Attribute reductions and concept lattices in interval-valued intuitionistic fuzzy rough set theory: Construction and properties

Abstract

The definition of interval-valued intuitionistic fuzzy (IVIF for short) rough sets is introduced and relative properties are developed. Furthermore, given different threshold values, we discuss the knowledge reduction of IVIF information systems by using the discernibility matrix based on the positive region of decision attributes with regard to some condition attributes set. Finally, the IVIF variable threshold concept lattice and one-sided IVIF concept lattice associated with IVIF formal contexts are given and various methods of construction of these concepts are discussed.

Keywords

Interval-valued intuitionistic fuzzy rough set knowledge reduction interval-valued intuitionistic fuzzy formal context variable threshold interval-valued intuitionistic fuzzy formal concept lattice rough set

1 Introduction

Rough set theory, proposed by Pawlak [17], is a new mathematical approach to deal with insufficient and incomplete information. Using the concepts of lower and upper approximations in rough set theory, knowledge hidden in information systems may be expressed in the form of decision rules. Just like fuzzy set theory, rough set theory may be seen as an extension of set theory for modeling vagueness and inaccuracy.

A key notion in Pawlak’s rough set model is equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations which are very useful in the analysis of data in complete information systems. By replacing the equivalence relation with an arbitrary binary relation, different kinds of generalizations in Pawlak’s rough set models were obtained. Meanwhile, partitions(or coverings) of the universe of discourse, neighborhood systems, and Boolean algebras are all primitive notions [5 , 22–24]. At the same time, the problem of a fuzzification of a rough set has been focused on by more and more authors. Dubois and Prade [8, 9] were among the first who proposed the concepts of rough fuzzy set and fuzzy rough set by replacing crisp binary relations with fuzzy relations in the universe [3 , 20]. Other than the above rough model which only deal with the symbolic values, real values, Gong et al. [10] discussed a specific interval-valued fuzzy set (IVF for short) and introduced the basic rough set theory into IVF information systems in which interval values of attributes could be treated. As another generalization of fuzzy set, Atanassov [1] introduced the concept of an intuitionistic fuzzy set (IF for short) which was characterized by a pair of functions valued in [0,1]: the membership function and the non-membership function. Combining the advantages of IVF sets and IF sets, Atanassov et al. [2] defined the concept of an interval-valued intuitionistic fuzzy set (IVIF for short) and interval values instead of real values were used to express the degree of membership and the degree of non-membership so as to better describe fuzzy phenomenon in details.

Knowledge reduction in [4] is performed in information systems by means of the notion of a reduction based on a specialization of the general notion of independence due to Maczewski [13]. The knowledge reduction of information systems based on the rough sets theory have many practical conclusions. In this paper, we are concerned with approaches to knowledge reduction based on IVIF rough sets, then give the algorithm of reduction by using discernibility matrixs of IVIF information systems.

Formal concept analysis (FCA for short) is an order-theoretic method for the mathematical analysis of scientific data, proposed by Wille and others [19] in 1982. Concept lattices are the core of the mathematical theory of FCA. A concept lattice is a partially ordered set consisting of formal concepts, each of which represents a subset of objects called extent and a subset of attributes called intent.

Fuzzy formal concept analysis which incorporates fuzzy set theory into FCA assumes that the objects and attributes can take not only crisp but also fuzzy values. However, the extent to which “object o has property a” may be sometimes hard to assess precisely. Then it is convenient to use a sub-interval rather than a precise value. Such formal contexts naturally lead to interval-valued fuzzy formal concept (IVF formal concept for short).

Operators which are constructed based on the lattice in traditional formal concept and IVF formal concept define only one relation between objects and attributes. In view of this shortcoming, an intuitionistic fuzzy formal concept (IF formal concept for short) is defined in which two operators called membership degree and non-membership degree with regard to the object set and the attribute set are constructed.

Combining the advantages of IVF formal concepts and IF formal concepts, interval-valued intuitionistic fuzzy formal concept (IVIF formal concept for short) is imported so as to better describe fuzzy phenomenon in details.

The structure of the rest of this paper is as follows. In Section 2, we introduce necessary notions of rough sets, IVF rough sets and IVIF rough sets. In Section 3, we firstly introduce the concept of the positive region of decision attributes with regard to some condition attributes set, then discuss knowledge reductions of the IVIF information systems by using the discernibility matrix. In Section 4, we introduce the variable threshold IVIF formal concept lattice based on Galois connection. There are several approaches and methods on how to construct IVIF formal concept lattices. Finally, we conclude the paper with a summary and an outlook for further research in Section 5.

