Attributes reduction based on intuitionistic fuzzy rough sets

Abstract

Intuitionistic fuzzy rough sets are the generalization of traditional rough sets by combining intuitionistic fuzzy set theory and rough set theory. The existing researches on intuitionistic fuzzy rough sets mainly concentrate on the establishment of lower and upper approximation operators by using constructive and axiomatic approaches. Less effort has been put on the attributes reduction of databases based on intuitionistic fuzzy rough sets. The aim of this paper is to focus on attributes reduction based on intuitionistic fuzzy rough sets. After reviewing attributes reduction with traditional rough sets, some equivalent conditions to describe the relative reduction with intuitionistic fuzzy rough sets are proposed, and the structure of reduction is completely examined. Based on discernibility matrix, an algorithm to compute all the attributes reductions is also developed. At last we employ an example to illustrate the concepts of attributes reduction.

Keywords

Intuitionistic fuzzy rough set rough set intuitionistic fuzzy set attributes reduction discernibility matrix

1 Introduction

The concept of the rough set was originally proposed by Pawlak [14] as a formal tool for modelling and processing incomplete information in information systems. The successful application of rough set theory in a variety of problems has been amply demonstrated its usefulness.

Atanassov [1, 2] firstly introduced the concept of intuitionistic fuzzy (IF, for short) set characterized by a membership function and a non-membership function, which is a natural generalization of Zadeh’s fuzzy set [30]. Since intuitionistic fuzzy sets take into account the three aspects of a membership degree, a non-membership degree and a hesitancy degree, compared with fuzzy sets, intuitionistci fuzzy sets is more accurate to describe and characterize the nature of the ambiguity of the objective world. Intuitionistic fuzzy set theory has been successfully applied to many practical fields, such as decision making [13 , 25–29], logic programming [3], medical diagnosis [7, 25], pattern recognition [11 , 23], machine learning and market prediction [24], etc. Many scholars have paid their attentions to combining IF set theory and rough set theory [4 , 15–17]. For example, by employing an approximation space constituted from an IF triangular norm $T$ , an IF $S$ -implicator $I$ , and an IF $T$ -equivalence relation, Cornelis et al. [5] defined the concept of IF rough sets. In [31], Zhou and Wu explored a general framework for the study of relation-based IF rough approximation operators when the IF triangular norm $T = min$ , IF triangular conorm $S = max$ , and IF negator $N = N_{S}$ . Furthermore, Zhou et al. [32] presented a general framework for the study of relation-based $(I, T)$ -intuitionistic fuzzy rough sets by employing IF logical operators, which are perhaps one of the most generalized IF rough approximation constructed on the basis of IF relations until now.

It is well known that any generalization of traditional rough set theory should address two important theoretical issues. The first one is to present reasonabledefinitions of set approximation operators, and the second one is to develop reasonable algorithms for attributes reduction. The concept of attributes reduction can be viewed as the strongest and most important results in rough sets theory to distinguish itself from other theories. It should be noted that the existing intuitionistic fuzzy rough sets mainly pay attention to constructing approximation operators. The study for the attributes reduction of IF rough sets is still blank. This paper provides a systematic study on attribute reduction with intuitionistic fuzzy rough sets. The structure of reduction is completely investigated and an algorithm using discernibility matrix to find all the attributes reductions is proposed.

The rest of the paper is organized as follows. Section 2 mainly reviews basic content about attributes reduction and intuitionistic fuzzy rough sets. In Section 3, we introduce the concepts of attributes reduction with intuitionistic fuzzy rough sets in detail, and develop an algorithm using discernibility matrix to compute all the attributes reductions. In the sequel, an illustrated example is proposed in Section 4. The paper is then concluded by a summarizing remark in Section 5.

2 Preliminaries

2.1 Rough sets attributes reduction

The following basic concepts about Pawlak’s rough sets can be found in [14, 18].

An information system is a pair A = (U, A), where U ={ x₁, x₂, …, x_n } is a nonempty finite set of objects and A ={ a₁, a₂, …, a_m } is a nonempty finite set of attributes. With every subset of attributes B ⊆ A we associate a binary relation IND (B), called B-indiscernibility relation, and defined as IND (B) ={ (x, y) ∈ U × U : a (x) = a (y) , ∀ a ∈ B }. IND (B) is obviously an equivalence relation and IND (B) = ∩ _a∈BIND ({ a }). By [x] _B we denote the equivalence class of IND (B) including x. For any subset X ⊆ U, $\underline{B} (X) = {x \in U : {[x]}_{B} \subseteq U}$ and $\bar{B} (X) = {x \in U : {[x]}_{B} \cap U \neq ø}$ are called B-lower and B-upper approximations of X in A, respectively.

By M (A) we denote a n × n matrix (c_ij), called the discernibility matrix of A, such that c_ij ={ a ∈ A : a (x_i) ≠ a (x_j) } for i, j = 1, 2, …, n. A discernibility function f (A) for an information system A = (U, A) is a Boolean function of m Boolean variables $\bar{a_{1}}, \bar{a_{2}}, \dots, \bar{a_{m}}$ corresponding to the attributes a₁, a₂, …, a_m, respectively, and defined as $f (A) (\bar{a_{1}}, \bar{a_{2}}, \dots, \bar{a_{m}}) = \land {\lor (c_{ij}) : 1 \leq j < i \leq n}$ where ∨ (c_ij) is the disjunction of all variables $\bar{a}$ such that a ∈ c_ij.

An attribute a ∈ B ⊆ A is superfluous in B if IND (B) = IND (B - { a }), otherwise a is indispensable in B.

The collection of all indispensable attributes in A is called the core of A. We say that B ⊆ A is independent in A if every attribute in B is indispensable in B. B ⊆ A is called a reduction in A if B is independent and IND (B) = IND (A). The set of all the reductions in A is denoted as Red (A). Let g (A) be the reduced disjunctive form of f (A) obtained from f (A) by applying the multiplication and absorption laws, then there exist l and X_k ⊆ A for k = 1, 2, …, l such that g (A) = (∧ X₁) ∨ (∧ X₂) ∨ ⋯ ∨ (∧ X_l) where each element in X_k appears only one time. We have Red (A) ={ X₁, …, X_l }.

A decision system is a pair A^* = (U, A ∪ { a^* }), where a^* is the decision attribute, A is condition attribute set. We say a ∈ B ⊆ A is relatively dispensable in B if POS_B (a^*) = POS_B-{a} (a^*), otherwise a is said to be relatively indispensable in B, where POS_B (a^*) is the union of B-lower approximation of all the equivalence classes induced by a^*, i.e., ${POS}_{B} (a^{*}) = \cup_{X \in U / a^{*}} \underline{B} (X)$ . If every attribute in B is relatively indispensable in B, we say that B ⊆ A is relatively independent in A^*. B ⊆ A is called a relative reduction in A^* if B is relatively independent in A^* and POS_B (a^*) = POS_A (a^*). The collection of all relatively indispensable attributes in A is called the relative core of A^*.

