Rough set model based on axiomatic fuzzy set

Abstract

Both fuzzy set and rough set are important mathematical tools to describe incomplete and uncertain information, and they are highly complementary to each other. What is more, most fuzzy rough sets are obtained by combining Zadeh fuzzy sets and Pawlak rough sets. There are few reports about the combination of axiomatic fuzzy sets and Pawlak rough sets. For this reason, we propose the axiomatic fuzzy rough sets (namely rough set model with respect to the axiomatic fuzzy set) establishing on fuzzy membership space. In this paper, we first present a similarity description method based on vague partitions. Then the concept of similarity operator is proposed to describe uncertainty in the fuzzy approximation space. Finally, some characterizations concerning upper and lower approximation operators are shown, including basic properties. Furthermore, we give a algorithm to verify the effectiveness and efficiency of the model.

Keywords

Rough sets axiomatic fuzzy rough sets residuated lattices fuzzy relations approximation operators

1 Introduction

Vagueness is everywhere, involving all aspects of real life and production practice. It is the external manifestation of the vagueness of objective things. In 1965, the famous American cybernetics expert Zadeh first proposed the concept of fuzzy sets [1]. So far, it has formed a relatively well-established branch of mathematics, and has been effectively applied in many fields [2 –5].

Through in-depth analysis of the nature of vague phenomena, Pan and Xu [6, 7] introduced a new understanding of the essence of vague phenomena from the perspective of axiomatization in 2018. They established an axiomatic system of membership, which is characterized by the ability to study fuzzy sets as a whole. On this basis, the concept of vague partition is defined, and then based on this vague partition, the axiomatic definition of fuzzy set membership function is given, which makes the establishment of fuzzy set membership function more objective and close to the fact. Naturally, all axiomatic fuzzy sets form a function space, which is called fuzzy membership space. The article points out that the so-called vague phenomenon is a phenomenon that reflects the difference between the qualitative and quantitative changes of things. The fuzzy sets studied in this paper are axiomatic fuzzy sets in fuzzy membership spaces.

Rough set theory (RST) [8] is a mathematical theory introduced by Polish mathematician Pawlak in 1982 to deal with incomplete and uncertain information, which can quantitatively analyze and deal with imprecise, inconsistent and incomplete information and knowledge. After more than 30 years of development, RST has been successfully applied in knowledge and data discovery, pattern recognition and classification, data mining and other fields [9 –16].

The key concept of rough set theory is a pair of approximation operators derived from the approximation space, that is, upper and lower approximation operators. With the aid of approximation operators, uncertainty concepts are described. Pawlak rough set model is suitable for processing symbolic data, but it can not solve the problem independently for real value data or incomplete data. To expand the application scope of RST, many researchers have extended Pawlak rough set model based on equivalence relations into various forms. The one category is practical applications of Pawlak rough sets, such as formal concept analysis, Yuan et al. [17] proposed an incremental learning mechanism for object classification based on progressive fuzzy three-way concept. The proposed algorithm based on progressive fuzzy concept can not only process real-valued data in life, but also achieve better classification performance. In concept-cognitive learning, two-way concept learning are influential studies of knowledge processing and cognitive learning. Due to concept-cognitive learning methods still face challenges, such as incomplete cognitive and weak generalization ability, Xu et al. [18] proposed a novel two-way concept-cognitive learning method for dynamic concept learning in a fuzzy context for these problems and challenges. Compared with two-way concept learning, fuzzy-based two-way concept-cognitive learning is more flexible and less time-consuming to learn granule concepts from the given clue, and meanwhile, it is good at dynamic concept learning. In addition, reference [19] also introduces the two way concept-cognitive learning via concept movement viewpoint and so on. The another category is the extended model researches on Pawlak rough sets, such as generalized rough set model based on general binary relations, fuzzy rough set model (FRS) based on universe covering, variable precision rough set (VPRS) model, L-FRS model, etc. Fuzzy sets and rough sets are both mathematical tools to describe incomplete and uncertain information. In 1990, Dubois and Prade [20] combined fuzzy sets and rough sets to establish FRS model, which can be used to process real value data. Later, Morsi and Yakou [21] introduced a new FRS model from the perspective of axiomatization, and discussed the axiomatization method of approximation operators in the FRS model. In addition, Radzikowska and Kerre [22] established a general FRS model through the use of triangular norms and fuzzy implication operators. Meanwhile, Yang and Hu [23, 24] proposed a rough set model based on fuzzy covering, studied the basic properties of the model, and further discussed the multi-granularity FRS based on covering and its application. Literature [25 –28] further studied the generalization and application of FRS model. In addition, the research on variable precision FRS model has also attracted extensive attention [29 –31]. Based on the VPRS model proposed by Ziako [29], Mieszkowicz-Rolka and Rolka [30] introduced the variable precision FRS model. Zhao [31] proposed a variable precision FRS model to process noise data. Radzikowska and Kerre [32] presented L-FRS model based on complete residuated lattices on the basis of residuated lattices. References [33, 34] studied the axiomatization of approximation operators in L-FRS models.

The FRS models described above are all obtained by combining Zadeh fuzzy sets and Pawlak rough sets. However, there are few reports about the combination of axiomatic fuzzy sets and Pawlak rough sets. For this reason, this paper combines axiomatic fuzzy sets with Pawlak rough sets to obtain axiomatic FRS, and further studies the properties and structure of axiomatic FRS. Based on the vague partition of axiomatic fuzzy sets, a similarity description method among objects in the universe is given. The approximation space is composed of fuzzy similarity relations among objects, and the approximation operators in the fuzzy approximation space are further introduced. The basic properties of approximation operators are discussed and some equivalent properties of approximation operators are given.

The remainder of this paper is organized as follows. In Section 2, we review the basic notions of axiomatic fuzzy sets and residual lattices. In Section 3, the properties of operators are discussed. In Section 4, some equivalent properties of approximation operators based on different binary relations are given. In Section 5, we briefly introduce some measures of rough approximation operators in data analysis. Furthermore, we give a algorithm to verify the effectiveness and efficiency of the model. The paper is completed with remarks.

2 Preliminaries

In this section, some basic concepts and properties of axiomatic fuzzy sets and residuated lattices are briefly recalled.

The concept of fuzzy sets was first proposed by Zadeh [1]. So far, it has formed a relatively well-established branch of mathematics, and has been effectively applied in many fields.

Based on the essence and characteristics of vague phenomena, Pan and Xu [7] put forward a new definition of vague partition from the axiomatic system of membership degree, and gave the axiomatic definition of fuzzy set on the basis of vague partition.

