# Multi-Target Rough Sets and Their Approximation Computation with Dynamic Target Sets

^{1}

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^{3}

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## Abstract

**:**

## 1. Introduction

- A rough set model considering the label correlation is proposed for multi-label learning. It provides a novel approach for handling multi-label information systems.
- The properties of the proposed models are investigated in this paper.
- An algorithm for calculating the approximations in the proposed rough set model is designed in this paper. It will boost the application of the proposed multi-target rough set model.
- Two algorithms for calculating the approximations in the proposed rough set model under the situation of adding (removing) a target concept to (from) the target group are proposed in this paper. It will improve the efficiency of calculating the approximations of the proposed model.
- Experiments are conducted to validate the efficiency and effectiveness of all proposed algorithms.

## 2. Global Multi-Target Rough Sets

#### 2.1. Definitions

**Definition**

**1**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

U | a_{1} | a_{2} | a_{3} | X_{1} | X_{2} |
---|---|---|---|---|---|

x_{1} | 1 | M | 1 | 0 | 1 |

x_{2} | 2 | F | 2 | 1 | 1 |

x_{3} | 2 | M | 2 | 1 | 0 |

x_{4} | 1 | M | 2 | 0 | 0 |

x_{5} | 2 | F | 2 | 1 | 1 |

#### 2.2. Properties

**Proposition**

**1.**

- Forall ${X}_{i}\in X,{X}_{i}=U$if and only if${\underset{\_}{R}}_{G}\left(X\right)=U$;
- There exists ${X}_{i}\in X,{X}_{i}=U$if and only if ${\overline{R}}_{G}\left(X\right)=U$;
- There exists ${X}_{i}\in X,{X}_{i}=\varphi $if and only if ${\underset{\_}{R}}_{G}\left(X\right)=\varphi $;
- Forall ${X}_{i}\in X,{X}_{i}=\varphi $if and only if ${\overline{R}}_{G}\left(X\right)=\varphi $.

- For all ${X}_{i}\in X,{X}_{i}=U$ implies, for all $x\in U$, for all $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{i}$, which implies ${\left[x\right]}_{A}\subseteq {X}_{1}\wedge {\left[x\right]}_{A}\subseteq {X}_{2}\wedge \cdots \wedge {\left[x\right]}_{A}\subseteq {X}_{r}$ if and only if $x\in {\underset{\_}{R}}_{G}\left(\mathit{X}\right)$, so ${\underset{\_}{R}}_{G}\left(X\right)=U$;
- There exists ${X}_{i}\in X,{X}_{i}=U$, which implies, for all $x\in U$, that there exists $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {X}_{i}\ne \varphi $, which implies ${\left[x\right]}_{A}\cap {X}_{1}\ne \varphi \vee \cdots \vee {\left[x\right]}_{A}\cap {X}_{r}\ne \varphi $ if and only if $x\in {\overline{R}}_{G}\left(X\right)$, so ${\overline{R}}_{G}\left(X\right)=U$;
- There exists ${X}_{i}\in X,{X}_{i}=\varphi $, which implies, for all $x\in U$, that there exists $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\not\subset {X}_{i}$ which implies $x\notin {\underset{\_}{R}}_{G}\left(X\right)$, so ${\underset{\_}{R}}_{G}\left(X\right)=\varphi $;
- For all ${X}_{i}\in X,{X}_{i}=\varphi $ implies, for all $x\in U$, for all $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {X}_{i}=\varphi $, which implies $x\notin {\overline{R}}_{G}\left(X\right)$, so ${\overline{R}}_{G}\left(X\right)=\varphi $.□

**Proposition**

**2.**

- For all${X}_{i}\in X,{\underset{\_}{R}}_{G}\left(X\right)\subseteq {X}_{i}$;
- For all${X}_{i}\in X,{X}_{i}\subseteq {\overline{R}}_{G}\left(X\right)$.

