# Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks
## 3 Finite-Time Set Reachability
### Finite-Time Set Reachability with Probability One  
The set reachability of PBMCNs is investigated and characterized in this section. First, we introduce a lemma for the set reachability of PBMCNs. **Lemma 4**. : _Assume that \(\mathcal{M}_{0}\subset\Delta_{2^{kN}}\) is an initial subset, \(\mathcal{M}_{d}\subset\Delta_{2^{kN}}\) is a overall target subset and \(\widetilde{\mathcal{M}}_{d}\subset\Delta_{2^{n}}\) is a global target subset._  
1. _An input sequence_ \(\{\mathbf{u}(t),t=0,1,\ldots,\gamma-1\}\) _exists such that_ \(\mathcal{M}_{d}\) _is overall reachable with probability one from_ \(\mathcal{M}_{0}\) _on_ \([0:\gamma-1]\) _for PBMCNs (4) if and only if_ \(\delta^{0}_{2^{kN}}\) _is overall reachable with probability one from_ \(\mathcal{M}_{0}\) _on_ \([1:\gamma]\) _for RLDS (_7_)._
2. _An input sequence_ \(\{\mathbf{u}(t),t=0,1,\ldots,\gamma-1\}\) _exists such that_ \(\widetilde{\mathcal{M}}_{d}\) _is global reachable with probability one from_ \(\mathcal{M}_{0}\) _on_ \([0:\gamma-1]\) _for PBMCNs (4) if and only if_ \(\delta^{0}_{2^{n}}\) _is global reachable with probability one from_ \(\mathcal{M}_{0}\) _on_ \([1:\gamma]\) _for RLDS (_7_)._  
Proof. : Assume that \(x_{0}\in\mathcal{M}_{0}\) is the initial state, \(\hat{x}(t,x_{0})\) is the overall state to RLDS (7), \(\hat{x}(t,x_{0})\) is the global state to RLDS (7) and there exists an input sequence \(\{\mathbf{u}(t),t=0,1,\ldots\gamma-1\}\). It holds that  
\[\Pr\{\exists\xi\in[0:t-1],\text{s.t.}\ x(\xi;\eta,x_{0})\in \mathcal{M}_{d}\}\] \[= \Pr\{\{x_{0}\in\mathcal{M}_{d}\}\cup\{x(1;\eta,x_{0})\in \mathcal{M}_{d}\}\cup\ldots\cup\{x(\xi;\eta,x_{0})\in\mathcal{M}_{d}\}\}\] \[= \Pr\{\cup_{i=1}^{t}\{\hat{x}(i;\eta,x_{0})=\delta^{0}_{2^{kN}}\}\}\] \[= \Pr\{\exists\xi\in[1:t],\text{s.t.}\ \hat{x}(\xi;\eta,x_{0})=\delta^{0}_{2^{kN}}\}\]  
Figure 2: STG of the RLDS (7) in Example 1. Then, we are able to easily verify that, for any \(x_{0}\in\mathcal{M}_{0}\), there exists an input sequence \(\{\mathbf{u}(t),t=0,1,\ldots,\gamma-1\}\), such that (5) and (6) are equivalent to  
\[\Pr\{\exists\xi\in[1:\gamma],s.t.\ \hat{x}(\xi;\eta,x_{0})=\delta^{0}_{2^{2K}}\}=1 \tag{12}\]  
and  
\[\Pr\{\exists\xi\in[1:\gamma],s.t.\ \hat{x}(\xi;\eta,x_{0})=\delta^{0}_{2^{2}}\}=1 \tag{13}\]  
respectively. Thus, the claims hold. **Lemma 5**. :
* _The_ \(\xi\) _step overall transition probability from any state to_ \(\delta^{0}_{2^{2K}}\) _is non-decreasing as_ \(\xi\) _increases, i.e.,_ \[\Pr\{\hat{x}(\xi+1)=\delta^{0}_{2^{KN}}\ |\ \hat{x}(0)=\delta^{i}_{2^{KN}}\}\geq \Pr\{\hat{x}(\xi)=\delta^{0}_{2^{KN}}\ |\ \hat{x}(0)=\delta^{i}_{2^{KN}}\}\] (14)
* _The_ \(\xi\) _step global transition probability from any state to_ \(\delta^{0}_{2^{2}}\) _is non-decreasing as_ \(\xi\) _increases, i.e.,_ \[\Pr\{\hat{x}(\xi+1)=\delta^{0}_{2^{2}}\ |\ \hat{x}(0)=\delta^{i}_{2^{KN}}\}\geq \Pr\{\hat{x}(\xi)=\delta^{0}_{2^{2}}\ |\ \hat{x}(0)=\delta^{i}_{2^{KN}}\}\] (15)  
Moreover, due to  
\[ \begin{array}{ll}\hat{x}(\gamma)&=P^{\hat{x}}u(\gamma-1)\hat{x}(\gamma-1)\\ &=P^{\hat{x}}W_{[2^{KN},2^{2m}]}\hat{x}(\gamma-1)u(\gamma-1)\\ &=(P^{\hat{x}}W_{[2^{KN},2^{m}]})^{2}\hat{x}(\gamma-2)u(\gamma-2)u(\gamma-1) \\ &=\ldots\\ &=(P^{\hat{x}}W_{[2^{KN},2^{m}]})^{\gamma}\hat{x}(0)u(0)u(1)\ldots u(\gamma-2 )u(\gamma-1)\\ &=(P^{\hat{x}}W_{[2^{KN},2^{m}]})^{\gamma}W_{[2^{m},2^{KN}]}u(0)u(1)\ldots u( \gamma-2)u(\gamma-1)\hat{x}(0)\end{array} \]  
and  
\[ \begin{array}{ll}\tilde{x}(\gamma)&=P^{\tilde{x}}u(\gamma-1)\hat{x}(\gamma-1 )\\ &=P^{\hat{x}}W_{[2^{KN},2^{m}]}\hat{x}(\gamma-1)u(\gamma-1)\\ &=(P^{\hat{x}}W_{[2^{KN},2^{m}]})(P^{\hat{x}}W_{[2^{KN},2^{m}]})\hat{x}( \gamma-2)u(\gamma-2)u(\gamma-1)\\ &=\ldots\\ &=(P^{\hat{x}}W_{[2^{KN},2^{m}]})(P^{\hat{x}}W_{[2^{KN},2^{m}]})^{\gamma-1} \hat{x}(0)u(0)u(1)\ldots u(\gamma-2)u(\gamma-1)\\ &=(P^{\hat{x}}W_{[2^{KN},2^{m}]})(P^{\hat{x}}W_{[2^{KN},2^{m}]})^{\gamma-1}W_ {[2^{m},2^{KN}]}u(0)u(1)\ldots u(\gamma-2)u(\gamma-1)\hat{x}(0)\end{array} \]  
we define \(\gamma\)-step nonzero state transition probability matrix, \(\hat{Q}\) and \(\hat{Q}\), by  
\[