Definition 1. [6] Suppose U = [φ, ψ], φ, ψ are real number. A vague partition of U is a collection in the following form $\tilde{U} = {μ_{A_{1}} (p), μ_{A_{2}} (p), \dots, μ_{A_{n}} (p)}, n \in N^{+},$ in which $A_{i} = {(p, μ_{A_{i}} (p)) ∣ p \in U} (i = 1, 2, \dots, n)$ , the degrees of memberships of the element p ∈ U to the class $A_{i}$ is defined by the functions $μ_{A_{i}} : U \to [0, 1]$ , respectively, and satisfy the following conditions:

For every i ∈ n, $μ_{A_{i}} (p)$ is continuous or has finite first-class broken points on U,

For every i ∈ n, at least a point p₀ ∈ U meet that $μ_{A_{i}} (p_{0}) = 1$ ,

For every i ∈ n, if $μ_{A_{i}} (p_{0}) = 1$ for p₀ ∈ U, then $μ_{A_{i}} (p)$ is non-decreasing on [φ, p₀], and is non-increasing on [p₀, ψ],

$0 < μ_{A_{1}} (p) + μ_{A_{2}} (p) + \dots + μ_{A_{n}} (p) \leq 1$ holds for every p ∈ U.

Based on vague partition, Pan and Xu [7] proposed the notion of axiomatic fuzzy sets.

Definition 2. [7] Suppose U = [φ, ψ], φ, ψ are real number and $\tilde{U} = {μ_{A_{1}} (p), μ_{A_{2}} (p)$ , $\dots, μ_{A_{n}} (p)}$ , n ∈ N⁺, is a vague partition of U, $N$ is a strong negation operator on [0, 1], $T$ is a triangular norm, $S$ is a triangular conorm, where $T$ is dual to $S$ with respect to $N$ . The set $F (\tilde{U})$ of fuzzy sets in U with respect to $\tilde{U}$ consists of the following elements:

If there exists i ∈ n such that $μ_{A} (p) = μ_{A_{i}} (p)$ for all p ∈ U, then $A = {(p, μ_{A} (p)) ∣ p \in U} \in F (\tilde{U})$ ,

If $μ_{A} (p) = \bar{μ} (p) = 1$ for all p ∈ U, then $A = {(p, μ_{A} (p)) ∣ p \in U} \in F (\tilde{U})$ ,

If $μ_{A} (p) = \underline{μ} (p) = 0$ for all p ∈ U, then $A = {(p, μ_{A} (p)) ∣ p \in U} \in F (\tilde{U})$ ,

If $A = {(p, μ_{A} (p)) ∣ p \in U} \in F (\tilde{U})$ , r∈ $ℚ^{+}$ , then $A^{r} = {(p, {(μ_{A} (p))}^{r})$ $∣ p \in U} \in F (\tilde{U})$ ,

If $A = {(p, μ_{A} (p)) ∣ p \in U} \in F (\tilde{U})$ , then $A^{N} = {(p, (μ_{A} (p))^{N}) ∣ p \in U} \in F (\tilde{U})$ ,

If $A = {(p, μ_{A} (p)) ∣ p \in U}, B = {(p, μ_{B}$ $(p)) ∣ p \in U} \in F (\tilde{U})$ , then $A \cap_{T} B = {(p, T (μ_{A} (p), μ_{B} (p))) ∣ p \in U} \in F (\tilde{U})$ ,

If $A = {(p, μ_{A} (p)) ∣ p \in U}, B = {(p, μ_{B}$ $(p)) ∣ p \in U} \in F (\tilde{U})$ , then $A \cup_{S} B = {(p, S (μ_{A} (p), μ_{B} (p))) ∣ p \in U} \in F (\tilde{U})$ ,

$F (\tilde{U})$ not contain other elements.

The set of all fuzzy sets in U with respect to $\tilde{U}$ will be denoted by $F (\tilde{U})$ . In fact, all the fuzzy sets in $F (\tilde{U})$ are recorded as axiomatic fuzzy sets.

Next, the related definitions and some basic properties of residuated lattices are recalled. In this paper, we choose the structure of the residuated lattice as L = [0, 1].

Definition 3. [35] A residuated lattice L is a structure $(L, \lor, \land, T, I, 0, 1)$ , where

(L, ∨ , ∧ , 0, 1) is a bounded lattice with the greatest element 1 and the smallest element 0,

$T : L \times L \to L$ satisfies for every φ, ψ, γ ∈ L,

$T (φ, ψ) = T (ψ, φ)$ ,

$T (T (φ, ψ), γ) = T (φ, T (ψ, γ))$ ,

$T (1, φ) = φ$ ,

$φ \leq ψ \Rightarrow T (φ, γ) \leq T (ψ, γ)$ ,

$I : L \times L \to L$ is a residual operation on $T$ , that is, for every φ, ψ, γ ∈ L, there are $T (φ, ψ) \leq γ \Leftrightarrow φ \leq I (ψ, γ) .$

Proposition 1. [35] Suppose $L = (L, \lor, \land, T, I,$ 0, 1) is a residuated lattice. Then the following properties hold

If φ ≤ ψ, then $I (γ, φ) \leq I (γ, ψ), I (φ, γ) \geq I (ψ, γ)$ ,

$T (φ, I (φ, 0)) = 0$ ,

$I (1, φ) = φ$ ,

φ ≤ ψ iff $I (φ, ψ) = 1$ ,

$I (φ, I (ψ, γ)) = I (T (φ, ψ), γ)$ ,

φ ≤ ¬ (¬ (φ)), there $\neg (φ) = I (φ, 0)$ ,

$\neg (\underset{i \in Λ}{\lor} φ_{i}) = \neg i \in Λ \land \neg (φ_{i})$ ,

Suppose L is a complete lattice, then

$T (\underset{i \in Λ}{\lor} φ_{i}, ψ) = \underset{i \in Λ}{\lor} T (φ_{i}, ψ)$ ,

$I (φ, \underset{i \in Λ}{\land} ψ_{i}) = \underset{i \in Λ}{\land} I (φ, ψ_{i}),$ $I (\underset{i \in Λ}{\lor} φ_{i}, ψ) = \underset{i \in Λ}{\land} I (φ_{i}, ψ)$ ,

$I (φ, \underset{i \in Λ}{\lor} ψ_{i}) \geq \underset{i \in Λ}{\lor} I (φ, ψ_{i}),$ $I (\underset{i \in Λ}{\land} φ_{i}, ψ) \geq \underset{i \in Λ}{\lor} I (φ_{i}, ψ)$ .

L is called a regular residuated lattice when the ¬ (¬ (φ)) = φ hold for every φ.

Remark 1. In this paper, let $\cap_{T}$ be a left continuous triangular norm and $\cup_{S}$ be a right continuous triangular conorm.

Remark 2. [36] A negation operator $N$ is one mapping $N : [0, 1] \to [0, 1]$ that satisfies $N (0) = 1, N (1) = 0$ . Specially, $N (p) = 1 - p$ expresses the strong negative operator. The fuzzy complement derived from $N$ is shorthand by the ¬. In short, when $A \in F (\tilde{U})$ , for all p ∈ U, $N (A (p))$ is abbreviated as $(\neg A) (p)$ .

Proposition 2. [35] Suppose $L = (L, \lor, \land, T, I, 0, 1)$ is a regular residuated lattice, for every φ, ψ ∈ L, then the following properties hold

$T (φ, ψ) = \neg (I (φ, \neg (ψ)))$ ,

$I (φ, ψ) = \neg (T (φ, \neg (ψ)))$ ,

$\neg \underset{i \in Λ}{\land} φ_{i} = \neg i \in Λ \lor \neg (φ_{i})$ ,

$I (φ, \neg (ψ)) = I (ψ, \neg (φ)), I (\neg (φ), ψ) = I (\neg (ψ), φ)$ .