**Proof.**

- For all $x\in {\underset{\_}{R}}_{G}\left(X\right)$, it is implied that ${\left[x\right]}_{A}\subseteq {X}_{i}$ if and only if ${\left[x\right]}_{A}\subseteq {X}_{1}\wedge {\left[x\right]}_{A}\subseteq {X}_{2}\wedge \cdots \wedge {\left[x\right]}_{A}\subseteq {X}_{r}$, which implies for all $i\in \left\{1,2,\cdots ,r\right\},x\subseteq {X}_{i}$, so ${\underset{\_}{R}}_{G}\left(X\right)\subseteq {X}_{i}$.
- For all $x\in {X}_{i}$, it is implied that there exists $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {X}_{i}=\varphi $, which implies $x\in {\overline{R}}_{G}\left(X\right)$, so ${X}_{i}\subseteq {\overline{R}}_{G}\left(X\right)$. □

**Proposition**

**3.**

- ${\underset{\_}{R}}_{G}\left(X\cap Y\right)\supseteq {\underset{\_}{R}}_{G}\left(X\right)\cap {\underset{\_}{R}}_{G}\left(Y\right)$;
- ${\overline{R}}_{G}\left(X\cap Y\right)\subseteq {\overline{R}}_{G}\left(X\right)\cap {\overline{R}}_{G}\left(Y\right)$;
- ${\underset{\_}{R}}_{G}\left(X\cup Y\right)\subseteq {\underset{\_}{R}}_{G}\left(X\right)\cup {\underset{\_}{R}}_{G}\left(Y\right)$;
- ${\overline{R}}_{G}\left(X\cup Y\right)={\overline{R}}_{G}\left(X\right)\cup {\overline{R}}_{G}\left(Y\right)$.

**Proof.**

- For all $x\in {\underset{\_}{R}}_{G}\left(X\right)\cap {\underset{\_}{R}}_{G}\left(Y\right)$, it is implied that, for all ${X}_{i}\in X\left(i\le \left|X\right|\right),{\left[x\right]}_{A}\subseteq {X}_{i}$ and, for all ${Y}_{j}\in Y\left(j\le \left|Y\right|\right),{\left[x\right]}_{A}\subseteq {Y}_{j}$, which implies, for all ${Z}_{k}\in X\cap Y\left(k\in \left\{1,2,\cdots ,r\right\}\right),{\left[x\right]}_{A}\subseteq {Z}_{k}$ if and only if $x\in {\underset{\_}{R}}_{G}\left(X\cap Y\right)$, so ${\underset{\_}{R}}_{G}\left(X\cap Y\right)\supseteq {\underset{\_}{R}}_{G}\left(X\right)\cap {\underset{\_}{R}}_{G}\left(Y\right)$;
- For all $x\in {\overline{R}}_{G}\left(X\cap Y\right)$, $x\in {\overline{R}}_{G}\left(X\cap Y\right)$ if and only if there exists ${Z}_{k}\in X\cap Y\left(k\in \left\{1,2,\cdots ,r\right\}\right),{\left[x\right]}_{A}\cap {Z}_{k}\ne \varphi $, which implies there exists ${X}_{i}\in X\left(i\le \left|X\right|\right),{\left[x\right]}_{A}\cap {X}_{i}\ne \varphi $ and there exists ${Y}_{j}\in Y\left(j\le \left|Y\right|\right),{\left[x\right]}_{A}\cap {Y}_{j}\ne \varphi $, which implies $x\in {\overline{R}}_{G}\left(X\right)\cap {\overline{R}}_{G}\left(Y\right)$, so ${\overline{R}}_{G}\left(X\cap Y\right)\subseteq {\overline{R}}_{G}\left(X\right)\cap {\overline{R}}_{G}\left(Y\right);$
- For all $x\in {\underset{\_}{R}}_{G}\left(X\cup Y\right)$, $x\in {\underset{\_}{R}}_{G}\left(X\cup Y\right)$ if and only if, for all ${Z}_{k}\in X\cup Y\left(k\in \left\{1,2,\cdots ,r\right\}\right),{\left[x\right]}_{A}\subseteq {Z}_{k}$, which implies, for all ${X}_{i}\in X\left(i\le \left|X\right|\right),{\left[x\right]}_{A}\subseteq {X}_{i}$ or, for all ${Y}_{j}\in Y\left(j\le \left|Y\right|\right),{\left[x\right]}_{A}\subseteq {Y}_{j}$, which implies $x\in {\underset{\_}{R}}_{G}\left(X\right)\text{or}x\in {\underset{\_}{R}}_{G}\left(Y\right),$ which implies $x\in \left({\underset{\_}{R}}_{G}\left(X\right)\cup {\underset{\_}{R}}_{G}\left(Y\right)\right),$ so ${\underset{\_}{R}}_{G}\left(X\cup Y\right)\subseteq {\underset{\_}{R}}_{G}\left(X\right)\cup {\underset{\_}{R}}_{G}\left(Y\right)$;
- For all $x\in {\overline{R}}_{G}\left(X\cup Y\right)$, if and only if there exists ${Z}_{k}\in X\cup Y\left(k\in \left\{1,2,\cdots ,r\right\}\right),{\left[x\right]}_{A}\cap {Z}_{k}\ne \varphi $, if and only if there exists ${X}_{i}\in X\left(i\le \left|X\right|\right),{\left[x\right]}_{A}\cap {X}_{i}\ne \varphi $ or there exists ${Y}_{j}\in Y\left(j\le \left|Y\right|\right),{\left[x\right]}_{A}\cap {Y}_{j}\ne \varphi $, which implies $x\in {\overline{R}}_{G}\left(X\right)\text{or}x\in {\overline{R}}_{G}\left(Y\right),$ if and only if ${\overline{R}}_{G}\left(X\cup Y\right)={\overline{R}}_{G}\left(X\right)\cup {\overline{R}}_{G}\left(Y\right)$.□

