Algorithm 1. Static algorithm in C-SODS

Input: (1) The set-valued information system;

(2) upward unions and downward unions: $C l_{n}^{\geq}$ , $C l_{n}^{\leq} (0 \leq n \leq t)$

Output: $\underline{R_{B}} (C l_{n}^{\geq})$ , $\bar{R_{B}} (C l_{n}^{\geq})$ , $\underline{R_{B}} (C l_{n}^{\leq})$ , $\bar{R_{B}} (C l_{n}^{\leq})$

1: for $\forall x \in U$

2: $\forall y \in U$

3: if $f (x, B) \subseteq f (y, B)$ then

4: ${[x]}_{B}^{+} \leftarrow {[x]}_{B}^{+} \cup {y}$

5: else

6: ${[x]}_{B}^{-} \leftarrow {[x]}_{B}^{-} \cup {y}$

7: if ${[x]}_{B}^{+} \subseteq C l_{n}^{\geq}$ then

8: $\underline{R_{B}} (C l_{n}^{\geq}) \leftarrow \underline{R_{B}} (C l_{n}^{\geq}) \cup {x}$

9: if ${[x]}_{B}^{-} \cap C l_{n}^{\geq} = \emptyset$ then

10: $\bar{R_{B}} (C l_{n}^{\geq}) \leftarrow \bar{R_{B}} (C l_{n}^{\geq}) \cup {x}$

11: if ${[x]}_{B}^{-} \subseteq C l_{n}^{\leq}$ then

12: $\underline{R_{B}} (C l_{n}^{\leq}) \leftarrow \underline{R_{B}} (C l_{n}^{\leq}) \cup {x}$

13: if ${[x]}_{B}^{+} \cap C l_{n}^{\leq} = \emptyset$ then

14: $\bar{R_{B}} (C l_{n}^{\leq}) \leftarrow \bar{R_{B}} (C l_{n}^{\leq}) \cup {x}$

15: Return $\underline{R_{B}} (C l_{n}^{\geq})$ , $\bar{R_{B}} (C l_{n}^{\geq})$ , $\underline{R_{B}} (C l_{n}^{\leq})$ , $\bar{R_{B}} (C l_{n}^{\leq})$