$F (\tilde{U})$ is able to be seen as a function space derived from $\tilde{U}$ . For every $A, B, A_{i} \in F (\tilde{U})$ , i ∈ Λ, Λ is an index set. Define the following operations: for every p ∈ U, $\begin{matrix} (A \cap_{T} B) (p) = T (A (p), B (p)) \\ (A \cup_{S} B) (p) = S (A (p), B (p)) \\ I (A, B) (p) = I (A (p), B (p)) \\ (\neg A) (p) = \neg A (p) \\ (\underset{i \in Λ}{\cup_{S}} A_{i}) (p) = \underset{i \in Λ}{S} A_{i} (p) \\ (\underset{i \in Λ}{\cap_{T}} A_{i}) (p) = \underset{i \in Λ}{T} A_{i} (p) \end{matrix}$

$A$ is called a fuzzy subset of $B$ , denoted as $A \subseteq B$ , when $A (p) \leq B (p)$ is true for every p ∈ U.

Definition 4. [20] Suppose that $R$ is a fuzzy binary relation on U. If q ∈ U hold true $R (p, q) = 1$ for every p ∈ U, then $R$ is serial. If $R (p, p) = 1$ hold true for every p ∈ U, then $R$ is reflexive. If $R (p, q) = R (q, p)$ hold true for every p, q ∈ U, then $R$ is symmetric. If it is true for every p, q, m ∈ U, $R (p, m) \geq \underset{q \in U}{\lor} T (R (p, q), R (q, m))$ , then $R$ is transitive. When $R$ satisfies a reflexive, symmetric, transitive, then $R$ is equivalent.

3 Some properties of approximation operators

In this section, most of our definitions will rely on the following common setting.

Assumption 1. Suppose L is a complete residuated lattice, $R$ is a fuzzy relation on U, and $(U, R)$ is a fuzzy approximation space.

First, we recap the notion of fuzzy rough sets. Let us bear in mind that we will provide convenient generalizations of this concept.

Definition 5. [22] Consider the situation of Assumption 1. The fuzzy sets ${\underline{Apr}}_{R}$ and ${\bar{Apr}}_{R}$ defined in the following ways are called $A$ and fuzzy lower approximation and fuzzy upper approximation respectively, ${\underline{Apr}}_{R} (A) (p) = \underset{q \in U}{\land} I (R (p, q), A (q)), {\bar{Apr}}_{R} (A) (p) = \underset{q \in U}{\lor} T (R (p, q), A (q)) .$

If ${\underline{Apr}}_{R} (A) = {\bar{Apr}}_{R} (A)$ , then $A$ is called fuzzy definable, otherwise $A$ is a fuzzy rough set.

Proposition 3. Let $R (p, q) = \lor (A_{i} (p) \land A_{i} (q))$ , $A_{i} \in F (\tilde{U}), i \in Λ$ . If $R$ is reflexive, symmetric, transitive, then $R$ is equivalent.

Proof. Verify that the binary relationship is reflexive, symmetric, transitive.

For every p ∈ U, $R (p, p) = \underset{1 \leq i \leq n}{\lor} (A_{i} (p) \land A_{i} (p)) = \underset{1 \leq i \leq n}{\lor} A_{i} (p) = 1$ ,

For every p, q ∈ U,

$R (p, q) = \underset{1 \leq i \leq n}{\lor} (A_{i} (p) \land A_{i} (q)), R (q, p) = \underset{1 \leq i \leq n}{\lor} (A_{i} (q) \land A_{i} (p))$ ,

$R (p, q) = R (q, p)$ ,

For every p, q, m ∈ U, $\begin{matrix} R (p, q) = \underset{1 \leq i \leq n}{\lor} (A_{i} (p) \land A_{i} (q)), R (q, p) = \underset{1 \leq i \leq n}{\lor} (A_{i} (q) \land A_{i} (p)), \\ \underset{q \in U}{\lor} T (R (p, q), R (q, m)) \leq \underset{q \in U}{\lor} (R (p, q) \land R (q, m)) \\ = \underset{q \in U}{\lor} (\underset{1 \leq i \leq n}{\lor} (A_{i} (p) \land A_{i} (q)) \land \underset{1 \leq i \leq n}{\lor} (A_{i} (q) \land A_{i} (m))) \\ = (\underset{q \in U}{\lor} \underset{1 \leq i \leq n}{\lor} A_{i} (q)) \land (\underset{q \in U}{\lor} \underset{1 \leq i \leq n}{\lor} (A_{i} (p) \land A_{i} (m))) \\ \leq \underset{1 \leq i \leq n}{\lor} (A_{i} (p) \land A_{i} (m)) = R (p, m) . \end{matrix}$

□

Next, we discuss the basic properties of approximation operators by giving some propositions.

Proposition 4. Consider the situation of Assumption 1. For all $A \in F (\tilde{U})$ , the following properties hold

If $R$ is serial, then ${\underline{Apr}}_{R} (A) \subseteq {\bar{Apr}}_{R} (A)$ ,

${\bar{Apr}}_{R} (\emptyset) = \emptyset, {\underline{Apr}}_{R} (U) = U$ ,

If $A \subseteq B$ , then ${\underline{Apr}}_{R} (A) \subseteq {\underline{Apr}}_{R} (B)$ and ${\bar{Apr}}_{R} (A) \subseteq {\bar{Apr}}_{R} (B)$ ,

${\underline{Apr}}_{R} (1_{U - {q}}) (p) = \neg R (p, q) = I (R (p, q), 0)$ .

Proof.

If $R$ is a serial binary relation, there exists one point q′ meet that $R (p, q^{'}) = 1$ for each p ∈ U, $\begin{matrix} {\underline{Apr}}_{R} (A) (p) = & \underset{q \in U}{\land} I (R (p, q), A (q)) \\ = & I (R (p, q^{'}), A (q^{'})) \land \underset{q \neq q^{'}}{\land} I (R (p, q), A (q)) \\ = & A (q^{'}) \land \underset{q \neq q^{'}}{\land} I (R (p, q), A (q)) \leq A (q^{'}) . \end{matrix}$ $\begin{matrix} {\bar{Apr}}_{R} (A) (p) = & \underset{q \in U}{\lor} T (R (p, q), A (q)) \\ = & T (R (p, q^{'}), A (q^{'})) \lor \underset{q \neq q^{'}}{\lor} T (R (p, q), A (q)) \\ = & A (q^{'}) \lor \underset{q \neq q^{'}}{\lor} T (R (p, q), A (q)) \geq A (q^{'}) . \end{matrix}$ Hence ${\underline{Apr}}_{R} (A) \subseteq {\bar{Apr}}_{R} (A)$ .

Easy to prove.

If $A \subseteq B$ , then $A (p) \leq B (p)$ for every p ∈ U, ${\underline{Apr}}_{R} (A) (p) = \underset{q \in U}{\land} I (R (p, q), A (q)) \leq \underset{q \in U}{\land} I (R (p, q), B (q)) = {\underline{Apr}}_{R} (B) (p) .$ Hence ${\underline{Apr}}_{R} (A) \subseteq {\underline{Apr}}_{R} (B)$ . Similarly, we can prove ${\bar{Apr}}_{R} (A) \subseteq {\bar{Apr}}_{R} (B)$ .