**Proposition**

**4.**

- ${\underset{\_}{R}}_{G}\left(X\right)\supseteq {\underset{\_}{R}}_{G}\left(Y\right)$;
- ${\overline{R}}_{G}\left(X\right)\subseteq {\overline{R}}_{G}\left(Y\right)$.

**Proof.**

- For all $x\in {\underset{\_}{R}}_{G}\left(Y\right)$, if and only if, for all ${Z}_{k}\in Y,k\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {Z}_{k},$ which implies for all ${Z}_{k}\in X,k\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {Z}_{k},$ which implies $x\in {\underset{\_}{R}}_{G}\left(X\right)$ so ${\underset{\_}{R}}_{G}\left(X\right)\supseteq {\underset{\_}{R}}_{G}\left(Y\right)$.
- For all $x\in {\overline{R}}_{G}\left(X\right)$, if and only if there exists ${Z}_{k}\in X,k\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {Z}_{k}\ne \varphi $, which implies there exists ${Z}_{k}\in Y,k\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {Z}_{k}\ne \varphi $ which implies $x\in {\overline{R}}_{G}\left(Y\right),$ so ${\overline{R}}_{G}\left(X\right)\subseteq {\overline{R}}_{G}\left(Y\right)$. □

## 3. Approximation Computation of GMTRSs

**Definition**

**5**

**Example**

**2.**

**Lemma**

**1.**

- $X\cap Y\ne \varphi \iff {\overrightarrow{X}}^{\prime}\cdot \overrightarrow{Y}>0$;
- $X\subseteq Y\iff {\overrightarrow{X}}^{\prime}\cdot \overrightarrow{\sim Y}=0$.

**Theorem**

**1.**

- $x\in {\underset{\_}{R}}_{G}\left(X\right)\iff \forall i\in \left\{1,2,\cdots ,r\right\},{\overrightarrow{\left[x\right]}}_{A}^{\prime}\cdot \overrightarrow{\sim {X}_{i}}=0$;
- $x\in {\overline{R}}_{G}\left(X\right)\iff \exists i\in \left\{1,2,\cdots ,r\right\},{\overrightarrow{\left[x\right]}}_{A}^{\prime}\cdot \overrightarrow{{X}_{i}}>0$.

**Proof.**

- For all $i\in \left\{1,2,\cdots ,r\right\},{\overrightarrow{\left[x\right]}}_{A}^{\prime}\cdot \overrightarrow{\sim {X}_{i}}=0,$ if and only if, for all $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{i},$ if and only if $x\in {\underset{\_}{R}}_{G}\left(X\right)$;
- There exists $i\in \left\{1,2,\cdots ,r\right\},{\overrightarrow{\left[x\right]}}_{A}^{\prime}\cdot \overrightarrow{{X}_{i}}>0$ if and only if there exists $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {X}_{i}\ne \varphi ,$ if and only if $x\in {\overline{R}}_{G}\left(X\right)$. □

**Definition**

**6.**

**Example**

**3.**

**Example**

**4.**

**Theorem**

**2.**

**X**is a GCTS. We have:

- ${x}_{j}\in {\underset{\_}{R}}_{G}\left(X\right)\iff {\wedge}_{i=1}^{r}{\underset{\_}{h}}_{ij}=1;$
- ${x}_{j}\in {\overline{R}}_{G}\left(X\right)\iff {\vee}_{i=1}^{r}{\overline{h}}_{ij}=1.$

