# Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks
## 2 Preliminaries and Problem Setting
### Problem Setting  
In this section, the model of PBMCs is introduced, and the network is transformed to an algebraic representation via the STP method. In the case of the PBMCs with \(K\) layers and \(N\) nodes in each layer, the total number of distinct nodes is \(n\)\((N\leq n\leq NK)\), which is described by  
\[ \begin{cases}X(t+1)=f^{\sigma(t)}(X(t),U(t)),\\ \widetilde{X}(t+1)=\widetilde{f}^{\sigma(t)}(X(t),U(t)),\end{cases}  \tag{1}\]  
where \(X(t)=[X_{1}(t)\ X_{2}(t)\ \ldots\ X_{KN}(t)]^{T}\in\mathcal{D}^{KN}\), \(\widetilde{X}(t)=[\widetilde{X}_{1}(t)\ \widetilde{X}_{2}(t)\ \ldots\ \widetilde{X}_{n}(t)]^{T}\!\!\in \mathcal{D}^{n}\), \(U(t)=[U_{1}(t)\ U_{2}(t)\ \ldots\ U_{m}(t)]^{T}\!\!\in\mathcal{D}^{m}\) are logical vectors, and \(X_{i},\widetilde{X}_{i},U_{k}\in\mathcal{D}\) denote the state of \(i\)-th state node, the state of \(i\)-th global state node and the \(k\)-th control input, respectively. \(\sigma(t)\in[\![\!1,\!O\!]\! ]\) denotes stochastic switching signal, where \(O\) represents the number of constituent Boolean multiplex control networks. Further, for any \(\sigma(t)=r\in[1,O]\), the \(r\)-th subnetwork of PBMCs is defined by \(f^{r}\): \(\mathcal{D}^{KN+m}\to\mathcal{D}^{KN+m}\), \(\widetilde{f}^{r}\): \(\mathcal{D}^{KN+m}\to\mathcal{D}^{n}\) is also the logic function, commonly known as the canalizing function. **Remark 1**. : _The concept of the global state layer was first put forward by [PERSON] et al. [43], expressed as follows:_  
\[\left\{ \begin{aligned} & X_{i}^{l}(t+1)=f_{i}^{l}(X_{j\in\Gamma_{ il(i)}(l)}^{l}(t),U_{1}(t),U_{2}(t),\dots,U_{m}(t)),\\ &\widetilde{X}_{i}(t+1)=\widetilde{Y}_{i}^{(X_{i}^{l_{1}}(t)},X_ {i}^{l_{2}}(t),\dots,X_{i}^{l_{k}}(t),U_{1}(t),U_{2}(t),\dots,U_{m}(t)).\end{aligned} \right. \tag{2}\]  
_Here, \(X_{i}^{l}\) and \(\widetilde{X}_{i}\) denote the state of the \(i\)-th state node in \(l\)-th layer and the global state of the \(i\)-th state node, respectively. \(\Gamma_{in(i)}(l)\) denotes that the incoming neighbor of node \(i\) is in the \(l\)-th layer. \(X_{i}^{l_{i}}\) denotes that the node \(i\) is in the \(l_{i_{i}}\)-th layer. There is a logical relationship between the global state node \(\widetilde{X}_{i}\) and the state node \(X_{i}^{l_{1}}\), \(X_{i}^{l_{2}},\dots,X_{i}^{l_{k}}\). In other words, \(\widetilde{X}_{i}\) is defined by the logical function \(\widetilde{f}_{i}\) and \(X_{i}^{l_{1}}\), \(X_{i}^{l_{2}},\dots,X_{i}^{l_{k}}\). For the representation of graph theory in the following paper, the following Equation (3) is converted into the form of Equation (1) in this paper._  
Let \(x(t)=\ltimes_{l=1}^{K}x^{l}(t)\), \(\widetilde{x}(t)=\ltimes_{i=1}^{\pi}\widetilde{x}_{i}(t)\) and \(u(t)=\ltimes_{k=1}^{m}u_{k}(t)\) denote the vector forms of \(X(t)\), \(\widetilde{X}(t)\) and \(U(t)\), respectively. Here, \(x^{l}(t)=\ltimes_{a_{l}=1}^{l}x_{i}^{l}(t)\) denotes the overall state of the \(l\)-th layer, \(a_{il}=1\) denotes that the node \(i\) is in the \(l\)-th layer, and \(x_{i}^{l}\) denotes that the state of node \(i\) is in the \(l\)-th layer. Then, by using Lemma 2, PBMCs (1) can be expressed as  
\[\left\{ \begin{aligned} & x(t+1)=L_{\sigma(t)}u(t)x(t),\\ &\widetilde{x}(t+1)=\widetilde{L}_{\sigma(t)}u(t)x(t),\end{aligned} \right. \tag{3}\]  
where \(x(t)\in\Delta_{2N\times},\widetilde{x}(t)\in\Delta_{2^{\pi}}\), \(u(t)\in\Delta_{2^{\pi}}\) and \(L_{j}\in\mathcal{L}_{2^{NK}\times 2^{m+NK},j}=\sigma(t)\in[1,O]\) and \(\widetilde{L}_{j}\in\mathcal{L}_{2^{n}\times 2^{m+NK},j}=\sigma(t)\in[1,O]\) denote the overall and global structure matrix of the \(j\)-th sub-network, respectively. The vector form of \(\sigma(t)\) is denoted by \(\eta(t):=\delta_{O}^{\sigma(t)}\in\Delta_{O}\). The PBMCs (3) can therefore be written similarly as  
\[\left\{ \begin{aligned} & x(t+1)=L\eta(t)u(t)x(t),\\ &\widetilde{x}(t+1)=\widetilde{L}\eta(t)u(t)x(t),\end{aligned} \right. \tag{4}\]  
where \(L=[L_{1}\ L_{2}\ \dots\ L_{O}]\), \(\widetilde{L}=[\widetilde{L}_{1}\ \widetilde{L}_{2}\ \dots\ \widetilde{L}_{O}]\).