$\begin{matrix} {\underline{Apr}}_{R} (1_{U - {q}}) (p) = & \neg m \in U \land I (R (p, m), 1_{U - {q}} (m)) \\ = & \underset{m \in U, m \neq q}{\land} I (R (p, m), 1_{U - {q}} (m)) \land I (R (p, q), 1_{U - {q}} (q)) \\ = & I (R (p, q), 1_{U - {q}} (q)) \\ = & I (R (p, q), 0) = \neg (R (p, q)) . \end{matrix}$

□

Then, we prove that the upper and lower approximation operators are dual through a theorem.

Theorem 5. Consider the situation of Assumption 1. For every $A \in F (\tilde{U})$ , the duality property holds $\begin{matrix} {\underline{Apr}}_{R} (A) = \neg {\bar{Apr}}_{R} (\neg A), {\bar{Apr}}_{R} (A) = \neg {\underline{Apr}}_{R} (\neg A) . \end{matrix}$

Proof. $\begin{matrix} \neg {\bar{Apr}}_{R} (\neg (A)) (p) = & \neg (\underset{q \in U}{\lor} T (R (p, q), \neg (A (q))) \\ = & \underset{q \in U}{\land} \neg (T (R (p, q), \neg (A (q))) (by Proposition 1 (7)) \\ = & \underset{q \in U}{\land} I (R (p, q), A (q)) (by Proposition 2 (2)) \\ = & {\underline{Apr}}_{R} (A) (p) . \end{matrix}$ Hence ${\underline{Apr}}_{R} (A) = \neg {\bar{Apr}}_{R} (\neg (A))$ .

$\begin{matrix} \neg {\underline{Apr}}_{R} (\neg (A)) (p) = & \neg (\underset{q \in U}{\land} I (R (p, q), \neg (A (q)))) \\ = & \underset{q \in U}{\lor} \neg (I (R (p, q), \neg (A (q)))) (by Proposition 2 (3)) \\ = & \underset{q \in U}{\lor} T (R (p, q), A (q)) (by Proposition 2 (1)) \\ = & {\bar{Apr}}_{R} (A) (p) . \end{matrix}$ Hence ${\bar{Apr}}_{R} (A) = \neg {\underline{Apr}}_{R} (\neg (A))$ .□

A t-norm or triangular norm [34] is a mapping $T : [0, 1] \times [0, 1] \to [0, 1]$ that is increasing, associative and commutative, and verifies the boundary condition (which states that for every p ∈ [0, 1], we have $T (p, 1) = p$ ).

Recall that a t-norm $T$ and a t-conorm $S$ satisfy weakened distributivity laws if they satisfy:

Definition 6. [36] A t-norm $T$ and a t-conorm $S$ satisfy weakened distributivity laws if the following conditions hold

$T (p, S (q, m)) \leq S (T (p, q), T (p, m))$ ,

$T (S (p, q), S (p, m)) \leq S (p, T (q, m))$ .

Next, we consider the relationship between implication operator $I$ , t-norm $T$ and t-conorm $S$ :

Definition 7. [36] Suppose $I$ is an implicator, $T$ is a t-norm and $S$ is a t-conorm. Then the following properties hold

$I (p, T (q, m)) \geq T (I (p, q), I (p, m))$ ,

$I (p, S (q, m)) \geq S (I (p, q), I (p, m))$ .

From the above assumptions and propositions, we give some theorems to illustrate the properties of approximation operators.

Theorem 6. Consider the situation of Assumption 1. For every $A, B \in F (\tilde{U})$ and p ∈ U, ${\bar{Apr}}_{R} (A \cup_{S} B) \subseteq {\bar{Apr}}_{R} (A) \cup_{S} {\bar{Apr}}_{R} (B), \underline{{Apr}_{R}} (A \cap_{T} B) \supseteq {\underline{Apr}}_{R} (A) \cap_{T} {\underline{Apr}}_{R} (B) .$

Proof. For every $A, B \in F (\tilde{U})$ and p ∈ U, $\begin{matrix} {\bar{Apr}}_{R} (A \cup_{S} B) (p) = & \underset{q \in U}{\lor} T (R (p, q), (A \cup_{S} B) (q)) \\ = & \underset{q \in U}{\lor} T (R (p, q), A (q) \cup_{S} B (q)) \\ \leq & \neg q \in U \lor (T (R (p, q), A (q)) \cup_{S} T (R (p, q), B (q))) (by Definition 6 (1)) \\ \leq & \underset{q \in U}{\lor} T (R (p, q), A (q)) \cup_{S} \underset{q \in U}{\lor} T (R (p, q), B (q)) \\ = & {\bar{Apr}}_{R} (A) (p) \cup_{S} {\bar{Apr}}_{R} (B) (p) . \end{matrix}$ Hence ${\bar{Apr}}_{R} (A \cup_{S} B) \subseteq {\bar{Apr}}_{R} (A) \cup_{S} {\bar{Apr}}_{R} (B)$ . $\begin{matrix} {\underline{Apr}}_{R} (A \cap_{T} B) (p) = & \underset{q \in U}{\land} I (R (p, q), (A \cap_{T} B) (q)) \\ = & \underset{q \in U}{\land} I (R (p, q), A (q) \cap_{T} B (q)) \\ \geq & \underset{q \in U}{\land} (I (R (p, q), A (q)) \cap_{T} I (R (p, q), B (q))) (by Definition 7 (1)) \\ \geq & \underset{q \in U}{\land} I (R (p, q), A (q)) \cap_{T} \underset{q \in U}{\land} I (R (p, q), B (q)) \\ = & {\underline{Apr}}_{R} (A) (p) \cap_{T} {\underline{Apr}}_{R} (B) (p) . \end{matrix}$ Hence ${\underline{Apr}}_{R} (A \cap_{T} B) \supseteq {\underline{Apr}}_{R} (A) \cap_{T} {\underline{Apr}}_{R} (B)$ .□

Corollary 1. If $\cap_{T}$ is ∧ operation and $\cup_{S}$ is ∨ operation, then

${\bar{Apr}}_{R} (A \cup_{S} B) = {\bar{Apr}}_{R} (A) \cup_{S} {\bar{Apr}}_{R} (B), {\underline{Apr}}_{R} (A \cap_{T} B) = {\underline{Apr}}_{R} (A) \cap_{T} {\underline{Apr}}_{R} (B) .$

Theorem 7. Consider the situation of Assumption 1. For every $A, B \in F (\tilde{U})$ and p ∈ U,

${\underline{Apr}}_{R} (A) \cup_{S} {\underline{Apr}}_{R} (B) \subseteq {\underline{Apr}}_{R} (A \cup_{S} B), {\bar{Apr}}_{R} (A \cap_{T} B) \subseteq {\bar{Apr}}_{R} (A) \cap_{T} {\bar{Apr}}_{R} (B) .$