**Proof.**

- ${x}_{j}\in {\underset{\_}{R}}_{G}\left(X\right)$ if and only if for all $i\in \left\{1,2,\cdots ,r\right\},{\overrightarrow{\left[{x}_{j}\right]}}_{A}^{\prime}\cdot \overrightarrow{\sim {X}_{i}}=0$, if and only if, for all $i\in \left\{1,2,\cdots ,r\right\},{\underset{\_}{h}}_{ij}=1,$ if and only if ${\wedge}_{i=1}^{r}{\underset{\_}{h}}_{ij}=1;$
- ${x}_{j}\in {\overline{R}}_{G}\left(X\right)$ if and only if there exists $i\in \left\{1,2,\cdots ,r\right\},{\overrightarrow{\left[{x}_{j}\right]}}_{A}^{\prime}\cdot \overrightarrow{{X}_{i}}>0$
**,**if and only if there exists $i\in \left\{1,2,\cdots ,r\right\},{\overline{h}}_{ij}=1,$ if and only if ${\vee}_{i=1}^{r}{\overline{h}}_{ij}=1.$□

**Example**

**5.**

Algorithm 1. Computing Approximations of GMTRSs (CAG). | |

Input: $\left(U,A\right),X=\left\{{X}_{1},{X}_{2},\cdots ,{X}_{r}\right\},{\left[x\right]}_{A}$ | |

Output:${\overline{R}}_{G}\left(X\right),{\underset{\_}{R}}_{G}\left(X\right)$ | |

1: | $n\leftarrow \left|U\right|,r\leftarrow \left|X\right|$ |

2: | for $i=1\to r$ |

3: | for$j=1\to n$ |

4: | if${\overrightarrow{\left[{x}_{j}\right]}}_{A}^{\prime}\cdot \overrightarrow{\sim {X}_{i}}=0$then |

5: | ${\underset{\_}{h}}_{ij}=1$ |

6: |
else |

7: | ${\underset{\_}{h}}_{ij}=0$ |

8: |
end if |

9: |
if${\overrightarrow{\left[{x}_{j}\right]}}_{A}^{\prime}\cdot \overrightarrow{{X}_{i}}>0$then |

10: | ${\overline{h}}_{ij}=1$ |

11: |
else |

12: | ${\overline{h}}_{ij}=0$ |

13: |
end if |

14: |
end for |

15: | end for |

16: | $\overrightarrow{{\underset{\_}{R}}_{G}\left(X\right)}={\left({\wedge}_{i=1}^{r}{\underset{\_}{h}}_{i\ast}\right)}^{\prime}={\left({\wedge}_{i=1}^{r}{\underset{\_}{h}}_{i1},{\wedge}_{i=1}^{r}{\underset{\_}{h}}_{i2},\cdots ,{\wedge}_{i=1}^{r}{\underset{\_}{h}}_{in}\right)}^{\prime}$ |

17: | $\overrightarrow{{\overline{R}}_{G}\left(X\right)}={\left({\vee}_{i=1}^{r}{\overline{h}}_{i\ast}\right)}^{\prime}={\left({\vee}_{i=1}^{r}{\overline{h}}_{i1},{\vee}_{i=1}^{r}{\overline{h}}_{i2},\cdots ,{\vee}_{i=1}^{r}{\overline{h}}_{in}\right)}^{\prime}$ |

18: | Return ${\overline{R}}_{G}\left(X\right),{\underset{\_}{R}}_{G}\left(X\right)$ |

## 4. Dynamical Approximation Computation

#### 4.1. Dynamical Approximation Computation while Adding a Target

**Theorem**

**3.**

- ${\underset{\_}{R}}_{G}\left(X\cup \left\{P\right\}\right)={\underset{\_}{R}}_{G}\left(X\right)-\left\{x\in U|x\in {\underset{\_}{R}}_{G}\left(X\right)\wedge {\left[x\right]}_{A}\not\subset P\right\}$;
- ${\overline{R}}_{G}\left(X\cup \left\{P\right\}\right)={\overline{R}}_{G}\left(X\right)\cup \left\{x\in U|x\in \sim {\overline{R}}_{G}\left(X\right)\wedge {\left[x\right]}_{A}\cap P\ne \varphi \right\}.$