Suppose M (A^*) = (c_ij). We denote a matrix M (A^*) = (c_ij) in the following way:

(1) c_ij = c_ij -{ a^* }, if (a^* ∈ c_ij and x_i, x_j ∈ POS_A (a^*)) or pos (x_i) ≠ pos (x_j);

(2) c_ij =ø, otherwise.

Here pos : U→ { 0, 1 } is defined as pos (x) = 1 if and only if x ∈ POS_A (a^*). All the relative reductions can be computed in an analogous way as reductions of M (A).

2.2 Intuitionistic fuzzy rough sets

This subsection mainly reviews some basic notions of intuitionisitc fuzzy sets and intuitionistic fuzzy rough sets to be used in the following sections. In the following discussion, the universe of discourse U is always considered to be finite and nonempty.

2.2.1 Intuitionistic fuzzy sets

Definition 2.1. [1] Let U be an ordinary nonempty set. An IF set A in U is an object having the form $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 | x \in U},$ where μ_A : U → [0, 1] and ν_A : U → [0, 1] satisfy 0 ≤ μ_A (x) + ν_A (x) ≤ 1 for all x ∈ U.

μ_A (x) and ν_A (x) are, respectively, called the degree of membership and the degree of non-membership of the element x ∈ U to A.

The complement of an IF set A is denoted by co (A) ={ 〈 x, ν_A (x) , μ_A (x) 〉 |x ∈ U }.

Let IF (U) denote the family of all IF sets in U.

Some basic operations on IF (U) are defined as follows: $\forall A, B \in IF (U),$

A ⊆ B if and only if (iff) μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) for all x ∈ U, $A \supseteq B iff B \subseteq A,$

A = B iff A ⊆ B and B ⊆ A, i.e., μ_A (x) = μ_B (x) and ν_A (x) = ν_B (x) for all x ∈ U, $\begin{matrix} A \cap B = {〈 x, μ_{A} (x) \land μ_{B} (x), ν_{A} (x) \lor ν_{B} (x) 〉 | x \in U}, \\ A \cup B = {〈 x, μ_{A} (x) \lor μ_{B} (x), ν_{A} (x) \land ν_{B} (x) 〉 | x \in U} . \end{matrix}$

Definition 2.2. [32] An IF relation R on U is an IF subset of U × U, namely, R is given by $R = {〈 (x, y), μ_{R} (x, y), ν_{R} (x, y) 〉 | (x, y) \in U \times U},$ where μ_R : U × U → [0, 1] and ν_R : U × U → [0, 1] satisfy the condition 0 ≤ μ_R (x, y) + ν_R (x, y) ≤ 1 for any (x, y) ∈ U × U. We denote the set of all IF relations on U by IF (U × U).

2.2.2 Intuitionistic fuzzy logical operators

In this subsection, we review a special lattice on [0, 1] × [0, 1] (where [0, 1] is the unit interval) and its logical operations originated by Cornelis et al. [6].

Definition 2.3. [6] Denote L = { (x₁, x₂) ∈ [0, 1] × [0, 1] |x₁+ x₂ ≤ 1 }. We define a relation ≤_L on L as follows: $\begin{matrix} \forall (x_{1}, x_{2}), (y_{1}, y_{2}) \in L, \\ (x_{1}, x_{2}) \leq_{L} (y_{1}, y_{2}) \Leftrightarrow x_{1} \leq y_{1} and x_{2} \geq y_{2} . \end{matrix}$

Then the relation ≤_L is a partial ordering on L and the pair (L, ≤ _L) is a complete lattice with the smallest element 0_L = (0, 1) and the greatest element 1_L = (1, 0). The meet operator ∧ and the join operator ∨ on (L, ≤ _L) which are linked to the ordering ≤_L are, respectively, defined as follows: $\begin{matrix} \forall (x_{1}, x_{2}), (y_{1}, y_{2}) \in L, \\ (x_{1}, x_{2}) \land (y_{1}, y_{2}) = (min (x_{1}, y_{1}), max (x_{2}, y_{2})), \\ (x_{1}, x_{2}) \lor (y_{1}, y_{2}) = (max (x_{1}, y_{1}), min (x_{2}, y_{2})) . \end{matrix}$

Meanwhile, we introduce the other relations on L as follows: $\begin{matrix} \forall (x_{1}, x_{2}), (y_{1}, y_{2}) \in L, \\ (x_{1}, x_{2}) \geq_{L} (y_{1}, y_{2}) \Leftrightarrow (y_{1}, y_{2}) \leq_{L} (x_{1}, x_{2}), \\ (x_{1}, x_{2}) = (y_{1}, y_{2}) \Leftrightarrow x_{1} = y_{1} and x_{2} = y_{2}, \\ (x_{1}, x_{2}) <_{L} (y_{1}, y_{2}) \Leftrightarrow \\ (x_{1}, x_{2}) \leq_{L} (y_{1}, y_{2}) and (x_{1}, x_{2}) \neq (y_{1}, y_{2}) \\ (x_{1}, x_{2}) >_{L} (y_{1}, y_{2}) \Leftrightarrow \\ (x_{1}, x_{2}) \geq_{L} (y_{1}, y_{2}) and (x_{1}, x_{2}) \neq (y_{1}, y_{2}) \end{matrix}$

Definition 2.4. [6] An IF negator on L is a decreasing mapping $N : L \to L$ satisfying $N (0_{L}) = 1_{L}$ and $N (1_{L}) = 0_{L}$ . The negator $N_{S} (x_{1}, x_{2}) = (x_{2}, x_{1})$ is usually referred to as the standard IF negator. An IF negator $N$ is called involutive if $N (N (x)) = x$ for all x ∈ L; it is called weakly involutive if $N (N (x)) \geq_{L} x$ for all x ∈ L.

Definition 2.5. [6] An IF triangular norm (IF t-norm, for short) on L is an increasing, commutative, associative mapping $T : L \times L \to L$ satisfying $T (1_{L}, x) = x$ for all x ∈ L.

Definition 2.6. [6] An IF triangular conorm (IF t-conorm, for short) on L is an increasing, commutative, associative mapping $S : L \times L \to L$ satisfying $S (0_{L}, x) = x$ for all x ∈ L.

Obviously, the greatest IF t-norm (respectively, the smallest IF t-conorm) with respect to (w.r.t., in short) the ordering ≤_L is min (respectively, max), defined by min(x, y) = x ∧ y (respectively, max(x, y) = x ∨ y) for all x, y ∈ L.

Definition 2.7. [6] A mapping $I : L \times L \to L$ is referred to as an IF implicator on L if it is decreasing in its first component (left monotonicity), increasing in its second component (right monotonicity), and satisfies following conditions: $I (0_{L}, 0_{L}) = 1_{L}$ , $I (1_{L}, 0_{L}) = 0_{L}$ , $I (0_{L}, 1_{L}) = 1_{L}$ , $I (1_{L}, 1_{L}) = 1_{L}$ .

Remark 2.1. According to the left monotonicity of $I$ , it is easy to prove that $I ((α, β), 1_{L}) = 1_{L}$ for all (α, β) ∈ L, similarly, by the right monotonicity of $I$ , we can conclude that $I (0_{L}, (α, β)) = 1_{L}$ for all (α, β) ∈ L.

