## 3 Improved Policy with Enhancement of Computational Efficiency
### Crucial Vector Covering Approach  
Under the process of calculating all PI controllers, the computational cost is very high if all the reachable markings need to be considered. Computational cost and computer time could be reduced once crucial legal markings and FBMs can be identified. Based on the above reason, a Crucial Vector Covering Approach (CVCA) is proposed and defined in this subsection. Firstly, the conventional Vector Covering Approach (VCA) is introduced as follows:  
**Definition 6** [1]. : _Let there be any two markings \(M_{a}\) and \(M_{b}\) in \(R(N,\,M_{0})\). The marking \(M_{a}\) covers \(M_{b}\) if \(\forall p\in P_{A},\,\,M_{a}(p)\geq M_{b}(p)\), which is denoted as \(M_{a}\geq M_{b}\)._  
**Theorem 1** [1]. : _Let \(M_{a}\) and \(M_{b}\) be any two markings in \(R(N,\,M_{0})\) with \(M_{a}\geq M_{b}\). If \(M_{b}\) is forbidden by one PI, \(M_{a}\) will be also forbidden. On the contrary, if \(M_{a}\) is not forbidden by a PI, \(M_{b}\) will not be forbidden either._  
Please note that Definition 6 seems imprecise according to our experimental results. For example (please refer to Figure 2), \(M_{5}\)\((p_{2}+p_{4})\) and \(M_{19}(p_{5}+p_{6}+p_{7})\) are any two markings, with \(M_{5}(p)=2\) and \(M_{19}(p)=3\), respectively. For comparison, the detailed information of all reachable markings in Figure 2 is presented in Table 1.  
Accordingly, \(M_{19}(p)\geq M_{5}(p)\) is checked. Therefore, according to Definition 6, \(M_{5}\) must be covered by \(M_{19}\) since \(M_{19}(p)\geq M_{5}(p)\). In addition, according to Theorem 1, if \(M_{5}\) is forbidden by the PI controller, \(M_{19}\) must be also forbidden. However, in fact, \(M_{19}\) is not forbidden in the situation that \(M_{5}\) is forbidden. Therefore, we revise Definition 6 so that it can be accurate and can fit Theorem 1. The new definition is presented as follows:  
**Definition 7**. : _Let \(M_{a}\) and \(M_{b}\) be any two markings in \(R(N,\,M_{0})\). The marking \(M_{a}\) is said to cover \(M_{b}\) if \(\forall p\in P_{A},\,(M_{a}(p)\geq M_{b}(p))\,\wedge\,(M_{a}\supseteq M_{b})\), which is denoted as \(M_{a}\geq M_{b}\)_ \begin{table}
\begin{tabular}{c c c} \hline
**Marking No.** & **Classification** & **Marking Information** \\ \hline \(M_{1}\) & Initial Marking & \(3p_{1}+3p_{8}+p_{9}+p_{10}+p_{11}\) \\ \(M_{2}\) & Live Marking & \(2p_{1}+p_{2}+3p_{8}+p_{10}+p_{11}\) \\ \(M_{3}\) & Live Marking & \(2p_{1}+p_{3}+3p_{8}+p_{9}+p_{11}\) \\ \(M_{4}\) & Live Marking & \(p_{1}+p_{2}+p_{3}+3p_{8}+p_{11}\) \\ \(M_{5}\) & Live Marking & \(p_{1}+p_{2}+p_{4}+3p_{8}+p_{10}\) \\ \(M_{6}\) & Live Marking & \(p_{1}+p_{3}+p_{4}+3p_{8}+p_{9}\) \\ \(M_{7}\) & Live Marking & \(p_{2}+p_{3}+p_{4}+3p_{8}\) \\ \(M_{8}\) & Deadlock Marking & \(p_{1}+p_{2}+p_{3}+p_{5}+2p_{8}\) \\ \(M_{9}\) & Live Marking & \(2p_{1}+p_{4}+3p_{8}+p_{9}+p_{10}\) \\ \(M_{10}\) & Quasi-deadlock Marking & \(2p_{1}+p_{3}+p_{5}+2p_{8}+p_{9}\) \\ \(M_{11}\) & Quasi-deadlock Marking & \(2p_{1}+p_{2}+p_{5}+2p_{8}+p_{10}\) \\ \(M_{12}\) & Quasi-deadlock Marking & \(2p_{1}+p_{2}+p_{6}+2p_{8}+p_{11}\) \\ \(M_{13}\) & Deadlock Marking & \(2p_{1}+p_{2}+p_{5}+p_{6}+p_{8}\) \\ \(M_{14}\) & Live Marking & \(3p_{1}+p_{5}+2p_{8}+p_{9}+p_{10}\) \\ \(M_{15}\) & Live Marking & \(3p_{1}+p_{6}+2p_{8}+p_{9}+p_{11}\) \\ \(M_{16}\) & Live Marking & \(3p_{1}+p_{5}+p_{6}+p_{8}+p_{9}\) \\ \(M_{17}\) & Live Marking & \(3p_{1}+p_{5}+p_{7}+p_{8}+p_{10}\) \\ \(M_{18}\) & Live Marking & \(3p_{1}+p_{6}+p_{7}+p_{8}+p_{11}\) \\ \(M_{19}\) & Live Marking & \(3p_{1}+p_{5}+p_{6}+p_{7}\) \\ \(M_{20}\) & Live Marking & \(3p_{1}+p_{7}+2p_{8}+p_{10}+p_{11}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The properties of all reachable markings in Figure 2.  
Please note that we add the extra condition \(M_{a}\supseteq M_{b}\) in Definition 7. Further, \((M_{a}\cap M_{b})=M_{b}\) so it can be precise and fit Theorem 1. Accordingly, we use Definition 7 instead of Definition 6 in this paper. We also provide an example so that the readers can follow (please also refer to Figures 1 and 2).