**Proof.**

- For all $x\in \left({\underset{\_}{R}}_{G}\left(X\right)-\left\{x\in U|x\in {\underset{\_}{R}}_{G}\left(X\right)\wedge {\left[x\right]}_{A}\not\subset P\right\}\right)$ if and only if for all $x\in {\underset{\_}{R}}_{G}\left(X\right)\cap \left\{x\in U|x\in {\underset{\_}{R}}_{G}\left(X\right)\wedge {\left[x\right]}_{A}\subseteq P\right\}$ if and only if for all $i\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{i}\wedge {\left[x\right]}_{A}\subseteq P,$ if and only if $x\in {\underset{\_}{R}}_{G}\left(X\cup \left\{P\right\}\right)$;
- For all $x\in \left\{{\overline{R}}_{G}\left(X\right)\cup \left\{x\in U|x\in \sim {\overline{R}}_{G}\left(X\right)\wedge {\left[x\right]}_{A}\cap P\ne \varphi \right\}\right\},$.$x\in {\overline{R}}_{G}\left(X\right)\cup \left\{x\in U|x\in \sim {\overline{R}}_{G}\left(X\right)\wedge {\left[x\right]}_{A}\cap P\ne \varphi \right\},$ if and only if there exists $x\in {\overline{R}}_{G}\left(X\right),$ such that ${\left[x\right]}_{A}\cap {X}_{i}\ne \varphi ,$ or there exists $x\in \left(\sim {\overline{R}}_{G}\left(X\right)\right)$ such that ${\left[x\right]}_{A}\cap P\ne \varphi ,$ if and only if there exists $Y\in \left(X\cup \left\{P\right\}\right)$ such that ${\left[x\right]}_{A}\cap Y\ne \varphi ,$ if and only if $x\in {\overline{R}}_{G}\left(X\cup \left\{P\right\}\right).$□

**Definition**

**7.**

**GCTS**,$P\subseteq U$and it is satisfied, for all${X}_{i}\in X,i\in \left\{1,2,\cdots ,r\right\}$, that${S}_{R}\left({X}_{i},P\right)>\alpha \text{and}{S}_{R}\left(P,{X}_{i}\right)\alpha .$The dynamical lower approximation matrix while adding a target concept of a GMTRS can be defined as:

**Example**

**6.**

**Example**

**7.**

**Theorem**

**4.**

- $\overrightarrow{{\underset{\_}{R}}_{G}\left(X\cup \left\{P\right\}\right)}=\overrightarrow{{\underset{\_}{R}}_{G}\left(X\right)}\wedge \left(\sim {\underset{\_}{L}}_{A}\right);$
- $\overrightarrow{{\overline{R}}_{G}\left(X\cup \left\{P\right\}\right)}=\overrightarrow{{\overline{R}}_{G}\left(X\right)}\vee \left({\overline{U}}_{A}\right).$

**Proof.**

**Example**

**8.**

Algorithm 2. Dynamic Computing Approximations of a GMTRS while Adding a Target Concept (DCAGA). | |

Input:$\left(U,A\right),X=\left\{{X}_{1},{X}_{2},\cdots ,{X}_{r}\right\},{\left[x\right]}_{A},P\subseteq U$ and satisfied for all${X}_{i}\in X,i\in \left\{1,2,\cdots ,r\right\},{S}_{R}\left({X}_{i},P\right)>\alpha and{S}_{R}\left(P,{X}_{i}\right)\alpha ,{\overline{R}}_{G}\left(X\right),{\underset{\_}{R}}_{G}\left(X\right)$. | |

Output:${\overline{R}}_{G}\left(X\cup \left\{P\right\}\right),{\underset{\_}{R}}_{G}\left(X\cup \left\{P\right\}\right)$ | |

1: | $n\leftarrow \left|U\right|$ |

2: | for$j=1\to n$ |

3: |
if $x\in {\underset{\_}{R}}_{G}\left(X\right)\wedge {\overrightarrow{\left[{x}_{j}\right]}}_{A}^{\prime}\cdot \overrightarrow{\sim P}>0$ then |