Definition 2.8. [6] Let $T$ be an IF t-norm on L. An IF residual implicator (R-implicator, in short) generated by $T$ is a mapping $I_{T}$ defined as follows: $I_{T} (x, y) = sup {γ \in L | T (x, γ) \leq_{L} y}, \forall x, y \in L .$

Some continuous IF t-norms are listed as follows [6]:

the standard min operator $T_{Min} (x, y) = min {x, y}$ (the largest IF t-norm),

$T (x, y) = (max (0, x_{1} + y_{1} - 1), min (1, x_{2} + y_{2}))$ ,

the Łukasiewicz IF t-norm $T_{W} (x, y) = (max (0, x_{1} + y_{1} - 1), min (1, x_{2} + 1 - y_{1}, y_{2} + 1 - x_{1}))$ .

Some IF R-implicators are listed as follows [6]:

(1) $I_{Min} (x, y) = {\begin{matrix} 1_{L}, if x_{1} \leq y_{1} and x_{2} \geq y_{2} \\ (1 - y_{2}, y_{2}) if x_{1} \leq y_{1} and x_{2} < y_{2} \\ (y_{1}, 0) if x_{1} > y_{1} and x_{2} \geq y_{2} \\ (y_{1}, y_{2}) if x_{1} > y_{1} and x_{2} < y_{2} \end{matrix}$ , based on $T_{Min}$ ;

(2) $I_{T} (x, y) = (min (1, 1 + y_{1} - x_{1}, 1 + x_{2} - y_{2}), max (0, y_{2} - x_{2}))$ based on $T (x, y) = (max (0, x_{1} + y_{1} - 1), min (1, x_{2} + y_{2}))$ ;

(3) $I_{T_{W}} (x, y) = (min (1, 1 + y_{1} - x_{1}, 1 + x_{2} - y_{2}), max (0, y_{2} + x_{1} - 1))$ based on $T_{W}$ .

Definition 2.9. [32] Let R ∈ IF (U × U). We say that R

Reflexive if R (x, x) = 1_L for all x ∈ U,

Symmetric if for all (x, y) ∈ U × U, R (x, y) = R (y, x),

$T$ -Transitive if $\lor_{y \in U} T (R (x, y), R (y, z)) \leq_{L} R (x, z)$ for all (x, z) ∈ U × U.

If an IF relation R on U is reflexive and symmetric, then R is called an IF tolerance relation. If an IF relation R is reflexive and $T$ -transitive, then R is called an IF $T$ -preordering. A symmetric IF $T$ -preordering relation R is called an IF $T$ -similarity relation.

The similarity class [x] _R (intuitionistic fuzzy equivalence class) with x ∈ U is an IF set on U defined by [x] _R (y) = R (x, y) for all y ∈ U.

The collection of all intuitionistic fuzzy similarity classes can be denoted as U/ R.

Proposition 2.1. Let $T$ be a continuous IF t-norm, $I$ the IF R-implicator based on $T$ , and $N = N_{I}$ . The following properties hold true.

For any a, b, c ∈ L and each index I with a_i, b_i ∈ L, i ∈ I:

$T (a, 0_{L}) = 0_{L}, T (a, b) \leq_{L} b$ ,

$T (a, I (a, b)) \leq_{L} b, b \leq_{L} I (a, T (a, b)), a \leq_{L} I (I (a, b), b)$

$a \leq_{L} b \Leftrightarrow I (a, b) = 1_{L}$ ,

$I (1_{L}, a) = a$ ,

$I (T (a, b), c) = I (a, I (b, c))$ ,

$T (a, I (b, c)) \leq_{L} I (I (a, b), c)$ ,

$a \leq_{L} N (N (a))$ ,

$N (\underset{i \in I}{\lor} a_{i}) = \underset{i \in I}{\land} N (a_{i})$ ,

$T (a, \underset{i \in I}{\lor} b_{i}) = \underset{i \in I}{\lor} T (a, b_{i})$ ,

$I (a, \underset{i \in I}{\land} b_{i}) = \underset{i \in I}{\land} I (a, b_{i})$ ,

$I (\underset{i \in I}{\lor} a_{i}, b) = \underset{i \in I}{\land} I (a_{i}, b)$ ,

$\underset{i \in I}{\lor} I (a_{i}, b) \leq_{L} I (\underset{i \in I}{\land} a_{i}, b)$ ,

$\underset{i \in I}{\lor} I (a_{i}, b_{i}) \leq_{L} I (\underset{i \in I}{\land} a_{i}, \underset{i \in I}{\lor} b_{i}), \underset{i \in I}{\lor} I (a, b_{i}) \leq_{L} I (a, \underset{i \in I}{\lor} b_{i})$ ,

$\underset{i \in I}{\lor} T (a_{i}, a_{i}) = T (\underset{i \in I}{\lor} a_{i}, \underset{i \in I}{\lor} a_{i})$ ,

$I (a, b) \leq_{L} I (T (a, c), T (c, b))$ ,

$\underset{a \in L}{\land} I (I (b, a), I (c, a)) = I (c, b)$ ,

$T (I (a, b), c) \leq_{L} I (a, T (b, c))$ ,

$\underset{a \in L}{\land} I (I (b, a), a) = b$ ,

$\underset{a \in L}{\land} I (T (c, I (b, a)), a) = I (c, b)$ ,

$T (I (T (a, b), c), a) \leq_{L} I (b, c)$ ,

$I (a, I (b, c)) = I (b, I (a, c))$ ,

$T (I (a, b), I (b, c)) \leq_{L} I (a, c)$ .

2.2.3 Intuitionistic fuzzy rough sets

We will review the IF rough approximation operators proposed by Zhou et al. [32], which are perhaps one of the most generalized IF rough approximation operators constructed on the basis of arbitrary IF relations.

Definition 2.10. [32] Assume that $T$ and $I$ are a continuous IF t- norm and an implicator on L, respectively. Let (U, R) be an IF approximation space and A ∈ IF (U), the $T$ -upper, $I$ -lower approximations of A, denoted as ${\bar{R}}^{T} (A)$ and ${\underline{R}}_{I} (A)$ , respectively, w.r.t. the approximation space (U, R), are IF sets of U and are, respectively, defined as follows: $\begin{matrix} {\bar{R}}^{T} (A) (x) = \underset{y \in U}{\lor} T (R (x, y), A (y)), \forall x \in U, \\ {\underline{R}}_{I} (A) (x) = \underset{y \in U}{\land} I (R (x, y), A (y)), \forall x \in U . \end{matrix}$

The operators ${\bar{R}}^{T}, {\underline{R}}_{I} : IF (U) \to IF (U)$ are, respectively, referred to as $T$ -upper, $I$ -lower IF rough approximation operators of (U, R), and are simply denoted without ambiguity as $\underline{R}$ and $\bar{R}$ . The pair $({\underline{R}}_{I} (A), {\bar{R}}^{T} (A))$ is called the $(I, T)$ -IF rough set of A w.r.t. (U, R).