Proof. $\begin{matrix} {\underline{Apr}}_{R} (A \cup_{S} B) (p) = & \underset{q \in U}{\land} I (R (p, q), (A \cup_{S} B) (q)) \\ = & \underset{q \in U}{\land} I (R (p, q), A (q) \cup_{S} B (q)) \\ \geq & \underset{q \in U}{\land} (I (R (p, q), A (q)) \cup_{S} I (R (p, q), B (q))) (by Definition 7 (2)) \\ \geq & \underset{q \in U}{\land} I (R (p, q), A (q)) \cup_{S} \underset{q \in U}{\land} I (R (p, q), B (q)) \\ = & {\underline{Apr}}_{R} (A) (p) \cup_{S} {\underline{Apr}}_{R} (B) (p) . \end{matrix}$ Hence ${\underline{Apr}}_{R} (A) \cup_{S} {\underline{Apr}}_{R} (B) \subseteq {\underline{Apr}}_{R} (A \cup_{S} B)$ . Similarly, ${\bar{Apr}}_{R} (A \cap_{T} B) \subseteq {\bar{Apr}}_{R} (A) \cap_{T} {\bar{Apr}}_{R} (B)$ can also be proved.□

Theorem 8. Consider the situation of Assumption 1. For every $A_{i} \in F (\tilde{U}), i \in Λ$ and p ∈ U,

${\bar{Apr}}_{R} (\underset{i \in \land}{\cup_{S}} A_{i}) \subseteq \underset{i \in \land}{\cup_{S}} {\bar{Apr}}_{R} (A_{i}), {\underline{Apr}}_{R} (\underset{i \in \land}{\cap_{T}} A_{i}) \supseteq \underset{i \in \land}{\cap_{T}} {\underline{Apr}}_{R} (A_{i}) .$

Proof. It follows immediately from Theorem 6.□

Theorem 9. Suppose L is a complete residuated lattice, then for every fuzzy approximation space $(U, R_{i})$ , i ∈ Λ and Λ is index set. For all $A \in F (\tilde{U})$ , every p ∈ U,

${\underline{Apr}}_{\neg i \in \land \cup R_{i}} (A) = \neg i \in \land \cap {\underline{Apr}}_{R_{i}} (A), {\bar{Apr}}_{\neg i \in \land \cup R_{i}} (A) = \neg i \in \land \cup {\bar{Apr}}_{R_{i}} (A) .$

Proof. For any p ∈ U, $\begin{matrix} {\underline{Apr}}_{\neg i \in \land \cup R_{i}} (A) (p) = & \underset{q \in U}{\land} I (\neg i \in \land \cup R_{i} (p, q), A (q)) \\ = & \underset{q \in U}{\land} I (\neg i \in \land \lor R_{i} (p, q), A (q)) \\ = & \underset{q \in U}{\land} \underset{i \in \land}{\land} I (R_{i} (p, q), A (q)) (by Proposition 1 (8)) \\ = & \underset{i \in \land}{\land} \underset{q \in U}{\land} I (R_{i} (p, q), A (q)) \\ = & (\underset{i \in \land}{\land} {\underline{Apr}}_{R_{i}} (A)) (p) = (\underset{i \in \land}{\cap} {\underline{Apr}}_{R_{i}} (A)) (p) . \end{matrix}$ Hence, ${\underline{Apr}}_{\neg i \in \land \cup R_{i}} (A) = \neg i \in \land \cap {\underline{Apr}}_{R_{i}} (A)$ . Similarly, ${\bar{Apr}}_{\neg i \in \land \cup R_{i}} (A) = \neg i \in \land \cup {\bar{Apr}}_{R_{i}} (A)$ can also be proved.□

Assumption 2.

Suppose $N$ is a strong negation operator on [0, 1],

Suppose r is mood operator, $r \in ℚ^{+}$ .

Proposition 10. Consider the situation of Assumption 2. For every $A, B \in F (\tilde{U})$ , the following properties hold

${(A \cup_{S} B)}^{r} = A^{r} \cup_{S} B^{r}, {(A \cup_{S} B)}^{N} = A^{N} \cap_{T} B^{N}$ ,

${(A \cap_{T} B)}^{r} = A^{r} \cap_{T} B^{r}, {(A \cap_{T} B)}^{N} = A^{N} \cup_{S} B^{N}$ ,

$((A)^{r})^{\frac{1}{r}} = A, ((A)^{N})^{N} = A$ .

Corollary 2. Consider the situation of Assumption 2 and remark 1. For every $A, B, C \in F (\tilde{U})$ , the following properties hold

${((A \cup B) \cap_{T} C)}^{r} = {(A \cap_{T} C)}^{r} \cup {(B \cap_{T} C)}^{r}$ ,

${((A \cup B) \cap_{T} C)}^{N} = {(A \cap_{T} C)}^{N} \cap {(B \cap_{T} C)}^{N}$ ,

${((A \cap B) \cup_{S} C)}^{r} = {(A \cup_{S} C)}^{r} \cap {(B \cup_{S} C)}^{r}$ ,

${((A \cap B) \cup_{S} C)}^{N} = {(A \cup_{S} C)}^{N} \cup {(B \cup_{S} C)}^{N}$ .

Theorem 11. Consider the situation of Assumption 1 and Assumption 2(2). For all $A, B \in F (\tilde{U})$ ,

${\bar{Apr}}_{R} ({(A \cup_{S} B)}^{r}) \subseteq {\bar{Apr}}_{R} (A^{r}) \cup_{S} {\bar{Apr}}_{R} (B^{r}),$ ${\underline{Apr}}_{R} ({(A \cap_{T} B)}^{r}) \supseteq {\underline{Apr}}_{R} (A^{r}) \cap_{T} {\underline{Apr}}_{R} (B^{r}) .$

Proof. For every p ∈ U, $\begin{matrix} {\bar{Apr}}_{R} ({(A \cup_{S} A)}^{r}) (p) = & {\bar{Apr}}_{R} (A^{r} \cup_{S} B^{r}) (p) \\ = & \underset{q \in U}{\lor} T (R (p, q), (A^{r} \cup_{S} B^{r}) (q)) \\ = & \underset{q \in U}{\lor} T (R (p, q), A^{r} (q) \cup_{S} B^{r} (q)) \\ \leq & \underset{q \in U}{\lor} (T (R (p, q), A^{r} (q)) \cup_{S} T (R (p, q), B^{r} (q))) (by Definition 6 (1)) \\ \leq & \underset{q \in U}{\lor} T (R (p, q), A^{r} (q)) \cup_{S} \underset{q \in U}{\lor} T (R (p, q), B^{r} (q)) \\ = & {\bar{Apr}}_{R} (A^{r}) (p) \cup_{S} {\bar{Apr}}_{R} (B^{r}) (p) . \end{matrix}$ Hence ${\bar{Apr}}_{R} ({(A \cup_{S} B)}^{r}) \subseteq {\bar{Apr}}_{R} (A^{r}) \cup_{S} {\bar{Apr}}_{R} (B^{r})$ .