4: | ${\underset{\_}{l}}_{j}^{A}=1$ |

5: |
else |

6: | ${\underset{\_}{l}}_{j}^{A}=0$ |

7: |
end if |

8: |
if $x\in \sim {\overline{R}}_{G}\left(X\right)\wedge {\overrightarrow{\left[{x}_{j}\right]}}_{A}^{\prime}\cdot \overrightarrow{P}>0$ then |

9: | ${\overline{u}}_{j}^{A}=1$ |

10: |
else |

11: | ${\overline{u}}_{j}^{A}=0$ |

12: |
end if |

13: | end for |

14: | $\overrightarrow{{\underset{\_}{R}}_{G}\left(X\cup \left\{P\right\}\right)}=\overrightarrow{{\underset{\_}{R}}_{G}\left(X\right)}\wedge \left(\sim {\underset{\_}{L}}_{A}\right)$ |

15: | $\overrightarrow{{\overline{R}}_{G}\left(X\cup \left\{P\right\}\right)}=\overrightarrow{{\overline{R}}_{G}\left(X\right)}\vee \left({\overline{U}}_{A}\right)$ |

16: | Return ${\overline{R}}_{G}\left(X\cup \left\{P\right\}\right),{\underset{\_}{R}}_{G}\left(X\cup \left\{P\right\}\right)$ |

#### 4.2. Dynamical Approximation Computation while Removing a Target

**Theorem**

**5.**

- ${\underset{\_}{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)={\underset{\_}{R}}_{G}\left(X\right)\cup \left\{x\in U|x\in \sim {\underset{\_}{R}}_{G}\left(X\right)\wedge \forall j\ne i,j\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{j}\right\};$
- ${\overline{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)={\overline{R}}_{G}\left(X\right)-\left\{x\in U|x\in {\overline{R}}_{G}\left(X\right)\wedge \forall j\ne i,j\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {X}_{j}=\varphi \right\}.$

**Proof.**

- For all $x\in \left\{{\underset{\_}{R}}_{G}\left(X\right)\cup \left\{x\in U|x\in \sim {\underset{\_}{R}}_{G}\left(X\right)\wedge \forall j\ne i,j\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{j}\right\}\right\}$ if and only if (for all $x\in {\underset{\_}{R}}_{G}\left(X\right)$, for all $j\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{j}$ or (for all $x\in \sim {\underset{\_}{R}}_{G}\left(X\right)$ and for all $j\ne i,j\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\subseteq {X}_{j}$) if and only if (for all $x\in U$, for all $j\in \left\{1,2,\cdots ,r\right\}$ and $j\ne i,{\left[x\right]}_{A}\subseteq {X}_{j}$) if and only if $x\in {\underset{\_}{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right);$
- For all $x\in \left\{{\overline{R}}_{G}\left(X\right)-\left\{x\in U|x\in {\overline{R}}_{G}\left(X\right)\wedge \forall j\ne i,j\in \left\{1,2,\cdots ,r\right\},{\left[x\right]}_{A}\cap {X}_{j}=\varphi \right\}\right\}$ if and only if $x\in \left\{{\overline{R}}_{G}\left(X\right)\cap \left\{x\in U|\exists j\ne i,j\in \left\{1,2,\cdots ,r\right\},s.t.{\left[x\right]}_{A}\cap {X}_{j}\ne \varphi \right\}\right\},$ if and only if $x\in {\overline{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right).$□

**Definition**

**8.**

**Theorem**

**6.**

- $\overrightarrow{{\underset{\_}{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)}=\overrightarrow{{\underset{\_}{R}}_{G}\left(X\right)}\vee \left({\underset{\_}{L}}_{D}\right);$
- $\overrightarrow{{\overline{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)}=\overrightarrow{{\overline{R}}_{G}\left(X\right)}\wedge \left(\sim {\overline{U}}_{D}\right).$

**Proof.**

**Example**

**9.**

Algorithm 3. Dynamic Computing Approximations of a GMTRS while Removing a Target Concept (DCAGR). | |

Input:$\left(U,A\right),X=\left\{{X}_{1},{X}_{2},\cdots ,{X}_{r}\right\},{\left[x\right]}_{A},\forall x\in U;P\subseteq U$ and satisfied for all${X}_{i}\in X,i\in \left\{1,2,\cdots ,r\right\},{S}_{R}\left({X}_{i},P\right)>\alpha and{S}_{R}\left(P,{X}_{i}\right)\alpha ,{\overline{R}}_{G}\left(X\right),{\underset{\_}{R}}_{G}\left(X\right),\overline{H},\underset{\_}{H}$ | |