Without further notification, we suppose that $T$ is a continuous IF t-norm, and $I$ is an IF R-implicator based on $T$ .

Theorem 2.1. Let R be an intuitionistic fuzzy $T$ -similarity relation. The following properties hold: ∀A, A_j ∈ IF (U), j ∈ J, J is an index set,

$\underline{R} (A) \subseteq A$ , $A \subseteq \bar{R} (A)$ ,

$\underline{R} (⋂_{j \in J} A_{j}) = ⋂_{j \in J} \underline{R} (A_{j})$ , $\bar{R} (⋃_{j \in J} A_{j}) = ⋃_{j \in J} \bar{R} (A_{j})$ ,

$\begin{matrix} {\underline{R}}_{I} ({\underline{R}}_{I} (A)) = {\bar{R}}^{T} ({\underline{R}}_{I} (A)) = {\underline{R}}_{I} (A), \\ {\bar{R}}^{T} ({\bar{R}}^{T} (A)) = {\underline{R}}_{I} ({\bar{R}}^{T} (A)) = {\bar{R}}^{T} (A), \end{matrix}$

both of ${\underline{R}}_{I}$ and ${\bar{R}}^{T}$ are monotone,

${\underline{R}}_{I} (A) = A \Leftrightarrow {\bar{R}}^{T} (A) = A$ ,

${\underline{R}}_{I} (A) = \cup {{\bar{R}}^{T} (x_{λ}) : {\bar{R}}^{T} (x_{λ}) \subseteq A}$ ,

${\bar{R}}^{T} (A) = \cup {{\bar{R}}^{T} (x_{λ}) : x_{λ} \subseteq A}$ , where x ∈ U, λ ∈ L, x_λ is an intuitionistic fuzzy set defined as $x_{λ} (y) = {\begin{matrix} λ, y = x, \\ 0_{L}, y \neq x . \end{matrix}$

Proof. (1) It directly follows from [32, Theorem 5 (2) and Theorem 8].

(2) It comes from [32, Theorem 4 (2) and Theorem 6 (2)]. $\begin{matrix} (3) {\bar{R}}^{T} ({\bar{R}}^{T} (A)) (x) \\ = \underset{y \in U}{\lor} T (R (x, y), {\bar{R}}^{T} (A) (y)) \\ = \underset{y \in U}{\lor} T (R (x, y), \underset{z \in U}{\lor} T (R (y, z), A (z))) \\ = \underset{y \in U}{\lor} \underset{z \in U}{\lor} T (R (x, y), T (R (y, z), A (z))) \\ = \underset{z \in U}{\lor} T (\underset{y \in U}{\lor} T (R (x, y), R (y, z)), A (z)) \\ = \underset{z \in U}{\lor} T (\underset{y \in U}{\lor} T (R (x, y), R (y, z)), A (z)) \\ \leq_{L} \underset{z \in U}{\lor} T (R (x, z), A (z)) = {\bar{R}}^{T} (A) (x), \end{matrix}$ hence, ${\bar{R}}^{T} ({\bar{R}}^{T} (A)) \subseteq {\bar{R}}^{T} (A)$ . So, by (1), we have ${\bar{R}}^{T} ({\bar{R}}^{T} (A)) = {\bar{R}}^{T} (A) .$ $\begin{matrix} {\underline{R}}_{I} ({\bar{R}}^{T} (A)) (x) = \underset{y \in U}{\land} I (R (x, y), {\bar{R}}^{T} (A) (y)) \\ = \underset{y \in U}{\land} I (R (x, y), \underset{z \in U}{\lor} T (R (y, z), A (z))) \\ \geq_{L} \underset{y \in U}{\land} \underset{z \in U}{\lor} I (R (x, y), T (R (y, z), A (z))) \\ \geq_{L} \underset{y \in U}{\land} \underset{z \in U}{\lor} I (R (x, y), T (T (R (y, x), R (x, z)), A (z))) \\ = \underset{y \in U}{\land} \underset{z \in U}{\lor} I (R (x, y), T (R (y, x), T (R (x, z), A (z)))) \\ \geq_{L} \underset{z \in U}{\lor} T (R (x, z), A (z)) = {\bar{R}}^{T} (A) (x), \end{matrix}$ i.e. ${\underline{R}}_{I} ({\bar{R}}^{T} (A)) \supseteq {\bar{R}}^{T} (A)$ . As the converse holds by (1), we obtain ${\underline{R}}_{I} ({\bar{R}}^{T} (A)) = {\bar{R}}^{T} (A)$ . $\begin{matrix} {\bar{R}}^{T} ({\underline{R}}_{I} (A)) (x) = \underset{y \in U}{\lor} T (R (x, y), {\underline{R}}_{I} (A) (y)) \\ = \underset{y \in U}{\lor} T (R (x, y), \underset{z \in U}{\land} I (R (y, z), A (z))) \\ = \underset{y \in U}{\lor} \underset{z \in U}{\land} T (R (x, y), I (R (y, z), A (z))) \\ \leq_{L} \underset{y \in U}{\lor} \underset{z \in U}{\land} T (R (x, y), I (T (R (y, x), R (x, z)), A (z))) \\ \leq_{L} \underset{z \in U}{\land} I (R (x, z), A (z)) = {\underline{R}}_{I} (A) (x), \end{matrix}$

that is ${\bar{R}}^{T} ({\bar{R}}^{T} (A)) \subseteq {\bar{R}}^{T} (A)$ . Combining this with (1), we obtain equality ${\bar{R}}^{T} ({\bar{R}}^{T} (A)) = {\bar{R}}^{T} (A) .$ $\begin{matrix} {\underline{R}}_{I} ({\underline{R}}_{I} (A)) (x) \\ = \underset{y \in U}{\land} I (R (x, y), {\underline{R}}_{I} (A) (y)) \\ = \underset{y \in U}{\land} I (R (x, y), \underset{z \in U}{\land} I (R (y, z), A (z))) \\ = \underset{y \in U}{\land} \underset{z \in U}{\land} I (R (x, y), I (R (y, z), A (z))) \\ = \underset{y \in U}{\land} \underset{z \in U}{\land} I (T (R (x, y), R (y, z)), A (z)) \\ \geq_{L} \underset{z \in U}{\land} I (R (x, z), A (z)) = {\underline{R}}_{I} (A) (x), \end{matrix}$ i.e. ${\underline{R}}_{I} ({\underline{R}}_{I} (A)) \supseteq {\underline{R}}_{I} (A)$ . But the opposite inequality holds by (1). Therefore, ${\underline{R}}_{I} ({\underline{R}}_{I} (A)) = {\underline{R}}_{I} (A)$ .

(4) It follows from [32, Theorem 4 (IFU7) andTheorem 6 (IFL7)].

(5) Suppose ${\underline{R}}_{I} (A) = A$ . Then ${\bar{R}}^{T} (A) = {\bar{R}}^{T} ({\underline{R}}_{I} (A)) = {\underline{R}}_{I} (A) = A$ . ${\underline{R}}_{I} (A) = A \Leftrightarrow {\bar{R}}^{T} (A) = A$

Conversely, suppose ${\bar{R}}^{T} (A) = A$ . Then ${\underline{R}}_{I} (A) = {\underline{R}}_{I} ({\bar{R}}^{T} (A)) = {\bar{R}}^{T} (A) = A$ .