$\begin{matrix} {\underline{Apr}}_{R} ({(A \cap_{T} B)}^{r}) (p) = & {\underline{Apr}}_{R} (A^{r} \cap_{T} B^{r}) (p) \\ = & \underset{q \in U}{\land} I (R (p, q), (A^{r} \cap_{T} B^{r}) (q)) \\ = & \underset{q \in U}{\land} I (R (p, q), A^{r} (q) \cap_{T} B^{r} (q)) \\ \geq & \underset{q \in U}{\land} (I (R (p, q), A^{r} (q)) \cap_{T} I (R (p, q), B^{r} (q))) (by Definition 7 (1)) \\ \geq & \underset{q \in U}{\land} I (R (p, q), A^{r} (q)) \cap_{T} \underset{q \in U}{\land} I (R (p, q), B^{r} (q)) \\ = & {\underline{Apr}}_{R} (A^{r}) (p) \cap_{T} {\underline{Apr}}_{R} (B^{r}) (p) . \end{matrix}$ Hence ${\underline{Apr}}_{R} ({(A \cap_{T} B)}^{r}) \supseteq {\underline{Apr}}_{R} (A^{r}) \cap_{T} {\underline{Apr}}_{R} (B^{r})$ .□

Theorem 12. Consider the situation of Assumption 1 and Assumption 2(2). For all $A, B \in F (\tilde{U})$ ,

${\underline{Apr}}_{R} (A^{r}) \cup_{S} {\underline{Apr}}_{R} (B^{r}) \subseteq {\underline{Apr}}_{R} ({(A \cup_{S} B)}^{r}),$ ${\bar{Apr}}_{R} ({(A \cap_{T} B)}^{r}) \subseteq {\bar{Apr}}_{R} (A^{r}) \cap_{T} {\bar{Apr}}_{R} (B^{r}) .$

Proof. $\begin{matrix} {\underline{Apr}}_{R} ({(A \cup_{S} B)}^{r}) (p) & = {\underline{Apr}}_{R} (A^{r} \cup_{S} B^{r}) (p) \\ = \underset{q \in U}{\land} I (R (p, q), (A^{r} \cup_{S} B^{r}) (q)) \\ = \underset{q \in U}{\land} I (R (p, q), A^{r} (q) \cup_{S} B^{r} (q)) \\ \geq \underset{q \in U}{\land} (I (R (p, q), A^{r} (q)) \cup_{S} I (R (p, q), B^{r} (q))) (by Definition 7 (2)) \\ \geq \underset{q \in U}{\land} I (R (p, q), A^{r} (q)) \cup_{S} \underset{q \in U}{\land} I (R (p, q), B^{r} (q)) \\ = {\underline{Apr}}_{R} (A^{r}) (p) \cup_{S} {\underline{Apr}}_{R} (B^{r}) (p) . \end{matrix}$ Hence ${\underline{Apr}}_{R} (A^{r}) \cup_{S} {\underline{Apr}}_{R} (B^{r}) \subseteq {\underline{Apr}}_{R} (A^{r} \cup_{S} B^{r})$ .

By Proposition 10, ${(A \cup_{S} B)}^{r} = A^{r} \cup_{S} B^{r}$ , Hence ${\underline{Apr}}_{R} (A^{r}) \cup_{S} {\underline{Apr}}_{R} (B^{r}) \subseteq {\underline{Apr}}_{R} ({(A \cup_{S} B)}^{r})$ . Similarly, ${\bar{Apr}}_{R} ({(A \cap_{T} B)}^{r}) \subseteq {\bar{Apr}}_{R} (A^{r}) \cap_{T} {\bar{Apr}}_{R} (B^{r})$ can also be proved.□

Theorem 13. Consider the situation of Assumption 1 and Assumption 2(1). For all $A, B \in F (\tilde{U})$ ,

${\underline{Apr}}_{R} ((A \cup_{S} B)^{N}) \subseteq {\underline{Apr}}_{R} (A^{N}) \cap_{T} {\underline{Apr}}_{R} (B^{N}),$ ${\bar{Apr}}_{R} ((A \cap_{T} B)^{N}) \supseteq {\bar{Apr}}_{R} (A^{N}) \cup_{S} {\bar{Apr}}_{R} (B^{N}) .$

Proof. Same as Theorem 11.□

Theorem 14. Consider the situation of Assumption 1 and Assumption 2(1). For all $A, B \in F (\tilde{U})$ ,

${\underline{Apr}}_{R} (A^{N}) \cup_{S} {\underline{Apr}}_{R} (B^{N}) \subseteq {\underline{Apr}}_{R} ((A \cap_{T} B)^{N}),$ ${\bar{Apr}}_{R} ((A \cup_{S} B)^{N}) \subseteq {\bar{Apr}}_{R} (A^{N}) \cap_{T} {\bar{Apr}}_{R} (B^{N}) .$

Proof. Same as Theorem 12.□

Corollary 3. Consider the situation of Assumption 1 and Assumption 2(2) and remark 1. For every $A, B, C \in F (\tilde{U})$ , the following conditions hold

${\bar{Apr}}_{R} ({((A \cup B) \cap_{T} C)}^{r}) = {\bar{Apr}}_{R} ({(A \cap_{T} C)}^{r}) \cup {\bar{Apr}}_{R} ({(B \cap_{T} C)}^{r})$ ,

${\underline{Apr}}_{R} ({((A \cap B) \cup_{S} C)}^{r}) = {\underline{Apr}}_{R} ({(A \cup_{S} C)}^{r}) \cap {\underline{Apr}}_{R} ({(B \cup_{S} C)}^{r})$ ,

${\underline{Apr}}_{R} ({(A \cap_{T} C)}^{r}) \cup {\underline{Apr}}_{R} ({(B \cap_{T} C)}^{r}) \subseteq {\underline{Apr}}_{R} ({((A \cup B) \cap_{T} C)}^{r})$ ,

${\bar{Apr}}_{R} ({((A \cap B) \cup_{S} C)}^{r}) \subseteq {\bar{Apr}}_{R} ({(A \cup_{S} C)}^{r}) \cap {\bar{Apr}}_{R} ({(B \cup_{S} C)}^{r})$ .

Corollary 4. Consider the situation of Assumption 1 and Assumption 2(1) and remark 1. For every $A, B, C \in F (\tilde{U})$ , the following conditions hold

${\underline{Apr}}_{R} (((A \cup B) \cap_{T} C)^{N}) = {\underline{Apr}}_{R} ((A \cap_{T} C)^{N}) \cap {\underline{Apr}}_{R} ((B \cap_{T} C)^{N})$ ,

${\bar{Apr}}_{R} (((A \cap B) \cup_{S} C)^{N}) = {\bar{Apr}}_{R} ((A \cup_{S} C)^{N}) \cup {\bar{Apr}}_{R} ((B \cup_{S} C)^{N})$ ,

${\bar{Apr}}_{R} (((A \cup B) \cap_{T} C)^{N}) \subseteq {\bar{Apr}}_{R} ((A \cap_{T} C)^{N}) \cap {\bar{Apr}}_{R} ((B \cap_{T} C)^{N})$ ,

${\underline{Apr}}_{R} ((A \cup_{S} C)^{N}) \cup {\underline{Apr}}_{R} ((B \cup_{S} C)^{N}) \subseteq {\underline{Apr}}_{R} (((A \cap B) \cup_{S} C)^{N})$ .