Output:${\overline{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right),{\underset{\_}{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)$ | |

1: | $n\leftarrow \left|U\right|$ |

2: | for $j=1\to n$ |

3: | if ${x}_{k}\in \left(U-{\underset{\_}{R}}_{G}\left(X\right)\right)\wedge \left({\wedge}_{j\ne i}{\underset{\_}{h}}_{jk}=1\right)$ then |

4: | ${\underset{\_}{l}}_{j}^{D}=1$ |

5: |
else |

6: | ${\underset{\_}{l}}_{j}^{D}=0$ |

7: |
end if |

8: | if ${x}_{k}\in \left(U-{\overline{R}}_{G}\left(X\right)\right)\wedge \left({\vee}_{j\ne i}{\overline{h}}_{jk}=0\right)$ then |

9: | ${\overline{u}}_{j}^{D}=1$ |

10: |
else |

11: | ${\overline{u}}_{j}^{D}=0$ |

12: |
end if |

13: | end for |

14: | $\overrightarrow{{\underset{\_}{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)}=\overrightarrow{{\underset{\_}{R}}_{G}\left(X\right)}\vee \left({\underset{\_}{L}}_{D}\right)$ |

15: | $\overrightarrow{{\overline{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)}=\overrightarrow{{\overline{R}}_{G}\left(X\right)}\wedge \left(\sim {\overline{U}}_{D}\right)$ |

16: | Return${\overline{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right),{\underset{\_}{R}}_{G}\left(X-\left\{{X}_{i}\right\}\right)$ |

## 5. Experimental Evaluations

#### 5.1. Comparison of Computational Time Using Matrix-Based Approach and Set-Operation-Based Approach

#### 5.1.1. Experimental Settings

#### 5.1.2. Discussions of the Experimental Results

#### 5.2. Comparison of Computational Time Using Elements in Target Concept with Different Sizes

#### 5.2.1. Experimental Settings

#### 5.2.2. Discussions of the Experimental Results

#### 5.3. Comparison of Computational Time Using Data Sets with Different Sizes

#### 5.3.1. Experimental Settings

#### 5.3.2. Discussions of the Experimental Results

#### 5.4. Parameter Analysis Experiments of α

#### 5.4.1. Experimental Settings

#### 5.4.2. Discussions of the Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison of the times of Algorithms 1–3 when the sizes of elements in X increase gradually.

**Figure 4.**The effluence of $\alpha $ on the lower approximation, the upper approximation and the computational time.

U | a_{1} | a_{2} | a_{3} | X_{1} | X_{2} | P |
---|---|---|---|---|---|---|

x_{1} | 1 | M | 1 | 0 | 1 | 0 |

x_{2} | 2 | F | 2 | 1 | 1 | 0 |

x_{3} | 2 | M | 2 | 1 | 0 | 0 |

x_{4} | 1 | M | 2 | 0 | 0 | 1 |

x_{5} | 2 | F | 2 | 1 | 1 | 1 |

No. | Data Sets | Samples | Attributes |

1 | Autism Screening Adult Data Set | 366 | 11 |

2 | Cargo2000 Freight Tracking and Tracing Data Set | 3943 | 98 |

3 | Mushroom | 8124 | 23 |

4 | Semeion Handwritten Digit Data Set | 1593 | 267 |

5 | Studentmat | 395 | 33 |

6 | Website Phishing Data Set | 1353 | 10 |

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**MDPI and ACS Style**

Zheng, W.; Li, J.; Liao, S.
Multi-Target Rough Sets and Their Approximation Computation with Dynamic Target Sets. *Information* **2022**, *13*, 385.
https://doi.org/10.3390/info13080385

**AMA Style**

Zheng W, Li J, Liao S.
Multi-Target Rough Sets and Their Approximation Computation with Dynamic Target Sets. *Information*. 2022; 13(8):385.
https://doi.org/10.3390/info13080385

**Chicago/Turabian Style**

Zheng, Wenbin, Jinjin Li, and Shujiao Liao.
2022. "Multi-Target Rough Sets and Their Approximation Computation with Dynamic Target Sets" *Information* 13, no. 8: 385.
https://doi.org/10.3390/info13080385