(6) Since A = ⋃ _{λ≤_LA(x)}x_λ, we have ${\bar{R}}^{T} (A) = \cup {{\bar{R}}^{T} (x_{λ}) : x_{λ} \subseteq A}$ . For any A ∈ IF (U), ${\underline{R}}_{I} (A) = {\bar{R}}^{T} ({\underline{R}}_{I} (A)) = \cup {{\bar{R}}^{T} (x_{λ}) : x_{λ} \subseteq {\underline{R}}_{I} (A)} \subseteq \cup {{\bar{R}}^{T} (x_{λ}) : {\bar{R}}^{T} (x_{λ}) \subseteq {\bar{R}}^{T} ({\underline{R}}_{I} (A)) = {\underline{R}}_{I} (A) \subseteq A} .$

Conversely, if ${\bar{R}}^{T} (x_{λ}) \subseteq A$ , then we have ${\underline{R}}_{I} ({\bar{R}}^{T} (x_{λ})) \subseteq {\underline{R}}_{I} (A)$ . This implies ${\underline{R}}_{I} (A) = \cup {{\bar{R}}^{T} (x_{λ}) : {\bar{R}}^{T} (x_{λ}) \subseteq A}$ .

Theorem 2.2. For two intuitionistic fuzzy $T$ -similarity relations R₁ and R₂, the following two statements are equivalent:

(1) R₁ ⊆ R₂; (2) ${\underline{R_{1}}}_{I} (A) \supseteq {\underline{R_{2}}}_{I} (A)$ , ${\bar{R_{1}}}^{T} (A) \subseteq {\bar{R_{2}}}^{T} (A)$ .

Proof. (1) ⇒ (2) If R₁ ⊆ R₂, for any A ∈ IF (U) , ptx ∈ U, we have $\begin{matrix} {\bar{R_{1}}}^{T} (A) (x) = \underset{y \in U}{\lor} T (R_{1} (x, y), A (y)) \\ \leq_{L} \underset{y \in U}{\lor} T (R_{2} (x, y), A (y)) \\ = {\bar{R_{2}}}^{T} (A) (x) \end{matrix}$

and $\begin{matrix} {\underline{R_{1}}}_{I} (A) (x) = \underset{y \in U}{\land} I (R_{1} (x, y), A (y)) \\ \geq_{L} \underset{y \in U}{\land} I (R_{2} (x, y), A (y)) \\ = {\underline{R_{2}}}_{I} (A) (x) . \end{matrix}$ (2) ⇒ (1) If for any A ∈ IF (U) , ${\underline{R_{1}}}_{I} (A) \supseteq {\underline{R_{2}}}_{I} (A)$ , ${\bar{R_{1}}}^{T} (A) \subseteq {\bar{R_{2}}}^{T} (A)$ , then for any x, y ∈ U, we have ${\bar{R_{1}}}^{T} (1_{y}) \subseteq {\bar{R_{2}}}^{T} (1_{y})$ . By [32, Theorems 4 (IFU9)], we have $R_{1} (x, y) = {\bar{R_{1}}}^{T} (1_{y}) (x) \leq_{L} {\bar{R_{2}}}^{T} (1_{y}) (x) = R_{2} (x, y)$ . Therefore, R₁ ⊆ R₂.

3 Attribute reduction based on intuitionistic fuzzy rough sets

In this section we will define attribute reduction based on intuitionistic fuzzy rough sets for intuitionistic fuzzy decision system and propose some equivalence conditions to describe the structure of attribute reduction. We also develop an algorithm using discernibility matrix to compute all the attribute reductions.

Suppose U is a finite universe of discourse, R is a finite set of intuitionistic fuzzy $T$ -similarity relations called conditional attributes set, D is an equivalence relation called decision attribute with symbolic values, then (U, R ∪ D) is called an intuitionistic fuzzy decision system. Denote Sim (R) =∩ { R : R ∈ R }, then Sim (R) is also an intuitionistic fuzzy $T$ -similarity relation. Suppose [x] _D is the equivalence class with respect to D for x ∈ U, then the positive region of D relative to Sim (R) is defined as ${POS}_{Sim (R)} (D) = \cup_{x \in U} {\underline{Sim (R)}}_{I} ({[x]}_{D})$ . We will say that R is dispensable relative to D in R if POS_Sim(R) (D) = POS_Sim(R-{R}) (D), otherwise we will say R is indispensable relative to D in R. The family R is independent relative to D if each R ∈ R is indispensable relative to D in R; otherwise R is dependent relative to D. P ⊆ R is an attributes reduction of relative to D if P is independent relative to D and POS_Sim(R) (D) = POS_Sim(P) (D), for short we call P a relative reduction of R. The collection of all the indispensable elements relative to D in R is called the core of R relative to D, denoted as Core_D (R). Similar to the result in traditional rough sets we have Core_D (R) = ∩ Red_D (R), Red_D (R) is the collection of all relative reductions of R. Following we study under what conditions that P ⊆ R could be a relative reduction of R.

By (6) of Theorem 2.1 we know that ${{\bar{R}}^{T} (x_{λ}) : x \in U, λ \in L}$ could be the basic IF granular set to construct lower and upper approximations of IF sets since every lower or upper approximation is just the union of IF sets with the form as ${\bar{R}}^{T} (x_{λ})$ . Thus the structure of lower approximation of every [x] _D is clear by ${\underline{R}}_{I} ({[x]}_{D}) = \cup {{\bar{R}}^{T} (y_{λ}) : {\bar{R}}^{T} (y_{λ}) \subseteq {[x]}_{D}}$ . For y ∉ [x] _D, clearly ${\underline{R}}_{I} ({[x]}_{D}) (y) = 0$ holds. For y ∈ [x] _D, the following theorem develops a sufficient and necessary condition for ${\bar{R}}^{T} (y_{λ}) \subseteq {\underline{R}}_{I} ({[x]}_{D})$ .

Theorem 3.1. Suppose y ∈ [x] _D, ${\bar{R}}^{T} (y_{λ}) \subseteq {\underline{R}}_{I} ({[x]}_{D})$ if and only if ${\bar{R}}^{T} (y_{λ}) (z) = 0_{L}$ for z ∉ [x] _D.

Proof. If ${\bar{R}}^{T} (y_{λ}) \subseteq {\underline{R}}_{I} ({[x]}_{D})$ , then ${\bar{R}}^{T} (y_{λ}) (z) = 0_{L}$ for z ∉ [x] _D is clear. Conversely, suppose ${\bar{R}}^{T} (y_{λ}) (z) = 0_{L}$ for z ∉ [x] _D. We have ${\bar{R}}^{T} (y_{λ}) \subseteq {\underline{R}}_{I} ({[x]}_{D})$ by [x] _D (u) = 1_L for every u ∈ [x] _D, this implies ${\bar{R}}^{T} (y_{λ}) \subseteq {\underline{R}}_{I} ({[x]}_{D})$ hold.