4 Equivalent properties of rough approximation operators

Suppose U is a nonempty set, L is a complete residuated lattice, R and V are two fuzzy relations on U. Define their composition $R \circ V : U \times U \to L$ as: $R \circ V (p, m) = \underset{q \in U}{\lor} T (V (p, q), R (q, m))$ , for every p, m ∈ U,

Proposition 15. Suppose $R$ and $V$ are two fuzzy relations on U. For every $A \in F (\tilde{U})$ ,

${\underline{Apr}}_{R \circ V} (A) = {\underline{Apr}}_{V} ({\underline{Apr}}_{R} (A)), {\bar{Apr}}_{R \circ V} (A) = {\bar{Apr}}_{V} ({\bar{Apr}}_{R} (A)) .$

Proof. For every p ∈ U, $\begin{matrix} {\underline{Apr}}_{V} ({\underline{Apr}}_{R} (A)) (p) = & \underset{q \in U}{\land} I (V (p, q), {\underline{Apr}}_{R} (A) (q)) \\ = & \underset{q \in U}{\land} I (V (p, q), \underset{m \in U}{\land} I (R (q, m), A (m))) \\ = & \underset{q \in U}{\land} \underset{m \in U}{\land} I (V (p, q), I (R (q, m), A (m))) (by Proposition 1 (8)) \\ = & \underset{q \in U}{\land} \neg m \in U \land I (T (V (p, q), R (q, m)), A (m)) . (by Proposition 1 (5)) \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{R \circ V} (A) (p) = & \underset{q \in U}{\land} I (R \circ V (p, q), A (q)) \\ = & \underset{q \in U}{\land} I (\underset{m \in U}{\lor} T (V (p, m), R (m, q)), A (q)) \\ = & \underset{q \in U}{\land} \neg m \in U \land I (T (V (p, m), R (m, q)), A (q)) (by Proposition 1 (8)) \\ = & \underset{q \in U}{\land} \neg m \in U \land I (T (V (p, q), R (q, m)), A (m)) . \end{matrix}$ Hence ${\underline{Apr}}_{R \circ V} (A) = {\underline{Apr}}_{V} ({\underline{Apr}}_{R} (A))$ . Similarly, ${\bar{Apr}}_{R \circ V} (A) = {\bar{Apr}}_{V} ({\bar{Apr}}_{R} (A))$ can also be proved.□

On the basis of the definition of approximation operators. We investigate the properties of approximation operators by giving some equivalent propositions.

Consider the situation of Assumption 1. For any $A \in F (\tilde{U})$ , the following equivalent properties hold:

Proposition 16. $R$ is serial if and only if the following properties hold

${\underline{Apr}}_{R} (\emptyset) = \emptyset$ ,

${\bar{Apr}}_{R} (U) = U$ ,

For every $A \in F (\tilde{U})$ , then ${\underline{Apr}}_{R} (A) \subseteq {\bar{Apr}}_{R} (A)$ .

Proposition 17. $R$ is reflexive if and only if the following properties hold

For every $A \in F (\tilde{U})$ , then ${\underline{Apr}}_{R} (A) \subseteq A$ ,

For every $A \in F (\tilde{U})$ , then $A \subseteq {\bar{Apr}}_{R} (A)$ .

Proposition 18. $R$ is symmetric if and only if the following properties hold

For every $A \in F (\tilde{U})$ , then ${\bar{Apr}}_{R} ({\underline{Apr}}_{R} (A)) \subseteq A$ ,

For every $A \in F (\tilde{U})$ , then $A \subseteq {\underline{Apr}}_{R} ({\bar{Apr}}_{R} (A))$ .

Proposition 19. $R$ is transitive if and only if the following properties hold

For every $A \in F (\tilde{U})$ , then $\underline{{Apr}_{R}} (A) \subseteq {\underline{Apr}}_{R} ({\underline{Apr}}_{R} (A))$ ,

For every $A \in F (\tilde{U})$ , ${\bar{Apr}}_{R} ({\bar{Apr}}_{R} (A)) \subseteq {\bar{Apr}}_{R} (A)$ .

Proposition 20. If $R$ is reflexive and transitive, then the following properties hold

For every $A \in F (\tilde{U})$ , then ${\underline{Apr}}_{R} (A) = {\underline{Apr}}_{R} ({\underline{Apr}}_{R} (A))$ ,

For every $A \in F (\tilde{U})$ , then ${\bar{Apr}}_{R} (A) = {\bar{Apr}}_{R} ({\bar{Apr}}_{R} (A))$ .

Proposition 21. $R$ is Euclidean if and only if the following properties hold

For every $A \in F (\tilde{U})$ , then ${\bar{Apr}}_{R} ({\underline{Apr}}_{R} (A)) \subseteq {\underline{Apr}}_{R} (A)$ ,

For every $A \in F (\tilde{U})$ , then ${\bar{Apr}}_{R} (A) \subseteq {\underline{Apr}}_{R} ({\bar{Apr}}_{R} (A))$ .

5 Application of axiomatic FRS in information system

In this section, we introduce some commonly used measurement methods and further discuss the application of axiomatic FRS in processing data sets. In addition, we give an algorithm to verify the effectiveness and efficiency of the model.

5.1 Application of axiomatic FRS in information system

The notion of information system has been studied by many authors as a simple knowledge representation method. In this subsection, we briefly review the definition of an information system. Furthermore, we give an algorithm to verify the effectiveness and efficiency of the model.

Definition 8. [37] An information system is a four-tuple $(U, A, V, f)$ , where $U = {x_{1}, x_{2}, \dots, x_{m}}$ is a finite non-empty set of objects called universe, A = {a₁, a₂, …, a_n} is a non-empty finite set of attributes, V = ∪ _a∈AV_a and V_a is the domain of attribute a, $f : U \times P \to V$ is the information function of information system, $f (x, a) \in V_{a} (\forall x \in U, a \in A)$ . More specially, $(U, A \cup {d}, V, f)$ is called a decision system.

For brevity, a decision system is denoted by $M = (U, A \cup {d})$ .

Definition 9. [38] Let $M = (U, A \cup {d})$ be a decision system. For any B ⊆ A, The fuzzy lower and upper approximations of the fuzzy decision d with respect to B are defined as $\begin{matrix} {\underline{Apr}}_{R} (d) (x) = \underset{y \in U}{\land} I (R (x, y), d (y)), \\ {\bar{Apr}}_{R} (d) (x) = \underset{y \in U}{\lor} T (R (x, y), d (y)) . \end{matrix}$

Definition 10. [38] Let $M = (U, A \cup {d})$ be a decision system. For any B ⊆ A, $x \in U$ , the fuzzy approximation accuracy of d with respect to B is defined as $Acc = \frac{| {\underline{Apr}}_{R} d (x) |}{| {\bar{Apr}}_{R} d (x) |} .$

Apparently, 0 ≤ Acc ≤ 1.

