## 4 Proposed Method
### Policy Optimization with Separation-of-Duty Constraints
#### 4.2.2 Construction of t-m MEAR Constraints from k-n SOAR Constraints
If the number of objects is less than 500, then the execution time is always below 335 s and tends to grow linearly. However, if the number is greater than 500, then the execution time increases exponentially to around 1100 s with 1000 users and 1000 objects, which is unacceptable. That is attributed to the fact that the dimension of an authorization matrix becomes larger and larger with an increasing number of users and objects, and a large-size matrix requires more time to be rearranged using Algorithm 1. To demonstrate the interpretability of the mining process, we employ synthetic datasets on which VisMAP has been executed, where the numbers of users and objects are 100, 200, 500, and 1000. In addition, the number of partitions for each rearranged matrix is 1, 2, and 4. We repeatedly implement the experiments 10 times on these synthetic datasets, output the number of rules and the execution time, and take their average values. Comparisons of the results for PEO_VR&SOD and VisMAP are shown in \[\exists|u_{1},u_{2},\ldots,u_{k-1}|\subset U:\mathop{\cup}\limits_{i=1}^{k-1 }\mathit{user\_rules_{\gamma}}(u_{i})\supseteq\{\mathit{ar}_{1},\mathit{ar}_{ 2},\ldots,\mathit{ar}_{n}\}\Rightarrow\mathit{safe}_{\text{sant}}(\gamma)\;=0. \tag{11}\]2

\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**[Partitions]**} \\ \cline{2-13}  & & \multicolumn{3}{c}{**1**} & \multicolumn{3}{c}{**2**} & \multicolumn{3}{c}{**4**} & \multicolumn{3}{c}{**1**} & \multicolumn{3}{c}{**2**} & \multicolumn{3}{c}{**4**} \\ \cline{2-13}
[II] & [OI] & \multicolumn{3}{c}{**VisMAP**} & \multicolumn{3}{c}{**PEO\_VR&SOD**} \\ \cline{2-13}  & & **[P]** & **T(s)** & **[P]** & **T(s)** & **[P]** & **T(s)** & **[P]** & **T(s)** & **[P]** & **T(s)** & **[P]** & **T(s)** \\ \hline
100 & 100 & 19 & 0.59 & 38 & 0.72 & 62 & 0.35 & 19 & 0.59 & 22 & 0.84 & 30 & 0.93 \\
200 & 100 & 26 & 1.77 & 55 & 1.11 & 84 & 1.03 & 26 & 1.77 & 31 & 1.98 & 42 & 2.14 \\
500 & 100 & 34 & 10.18 & 76 & 5.38 & 108 & 5.9 & 34 & 10.18 & 42 & 10.73 & 53 & 11.95 \\
1000 & 100 & 47 & 43.28 & 96 & 20.46 & 197 & 10.92 & 47 & 43.28 & 56 & 43.61 & 74 & 44.17 \\
100 & 200 & 23 & 1.63 & 48 & 1.57 & 78 & 0.95 & 23 & 1.63 & 26 & 1.87 & 34 & 2.02 \\
200 & 200 & 37 & 4.25 & 77 & 3.33 & 114 & 2.14 & 37 & 4.25 & 42 & 5.01 & 53 & 5.61 \\
500 & 200 & 54 & 21.54 & 96 & 12.36 & 146 & 10.78 & 54 & 21.54 & 56 & 22.34 & 62 & 23.24 \\
1000 & 200 & 70 & 100.4 & 145 & 43.01 & 197 & 39.77 & 70 & 100.4 & 81 & 105.34 & 98 & 106.11 \\
100 & 500 & 28 & 8.27 & 50 & 9.78 & 89 & 4.72 & 28 & 8.27 & 33 & 8.77 & 43 & 9.07 \\
200 & 500 & 45 & 17.64 & 91 & 16.41 & 156 & 9.39 & 45 & 17.64 & 50 & 18.13 & 65 & 18.55 \\
500 & 500 & 67 & 75.23 & 141 & 49.92 & 224 & 33.57 & 67 & 75.23 & 74 & 76.02 & 87 & 76.93 \\
1000 & 500 & 89 & 334.98 & 207 & 163.96 & 304 & 115.50 & 89 & 334.98 & 99 & 335.44 & 1 Table 18 shows that, for both methods, as the number of users and objects increases, the number of rules increases slightly, while the execution time increases more clearly whenever \(|\)Partitions\(|\) is equal to 1, 2, or 4, which is consistent with the experimental analyses shown in Figures 3 and 4. Second, as the number of partitions increases, the number of rules increases clearly, while the execution time decreases in VisMAP; however, both the number of rules and the execution time increase slightly in PEO_VR&SOD. Specifically, when the number of users and objects is fixed at 1000 and 100, respectively, for the former, the number of rules varies from 47 to 197 with an increase of more than 300%, while the time varies from 43.28 to 10.92 with a decrease of more than 75%; for the latter, the number of rules increases from 47 to 74, and the time increases from 43.28 to 44.17. This is attributed to the fact that the original matrix in VisMAP is separated into several partitions before policy mining and spends little time on the small datasets. However, the rearranged matrix is separated during mining and spends more time with the number of partitions according to Algorithm 2.