According to Theorem 3.1, $y_{λ} \subseteq {\underline{R}}_{I} ({[x]}_{D})$ if and only if ${\bar{R}}^{T} (y_{λ}) (z) = 0$ for z ∉ [x] _D. This statement is the key point to develop sufficient and necessary condition for relative reduction with IF rough sets.

Theorem 3.2. ${POS}_{Sim (R)} (D) (x) = {\underline{Sim (R)}}_{I} ({[x]}_{D}) (x)$ .

Proof. Since ${POS}_{Sim (R)} (D) = \cup_{z \in U} {\underline{Sim (R)}}_{I} ({[z]}_{D})$ , ${POS}_{Sim (R)} (D) (x) = (\underset{z \in U}{\lor} μ_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x), \underset{z \in U}{\land} ν_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x))$ and U is finite, we know $\underset{z \in U}{\lor} μ_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x)$ can get its max value at some $μ_{{\underline{Sim (R)}}_{I} ({[z_{1}]}_{D})} (x)$ , and $\underset{z \in U}{\land} ν_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x)$ can get its min value at some $ν_{{\underline{Sim (R)}}_{I} ({[z_{2}]}_{D})} (x)$ .

We firstly prove that $\underset{z \in U}{\lor} μ_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x) = μ_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ . It is clear that ∀y ∈ U, $μ_{{\underline{Sim (R)}}_{I} ({[z_{1}]}_{D})} (x) \geq μ_{{\underline{Sim (R)}}_{I} ({[y]}_{D})} (x)$ . If $μ_{{\underline{Sim (R)}}_{I} ({[z_{1}]}_{D})} (x) = μ_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ , then it is straightforward to obtain that $\underset{z \in U}{\lor} μ_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x) = μ_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ . Assume that $μ_{{\underline{Sim (R)}}_{I} ({[z_{1}]}_{D})} (x) > μ_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ , then we obtain [x] _D ≠ [z₁] _D and $\underset{{[z_{1}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L}) = 0_{L}$ .

Since ${\underline{Sim (R)}}_{I} ({[z_{1}]}_{D}) (x) = \underset{{[z_{1}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L})$ and ${\underline{Sim (R)}}_{I} ({[x]}_{D}) (x) = \underset{{[x]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L})$ , we obtain $μ_{{\underline{Sim (R)}}_{I} ({[z_{1}]}_{D})} (x) > μ_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x) \geq 0$ and $\underset{{[z_{1}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L}) \neq 0_{L}$ . This contradicts $\underset{{[z_{1}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L}) = 0_{L}$ . Thus [x] _D = [z₁] _D holds, i.e. $\underset{z \in U}{\lor} μ_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x) = μ_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ .

On the other hand, it is clear that ∀y ∈ U, $ν_{{\underline{Sim (R)}}_{I} ({[z_{2}]}_{D})} (x) \leq ν_{{\underline{Sim (R)}}_{I} ({[y]}_{D})} (x)$ .

If $ν_{{\underline{Sim (R)}}_{I} ({[z_{2}]}_{D})} (x) = ν_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ , then it is straightforward to obtain that $\underset{z \in U}{\land} ν_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x) = ν_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ . Assume that $ν_{{\underline{Sim (R)}}_{I} ({[z_{2}]}_{D})} (x) < ν_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ , then we obtain [x] _D ≠ [z₂] _D and $\underset{{[z_{2}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L}) = 0_{L}$ .

Since ${\underline{Sim (R)}}_{I} ({[z_{2}]}_{D}) (x) = \underset{{[z_{2}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L})$ and ${\underline{Sim (R)}}_{I} ({[x]}_{D}) (x) = \underset{{[x]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L})$ , we obtain $ν_{{\underline{Sim (R)}}_{I} ({[z_{2}]}_{D})} (x) < ν_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x) \leq 1$ and $\underset{{[z_{2}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L}) \neq 0_{L}$ . This contradicts $\underset{{[z_{2}]}_{D} (u) = 0}{\land} I (Sim (R) (x, u), 0_{L}) = 0_{L}$ . Thus [x] _D = [z₂] _D holds, i.e. $\underset{z \in U}{\land} ν_{{\underline{Sim (R)}}_{I} ({[z]}_{D})} (x) = ν_{{\underline{Sim (R)}}_{I} ({[x]}_{D})} (x)$ .

Therefore, ${POS}_{Sim (R)} (D) (x) = {\underline{Sim (R)}}_{I} ({[x]}_{D}) (x)$ .

Theorem 3.3. Suppose P ⊂ R, POS_Sim(R) (D) = POS_Sim(P) (D) if and only if ${\bar{Sim (P)}}^{T} (x_{λ (x)}) \subseteq {[x]}_{D}$ for every x ∈ U, here $λ (x) = {\underline{Sim (R)}}_{I} ({[x]}_{D}) (x)$ .

Proof. Since every two different decision classes have empty overlap, POS_Sim(R) (D) = POS_Sim(P) (D) $\Leftrightarrow {\underline{Sim (R)}}_{I} ({[x]}_{D}) = {\underline{Sim (P)}}_{I} ({[x]}_{D})$ for every $x \in U \Leftrightarrow {\bar{Sim (P)}}^{T} (y_{λ (y)}) \subseteq {[x]}_{D}$ for $y \in {[x]}_{D} \Leftrightarrow {\bar{Sim (P)}}^{T} (x_{λ (x)}) \subseteq {[x]}_{D}$ for every x ∈ U since y ∈ [x] _D implies [x] _D = [y] _D.

Thus we have the following theorem to characterize the relative reduction by Theorems 3.1 and 3.3.

Theorem 3.4. Suppose P ⊂ R, then P contains a relative reduction of R if and only if ${\bar{Sim (P)}}^{T} (x_{λ (x)}) (z) = 0_{L}$ for every x, z ∈ U and z ∉ [x] _D, here $λ (x) = {\underline{Sim (R)}}_{I} ({[x]}_{D}) (x)$ .

Theorem 3.5. Suppose P ⊂ R, then P contains a relative reduction of R if and only if there exists R ∈ P such that $T (R (x, z), λ (x)) = 0_{L}$ for every x, z ∈ U and z ∉ [x] _D.

Proof. For z ∉ [x] _D, we have $\begin{matrix} {\bar{Sim (P)}}^{T} (x_{λ (x)}) (z) = \underset{y \in U}{\lor} T (Sim (P) (z, y), x_{λ (x)} (y)) \\ = T (Sim (P) (z, x), x_{λ (x)} (x)) \\ = \underset{R \in P}{\land} T (R (z, x), λ (x)) . \end{matrix}$

Thus we finish the proof.

It is clear that P is a relative reduction of R if and only if P is the minimal subset of R satisfying conditions in Theorem 3.4 and Theorem 3.5. And condition in Theorem 3.5 is can be employed to design algorithm to compute reductions.

