{KN}}\cup\{\delta^{0}_{2^{KN}}\}\), \(\Delta^{0}_{2^{n}}:=\Delta_{2^{n}}\cup\{\delta^{0}_{2^{n}}\}\), and construct the RLDS as  
\[\left\{ \begin{array}{ll}\hat{x}(t+1)&=L\eta(t)\hat{\mathbf{D}}_{\mathcal{M}^{c}_ {d}}u(t)\hat{x}(t),\\ \check{x}(t+1)&=\widetilde{L}\eta(t)\hat{\mathbf{D}}_{\widetilde{\mathcal{M}^{c}_ {d}}}u(t)\hat{x}(t),\end{array} \right. \tag{7}\]  
where \(\hat{x}(t)\in\Delta^{0}_{2^{KN}}\), \(\check{x}(t)\in\Delta^{0}_{2^{n}}\), and for any \(u(t)\in\Delta_{2^{n}}\), it follows that \(L\eta(t)\hat{\mathbf{D}}_{\mathcal{M}^{c}_{d}}u(t)\hat{x}(t)\
eq\delta^{0}_{2^{KN}}\) iff \(\hat{x}(t)\in\mathcal{M}^{c}_{d}\) and \(\widetilde{L}\eta(t)\hat{\mathbf{D}}_{\widetilde{\mathcal{M}^{c}_{d}}}u(t)\hat{x} (t)\
eq\delta^{0}_{2^{n}}\) iff \(x(t)\in\widetilde{\mathcal{M}}^{c}_{d}\). **Remark 2**. : _Different from [18], the RLDS proposed in this paper is an extension of PMBCN rather than one-layer probabilistic Boolean network. Moreover, the definition of indicator matrix is also different from [18]._  
The PBMCNs (4) can be transformed into RLDS (7) by the indicator matrices \(\hat{\mathbf{D}}_{\mathcal{M}^{c}_{d}}\) and \(\hat{\mathbf{D}}_{\widehat{\mathcal{M}^{c}_{d}}}\). Next, the transformation relationship between the PBMCNs (4) and RLDS (7) is given by using the STG reconstruction technique. Since \(\mathrm{p}^{\sigma}\) is the probability distribution vector of \(\eta(t)\), using (7), we define the one-step nonzero state transition probability matrix of RLDS (7), \(\mathrm{p}^{\hat{x}}\) and \(\mathrm{p}^{\hat{x}}\)  
\[ \begin{array}{l}P^{\hat{x}}=L\ltimes\mathrm{p}^{\sigma}\ltimes\hat{\mathbf{D}}_ {\mathcal{M}^{c}_{d}}=P^{x}\hat{\mathbf{D}}_{\mathcal{M}^{c}_{d}},\\ P^{\hat{x}}=\widetilde{L}\ltimes\mathrm{p}^{\sigma}\ltimes\hat{\mathbf{D}}_{ \widetilde{\mathcal{M}^{c}_{d}}}=P^{x}\hat{\mathbf{D}}_{\widetilde{\mathcal{M}^ {c}_{d}}}.\end{array}