Example 1. From Table 1, $U = {x_{1}, x_{2}, x_{3}}$ , the fuzzy decision attribute d can induce a fuzzy set on $U$ such as $d = \frac{0.5}{x_{1}} + \frac{0.2}{x_{2}} + \frac{0.8}{x_{3}} .$

Table 1

$U$ a ₁ a ₂ a ₃ d

x ₁ 0.3 1 0.5 0.5

x ₂ 1 0.2 0.3 0.2

x ₃ 0.8 0.6 1 0.8

$U$	a ₁	a ₂	a ₃	d
x ₁	0.3	1	0.5	0.5
x ₂	1	0.2	0.3	0.2
x ₃	0.8	0.6	1	0.8

Furthermore, an example is given to calculate the fuzzy approximation precision of d with respect to A.

Example 2. (Continuation of Example 1) By using Proposition 3, the fuzzy relation $R$ with respect to A is computed as $\begin{matrix} R = [\begin{matrix} 1 & 0.3 & 0.6 \\ 0.3 & 1 & 0.8 \\ 0.6 & 0.8 & 1 \end{matrix}] \end{matrix}$

By Definition 9, $\begin{matrix} {\underline{Apr}}_{R} (d) (x) = \\ {\underline{Apr}}_{R} (d) (x_{1}) + {\underline{Apr}}_{R} (d) (x_{2}) + {\underline{Apr}}_{R} (d) (x_{3}) \\ = & \frac{0.5}{x_{1}} + \frac{0.2}{x_{2}} + \frac{0.2}{x_{3}}, \end{matrix}$ $\begin{matrix} {\bar{Apr}}_{R} (d) (x) = \\ {\bar{Apr}}_{R} (d) (x_{1}) + {\bar{Apr}}_{R} (d) (x_{2}) + {\bar{Apr}}_{R} (d) (x_{3}) \\ = & \frac{0.6}{x_{1}} + \frac{0.8}{x_{2}} + \frac{0.8}{x_{3}} . \end{matrix}$

Therefore, $\sum {\underline{Apr}}_{R} (d) (x) = 0.9$ , $\sum {\bar{Apr}}_{R} (d) (x) = 2.2$ , then $Acc = \frac{0.9}{2.2} = 0.409$ .

It can be seen from Definition 9 that if we want to obtain the fuzzy lower and upper approximations in Axiomatic FRS, we must first acquire the fuzzy relation $R$ . Next, according to the above discussion, we design a static algorithm (Algorithm 1) to calculate the fuzzy approximations in AxiomaticFRS.

5.2 Some measures of rough approximation operators in data analysis

In this subsection, we introduce some commonly used metrics.

Algorithm 1: A static algorithm for calculating the fuzzy approximations in Axiomatic FRS
Input: $(U, A \cup {d}, V, f)$ .
Output:Acc.
1: for i=1 to $\| U \|$
2: for j=1 to $\| U \|$
3: calculate $R$ according to Proposition 3;
4: end for
5: end for
6: for i=1 to $\| U \|$
7: for j=1 to $\| U \|$
8: calculate ${\underline{Apr}}_{R} d (x), {\underline{Apr}}_{R} d (x)$ according to Definition 9;
9: end for
10: end for
11: return $Acc = \frac{\| {\underline{Apr}}_{R} d (x) \|}{\| {\bar{Apr}}_{R} d (x) \|}$ .

Consider the situation of Assumption 1. For every $A \in F (\tilde{U})$ , the approximate mass about $A$ is denoted by $θ_{R} (A) = \frac{| {\underline{Apr}}_{R} (A) |}{| U |} .$

Approximate quality reflects the percentage of the knowledge $A$ that must be in the knowledge base in the existing knowledge. The approximate precision of $A$ with respect to $(U, R)$ is denoted by $σ_{R} (A) = \frac{| {\underline{Apr}}_{R} (A) |}{| {\bar{Apr}}_{R} (A) |} .$ It reflects the level of understanding of $A$ based on existing knowledge.

The roughness measure of $A$ with respect to $(U, R)$ is denoted by $ρ_{R} (A) = 1 - \frac{| {\underline{Apr}}_{R} (A) |}{| {\bar{Apr}}_{R} (A) |} .$ Obviously, $0 \leq ρ_{R} (A) \leq 1$ . $A$ is definable if and only if $ρ_{R} (A) = 0$ . The roughness measure reflects the incomplete degree of knowledge.

RST also defines the upper approximation and the lower approximation for the approximation space of function classes, so the theory can be applied to classification problems. Suppose $F = {A_{1}, A_{2}, \dots, A_{n}}$ is a set class composed of subsets of U. Then the lower approximation ${\underline{Apr}}_{R} (F)$ and the upper approximation ${\bar{Apr}}_{R} (F)$ of $F$ with respect to the approximation space $(U, R)$ are denoted by $\begin{matrix} {\underline{Apr}}_{R} (F) = {{\underline{Apr}}_{R} (A_{1}), {\underline{Apr}}_{R} (A_{2}), \dots, {\underline{Apr}}_{R} (A_{n})}, \\ {\bar{Apr}}_{R} (F) = {{\bar{Apr}}_{R} (A_{1}), {\bar{Apr}}_{R} (A_{2}), \dots, {\bar{Apr}}_{R} (A_{n})} \end{matrix}$ In this case, the approximate precision $σ_{R} (F)$ and approximate mass $θ_{R} (F)$ of $F$ with respect to $(U, R)$ are denoted by $σ_{R} (F) = \frac{| ⋃_{i = 1}^{n} {\underline{Apr}}_{R} (A_{i}) |}{| ⋃_{i = 1}^{n} {\bar{Apr}}_{R} (A_{i}) |},$ $θ_{R} (F) = \frac{| ⋃_{i = 1}^{n} {\underline{Apr}}_{R} (A_{i}) |}{| U |} .$

In particular, when $F$ is also the partition of U, the approximate quality of $F$ with respect to $(U, R)$ has important applications in judging whether a decision table is coordinated in rule extraction.

6 Conclusions

This paper discusses the rough set model of axiomatic fuzzy sets. Based on vague partition, a similarity description method among objects in the universe is given. The approximation space is composed of fuzzy similarity relations among objects. The concept of uncertainty described by approximation operators in the fuzzy approximation space is further proposed. Some basic properties of upper and lower approximation operators are discussed, and some equivalent descriptions of approximation operators are given. In future studies, we will further discuss the topological structure of the model and its compactness, countability, separability and other topological properties.

Footnotes

Acknowledgment

This work has been partially supported by the National Natural Science Foundation of China (Grant No. 61976130; 12271319).

Declarations

Competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Rough set model based on axiomatic fuzzy set

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Some properties of approximation operators

4 Equivalent properties of rough approximation operators

5 Application of axiomatic FRS in information system

5.1 Application of axiomatic FRS in information system

Table 1 U a 1 a 2 a 3 d x 1 0.3 1 0.5 0.5 x 2 1 0.2 0.3 0.2 x 3 0.8 0.6 1 0.8

6 Conclusions

Footnotes

Acknowledgment

Declarations

Competing interest

References

Table 1

$U$ a ₁ a ₂ a ₃ d

x ₁ 0.3 1 0.5 0.5

x ₂ 1 0.2 0.3 0.2

x ₃ 0.8 0.6 1 0.8