With the previous discussion, we can develop an algorithm to compute the relative reductions. Suppose U ={ x₁, …, x_n }, R ={ R₁, …, R_m }. We denote a n × n matrix (c_ij) by M_D (U, R), called the discernibility matrix of (U, R ∪ D), such that

(1) $c_{ij} = {R \in R : T (R (x_{i}, x_{j}), λ (x_{i})) = 0_{L}}$ , $λ (x_{i}) = {\underline{Sim (R)}}_{I} ({[x_{i}]}_{D}) (x_{i})$ , if x_j ∉ [x_i] _D;

(2) c_ij =ø, otherwise.

M_D (U, R) may not be symmetric and c_ii =ø. R ∈ c_ij implies ${\bar{R}}^{T} ({(x_{i})}_{λ (x_{i})}) (x_{j}) = 0_{L}$ , thus c_ij is the collection of conditional attributes to ensure ${\bar{R}}^{T} ({(x_{i})}_{λ (x_{i})}) (x_{j}) = 0_{L}$ for x_j ∉ [x_i] _D.

A discernibility function f_D (U, R) for (U, R ∪ D) is a Boolean function of m Boolean variables m $\bar{R_{1}}, \bar{R_{2}}, \dots, \bar{R_{m}}$ corresponding to the IF attributes R₁, R₂, …, R_m, respectively, and is defined as follows: $f_{D} (U, R) (\bar{R_{1}}, \bar{R_{2}}, \dots, \bar{R_{m}}) = \land {\lor (c_{ij}) : c_{ij} \neq ø},$

where ∨ (c_ij) is the disjunction of all variables $\bar{R}$ such that R ∈ c_ij. In the sequel, $\bar{R_{i}}$ is simply denoted without ambiguity as R_i.

We have the following theorem for the relative core.

Theorem 3.6. Core_D (R) ={ R ∈ R : c_ij = { R }} for some 1 ≤ i, j ≤ n.

Proof. R∈ Core_D (R) ⇔ POS_Sim(R) (D)≠_{POSSim (R - {R })} (D) ⇔ There exists x_i, x_j ∈ U, x_j ∉ [x_i] such that $T (R (x_{i}, x_{j}), λ (x_{i})) = 0_{L}$ and $T (R^{'} (x_{i}, x_{j}), λ (x_{i})) \neq 0_{L}$ holds for any R′ ≠ R. ⇔c_ij ={ R }.

The statement c_ij ={ R } implies that R is the unique attribute to ensure $T (R (x_{i}, x_{j}), λ (x_{i})) = 0_{L}$ .

Theorem 3.7. If P ⊆ R, then P contains a relative reduction of R if and only if P∩ c_ij ≠ ø for every c_ij≠ ø.

Proof. It is straightforward by Theorem 3.5 anddefinition of c_ij.

Corollary 3.1. Suppose P ⊆ R, then P is a relative reduction of R if and only if P is the minimal set satisfying P∩ c_ij ≠ ø for every c_ij≠ ø.

Let g_D (U, R) be the reduced disjunctive form of f_D (U, R) obtained from f_D (U, R) by applying the multiplication and absorption laws as many times as possible, then there exist l and R_k ⊆ R for k = 1, 2, …, l such that g_D (U, R) = (∧ R₁) ∨ (∧ R₂) ∨ ⋯ ∨ (∧ R_l) where each element in R_k appears only one time. We have the following theorem.

Theorem 3.8. Red_D (R) ={ R₁, …, R_l }.

Proof. For every k = 1, 2, …, l, we have ∧R_k ≤ ∨ c_ij. By the disjunction and conjunction laws, R_k∩ c_ij ≠ ø for any c_ij≠ ø. Since g_D (U, R) = (∧ R₁) ∨ (∧ R₂) ∨ ⋯ ∨ (∧ R_l), it follows that for arbitrary R_k if we reduce an element R from R_k, let $R_{k}^{'} = R_{k} - {R}$ , then $g_{D} (U, R) \neq \lor_{r = 1}^{k - 1} (\land R_{r}) \lor (\land R_{k}^{'}) \lor \lor_{r = k + 1}^{l} (\land R_{r})$ and $g_{D} (U, R) < \lor_{r = 1}^{k - 1} (\land R_{r}) \lor (\land R_{k}^{'}) \lor \lor_{r = k + 1}^{l} (\land R_{r})$ . If we still have $R_{k}^{'} \cap c_{ij} \neq ø$ for any c_ij≠ ø, then $\land R_{k}^{'} \leq \lor c_{ij}$ for any c_ij≠ ø. This implies that $g_{D} (U, R) \geq \lor_{r = 1}^{k - 1} (\land R_{r}) \lor (\land R_{k}^{'}) \lor \lor_{r = k + 1}^{l} (\land R_{r})$ and $g_{D} (U, R) = \lor_{r = 1}^{k - 1} (\land R_{r}) \lor (\land R_{k}^{'}) \lor \lor_{r = k + 1}^{l} (\land R_{r})$ , which is a contradiction. Hence, there exists $g_{D} (U, R) \neq \lor_{r = 1}^{k - 1} (\land R_{r}) \lor (\land R_{k}^{'}) \lor \lor_{r = k + 1}^{l} (\land R_{r})$ and c_{i
₀
j
₀}≠ ø such that $R_{k}^{'} \cap c_{i_{0} j_{0}} = ø$ , which implies that R_k is a relative reduction of R.

For any X ∈ Red_D (A), we have X∩ c_ij ≠ ø for any c_ij≠ ø. Thus f_D (U, R) ∧ (∧ X) = ∧ (∨ c_ij) ∧ (∧ X) = (∧ X). This implies ∧X ≤ f_D (U, R) = g_D (U, R). If R_k - X≠ ø for each k, we can find R_k ∈ R_k - X for each k. By rewriting $g_{D} (U, R) = (\lor_{k = 1}^{l} R_{k}) \land ø$ , we have $\land X \leq \lor_{k = 1}^{l} R_{k}$ . So there must be R_{k
₀} such that ∧X ≤ R_{k
₀}, this implies R_{k
₀} ∈ X. This is a contradiction. So R_{k
₀} ∈ X for some k₀. Since both X and R_{k
₀} are relative reductions, we have X = R_{k
₀}. Hence Red_D (R) ={ R₁, …, R_l }.

It should be noted that if c_ij∩ Core_D (R) ≠ ø, then {R} ∧ (∨ c_ij) = { R } for R ∈ c_ij ∩ Core_D (R). When computing g_D (U, R) by f_D (U, R) we can onlyconsider elements in Core_D (R) and c_ij satisfying c_ij∩ Core_D (R) = ø so that the computational load may be reduced. We can design an algorithm to compute reductions for IF decision systems.

Suppose U ={ x₁, x₂, …, x_n }, U/ - D = {D₁, D₂, …, D_s}.

Step 1: Compute Sim (R).

Step 2: Compute $\underline{Sim (R)} (D_{k})$ for each D_k ∈ U/ - D.

Step 3: Compute c_ij: if x_j ∉ [x_i] _D, then $c_{ij} = {R \in R : T (R (x_{i}, x_{j}), λ (x_{i})) = 0_{L}}$ , other-wise c_ij =ø.

Step 4: Compute core as collection of those c_ij with single element.

Step 5: Delete those c_ij =ø or c_ij with nonempty overlap with the core.

