# Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks
## 2 Preliminaries and Problem Setting  
### Preliminaries  
In this section, some concepts and properties of STP of matrices are introduced, which are used in a later section.  
Definition 1 ([9]): For \(X\in\mathcal{L}_{n\times m}\), \(Y\in\mathcal{L}_{p\times q}\), \(l\) is the least common multiple of \(m\) and \(p\). Then, the semi-tensor product of \(X\) and \(Y\) is  
\[X\ltimes Y=(X\otimes I_{l/m})(Y\otimes I_{l/p}).\]  
The STP of matrices may be considered an expansion of the standard matrix product since \(X\ltimes Y=XY\) when \(n=p\).  
Definition 2 ([9]): A matrix \(W_{[m,n]}\in\mathcal{L}_{mn\times mn}\) is called a swap matrix if its columns by \((11,12,\ldots,\)\(1n,\ldots,m1,m2,\ldots,mn)\), its rows by \((11,21,\ldots,m1,\ldots,1n,2n,\ldots,mn)\), and its element in the position \(((I,J),(i,j))\) are assigned to  
\[\omega_{(I,J),(i,j)}=\delta_{i,j}^{I,J}=\left\{\begin{array}{ll}1,&I=i\;\text {and}\;J=j,\\ 0,&\quad\text{otherwise.}\end{array}\right.\]  
In particular, we denote \(W_{[n]}=W_{[n,n]}\) when \(m=n\).  
Lemma 1 ([9]): For \(A\in\mathbb{R}^{m}\), \(B\in\mathbb{R}^{n}\), and \(M\) is an arbitrary matrix. Then  
\[\begin{array}{l}W_{[m,n]}\ltimes A\ltimes B=B\ltimes A,\\ W_{[n,m]}\ltimes B\ltimes A=A\ltimes B,\\ AM=(I_{m}\otimes M)A.\end{array}\]Let \(\mathcal{D}_{k}=\{0,1,\ldots,k-1\}\), where \(i(0\leq i\leq k-1)\) represents the \(i+1\)-th elements in \(\mathcal{D}_{k}\). In particular, \(\mathcal{D}_{2}=\{1,0\}\) is used to represent a set of two elements. For the sake of clarity, \(\mathcal{D}_{2}\) is symbolized by \(D\).  
**Definition 3** ([9]).: _A mapping \(f:\mathcal{D}_{k_{1}}\times\mathcal{D}_{k_{2}}\times\ldots\times\mathcal{D}_{ k_{n}}\to\mathcal{D}_{k_{0}}\) is called a multi-valued logical function. If \(k_{0}=k_{1}=\ldots=k_{n}:=k\), then \(f\) is a \(k\)-valued logical function. If \(k=2,f\) is called a logical function._  
Let the element of \(\mathcal{D}_{2}\) correspond with the element of \(\Delta_{2}\) one by one. That is, \(1\sim\delta_{2}^{1}\), \(0\sim\delta_{2}^{2}\), and \(\Delta:=\Delta_{2}=\{\delta_{2}^{1},\delta_{2}^{2}\}\sim\mathcal{D}\). Then, the logical function with \(n\) arguments \(f:\mathcal{D}\times\mathcal{D}\times\ldots\times\times\mathcal{D}\to\mathcal{D}\) can be converted into an algebraic form using the STP method.  
**Lemma 2** ([9]).: _Any logical function \(f(X_{1},X_{2},\ldots,X_{n})\) with logical arguments \(X_{1},X_{2},\ldots,X_{n}\in\mathcal{D}\) can be expressed in a multi-linear form as_  
\[f(X_{1},X_{2},\ldots,X_{n})=M_{f}\ltimes_{i=1}^{n}x_{i},\]  
_where \(x_{i}\in\Delta\), \(M_{f}\in\mathbb{R}^{2\times 2^{n}}\) is the structure matrix uniquely determined by logical function \(f\), and \(Col_{i}(M_{f})=f(\delta_{2^{n}}^{i}),i=1,2,\ldots,2^{n}\)._  
**Lemma 3** ([9]).: _Assume \(X_{n}=x_{1}x_{2}\ldots x_{n}=\ltimes_{i=1}^{n}x_{i}\), then_  
\[X_{n}^{2}=\Phi_{n}X_{n},\]  
_where_  
\[\Phi_{n}=\prod_{i=1}^{n}I_{2^{i-1}}\otimes[(I_{2}\otimes W_{[2,2^{n-i}]})M_{r}],\]  
_and \(M_{r}=\delta_{4}[1,4]\) is a power-reducing matrix that can be proven by \(x^{2}=M_{r}x,\forall x\in\Delta\)._  
**Proposition 1** ([9]).: _For each \(i\in\{1,2,\ldots,n\}\), if \(\Gamma_{\mathrm{in}(i)}=\{j_{1},j_{2},\ldots,j_{r}\}(r\leqslant n)\), we can find an appropriate dimension matrix \(R_{i}\) such that_  
\[R_{i}\ltimes_{i=1}^{n}x_{i}=x_{j_{1}}x_{j_{2}}\ldots x_{j_{r}}.\]