Algorithm 2: ARIPA-IVPR.

Input: $\mathrm{IDT}=\left(U,C\cup D\right)$, threshold $\beta \le 0.5$ Output: Attribute reduct $red$ 1: Initialize $red$ as ∅, i.e., $red\leftarrow \varnothing$, where $red$ indicates    condition attribute subset which has been selected. 2: Evaluate ${\mathrm{SIG}}_3^{\mathrm{inner}}\left(a_k,C,D,U\right)$, where $k\le \left|C\right|$. 3: If ${\mathrm{SIG}}_3^{\mathrm{inner}}\left(a_k,C,D,U\right)>0$, then add $a_k$ into $red$.    IDT’s kernel partly consists of condition attributes    in $red$ at this step. 4: $i\leftarrow 1$, $U_1\leftarrow U$, $R_1=red$, ${\mathbf{P}}_1=\left\{R_1\right\}$. 5: While $U_i\ne \varnothing$ and ${\gamma }_{red}^{\beta U_i}\left(D\right)\ne {\gamma }_C^{\beta U_i}\left(D\right)$, do 6:       {Evaluate the positive region of the positive          approximation set ${\mathrm{POS}}_{{\mathbf{P}}_i}^{\beta U}\left(D\right)$, 7:       $U_i=U-{\mathrm{POS}}_{{\mathbf{P}}_i}^{\beta U}\left(D\right)$, 8:       $i\leftarrow i+1$, 9:       $red\leftarrow red\cup \left\{a_0\right\}$, where          ${\mathrm{SIG}}_3^{\mathrm{outer}}\left(a_0,red,D,U_i\right)=max\left\{{\mathrm{SIG}}_3^{\mathrm{outer}}\left(a_k,red,D,U_i\right)\right\}$          $a_k\in C-red$}, End. 10: $R_i\leftarrow R_i\cup \left\{a_0\right\},{\mathbf{P}}_i\leftarrow \left\{R_1,R_2,\dots ,R_i\right\}$. 11: Return $red$.