Step 6: Define f_D (U, R) =∧ { ∨ (c_ij) } with c_ij left after Step 5.

Step 7: Compute g_D (U, R) = (∧ R₁) ∨ (∧ R₂) ∨ ⋯ ∨ (∧ R_l) by f_D (U, R) =∧ { ∨ (c_ij) }.

Step 8: Output all reductions Red_D (R) ={ R₁, …, R_l }.

4 An illustrated example

To illustrate our idea of attributes reduction based on IF rough sets, an example of data set (adopted from the [28]) is given as follows.

Example 4.1. The car data set contains the information of ten new cars in the Guangzhou car market in Guangdong Province, China. Let U ={ x₁, x₂, …, x₁₀ } be the cars, each of which is described by six attributes: (1) C₁: fuel economy; (2) C₂: aerod degree; (3) C₃: price; (4) C₄: comfort; (5) C₅: design; and (6) C₆: safety. The characteristics of the ten new cars under the six attributes are represented by the IF sets, as shown in Table 1.

Every IF attribute C_k can define an IF similarity relation R_k as $\begin{matrix} R_{k} (x_{i}, x_{j}) \\ = {\begin{matrix} min {C_{k} (x_{i}), C_{k} (x_{j})}, C_{k} (x_{i}) \neq C_{k} (x_{j}) \\ (1, 0), C_{k} (x_{i}) = C_{k} (x_{j}) \end{matrix} \end{matrix}$

Sim (R) can be computed as $\begin{matrix} Sim (R) (x_{i}, x_{j}) \\ = (\begin{matrix} (1, 0) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 2, 0 . 7) \\ (0 . 2, 0 . 7) & (1, 0) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 2, 0 . 7) \\ (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (1, 0) & (0 . 1, 0 . 8) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 6) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 2, 0 . 7) \\ (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (1, 0) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) \\ (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (1, 0) & (0 . 2, 0 . 6) & (0 . 3, 0 . 7) & (0 . 4, 0 . 6) & (0 . 1, 0 . 8) & (0 . 4, 0 . 6) \\ (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 2, 0 . 6) & (1, 0) & (0 . 2, 0 . 7) & (0 . 2, 0 . 6) & (0 . 1, 0 . 8) & (0 . 2, 0 . 6) \\ (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 6) & (0 . 1, 0 . 8) & (0 . 3, 0 . 7) & (0 . 2, 0 . 7) & (1, 0) & (0 . 3, 0 . 7) & (0 . 1, 0 . 8) & (0 . 3, 0 . 7) \\ (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 4, 0 . 6) & (0 . 2, 0 . 6) & (0 . 3, 0 . 7) & (1, 0) & (0 . 1, 0 . 8) & (0 . 4, 0 . 5) \\ (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (0 . 1, 0 . 8) & (1, 0) & (0 . 1, 0 . 8) \\ (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 2, 0 . 7) & (0 . 1, 0 . 8) & (0 . 4, 0 . 6) & (0 . 2, 0 . 6) & (0 . 3, 0 . 7) & (0 . 4, 0 . 5) & (0 . 1, 0 . 8) & (1, 0) \end{matrix}) \end{matrix}$

Suppose a decision partition is A = {x₁, x₃, x₆, x₈,x₉}, B ={ x₂, x₄, x₅, x₇, x₁₀ }, $T (x, y) = (max (0, x_{1} +$ y₁ - 1) , min(1, x₂ + y₂)) and $I_{T} (x, y) = (min (1, 1 + y_{1} - x_{1}, 1 + x_{2} - y_{2}), max (0, y_{2} - x_{2}))$ , then $\underline{Sim (R)} (A) (x) = {\begin{matrix} (0.7, 0.3), x = x_{1} \\ (0.6, 0.4), x = x_{3} \\ (0.6, 0.4), x = x_{6} \\ (0.5, 0.5), x = x_{8} \\ (0.8, 0.2), x = x_{9} \\ (0, 1), other \end{matrix}$ $\underline{Sim (R)} (B) (x) = {\begin{matrix} (0.7, 0.3), x = x_{2} \\ (0.8, 0.2), x = x_{4} \\ (0.6, 0.4), x = x_{5} \\ (0.6, 0.4), x = x_{7} \\ (0.5, 0.5), x = x_{10} \\ (0, 1), other \end{matrix}$ and the discernibility matrix of (c_ij) is as follows: $\begin{matrix} M_{D} (U, R) \\ = (\begin{matrix} ø & {2, 4, 6} & ø & {2, 5, 6} & {2, 6} & ø & {2, 5, 6} & ø & ø & {2, 6} \\ {2, 4, 6} & ø & {4, 5} & ø & ø & {4} & ø & {4} & {4, 5, 6} & ø \\ ø & {4, 5} & ø & {4, 5, 6} & {4, 5, 6} & ø & {4} & ø & ø & {4, 5} \\ {5, 6} & ø & {5, 6} & ø & ø & {5, 6} & ø & {5, 6} & {5, 6} & ø \\ {2, 6} & ø & {4, 5, 6} & ø & ø & {3, 6} & ø & {6} & {5, 6} & ø \\ ø & {3, 4, 5, 6} & ø & {3, 5, 6} & {3, 6} & ø & {3, 4, 5, 6} & ø & ø & {3, 6} \\ {2, 4, 5, 6} & ø & {4} & ø & ø & {3, 4, 5, 6} & ø & {4, 5} & {4, 5, 6} & ø \\ ø & {4, 5} & ø & {4, 5, 6} & {4, 5, 6} & ø & {4, 5} & ø & ø & {4, 5} \\ ø & {6} & ø & {5, 6} & {6} & ø & {6} & ø & ø & {6} \\ {2, 3, 5, 6} & ø & {4, 5} & ø & ø & {2, 3, 6} & ø & {4, 5} & {5, 6} & ø \end{matrix}) \end{matrix}$

We can get that Core_D (R) ={ C₄, C₆ } and Red_D(R) ={ C₄, C₆ }.

5 Concluding remarks

This paper systematically studies attributes reduction based on intuitionistic fuzzy rough sets. We introduce some concepts and theorems of attributes reduction with intuitionistic fuzzy rough sets, and completely investigate the structure of attributes reduction. By employing the approach of discernibility matrix, an algorithm to find all the attributes reductions is also presented. At last an example is proposed to illustrate our idea and method. Altogether these findings lay the solid theoretical foundation for attributes reduction based on intuitionistic fuzzy rough sets. In the future, our work will focus on the two facets. On one hand, we will study computational complexity of the proposed algorithm in this paper, and develop some fast algorithms to compute attributes reduction. On the other hand, different types of IF approximation operators can be defined by using various IF t-norms and IF implicators. One issue whether and how the different IF approximation operators affect the result of attribute reduction is then arisen. Our future work is to identify the impact on the performance of attribute reduction of using various IF t-norms and IF implicators.

Footnotes

Acknowledgments

The author would like to thank the anonymous referees for their valuable suggestions in improving this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 61375075)

and the Natural Science Foundation of Hebei Province of China (Grant Nos. F2012201020 and F2015402033).

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