2 Preliminaries

In this section, we recall briefly the special lattice L^I originated by Gong et al. [10] and introduce some basic definitions and properties of IVIF sets which will be used in this paper.

Definition 1. Denote L^I = {[μ, ν] | μ ≤ ν, μ, ν∈ [0, 1]}. ∀ [μ₁, ν₁] , [μ₂, ν₂] ∈ L^I, (L^I, ≤ _{L
^I}) is a complete, bounded lattice defined by $[μ_{1}, ν_{1}] \leq_{L^{I}} [μ_{2}, ν_{2}] \Leftrightarrow μ_{1} \leq μ_{2}, ν_{1} \leq ν_{2} .$

The operators ∧ and ∨ on (L^I, ≤ _{L
^I}) are defined as follows: $\begin{matrix} [μ_{1}, ν_{1}] \land [μ_{2}, ν_{2}] & = & [\min {μ_{1}, μ_{2}}, \min {ν_{1}, ν_{2}}], \\ [μ_{1}, ν_{1}] \lor [μ_{2}, ν_{2}] & = & [\max {μ_{1}, μ_{2}}, \max {ν_{1}, ν_{2}}] . \end{matrix}$

The units of this lattice are 0_{L
^I} = [0, 0] and 1_{L
^I} = [1, 1].

Definition 2. An IVF set in U is an expression A denoted by

A = {〈x, A (x) 〉|x ∈ U},

where A : U → L^I, x → A (x) = [μ_A (x) , ν_A (x)] ∈ L^I.

We denote by IVF(U) the set of all IVF sets in U. It is clear that IVF(U) is a complete lattice with the greatest element 1_{L
^I} and the least element 0_{L
^I}.

Definition 3. An IVF relation from U to W is an IVF set on U × W, i.e., R is given by R = {[μ_R (x, y) , ν_R (x,

y)] | (x, y) ∈ U × W}, for simplicity, R = [μ_R, ν_R], where μ_R and ν_R are two fuzzy relations on U × W satisfying μ_R (x, y) ≤ ν_R (x, y) for all (x, y) ∈ U × W.

Definition 4. Let R be an IVF relation from U to W, (U,W,R) be the IVF approximation space. For all A∈IVF(W), we define the upper and lower IVF rough approximations of A w.r.t. (U,W,R), denoted by $\bar{R} (A)$ and $\underline{R} (A)$ respectively, as follows: ∀x ∈ U,

$\begin{matrix} \bar{R} (A) (x) & = & ⋁_{y \in U} (R (x, y) \land A (y)), \\ \underline{R} (A) (x) & = & ⋀_{y \in U} ((1_{L^{I}} - R (x, y)) \lor A (y)) . \end{matrix}$

The pair $(\underline{R} (A), \bar{R} (A))$ is referred to as an IVF rough set of A w.r.t. (U,W,R), and $\underline{R}, \bar{R} : IVF (W) \to IVF (U)$ are, respectively, called lower and upper IVF rough approximation operators.

Definition 5. Denote L = {(α, β) |α = [α₁, α₂] ∈ L^I, β = [β₁, β₂] ∈ L^I, α₂ + β₂ ≤ 1} . ∀ (α, β) , (ξ, η) ∈ L, (L, ≤ _L) is a complete, bounded lattice defined as follows: $(α, β) \leq_{L} (ξ, η) \Leftrightarrow α \leq_{L^{I}} ξ, β \geq_{L^{I}} η .$

The operators ∧ and ∨ on (L, ≤ _L) are defined as follows: $\begin{matrix} (α, β) \land (ξ, η) = (α \land ξ, β \lor η), \\ (α, β) \lor (ξ, η) = (α \lor ξ, β \land η) . \end{matrix}$

The units of this lattice are 0_L = (0_{L
^I}, 1_{L
^I}) and 1_L = (1_{L
^I}, 0_{L
^I}).

Definition 6. An IVIF set in U is an expression A denoted by

A = {〈x, A (x) 〉|x ∈ U}

where A : U → L, x → A (x) = (μ_A (x) , ν_A (x)) = ([μ_AL (x) , μ_AU (x)] , [ν_AL (x) , ν_AU (x)]) ∈ L . We denote by IVIF(U) the set of all IVIF sets in U.

Definition 7. An IVIF relation from U to W is an IVIF set on U × W, i.e., R is given by R = {(μ_R (x, y) , ν_R (x,

y)) | (x, y) ∈ U × W}, for simplicity, R = (μ_R, ν_R), where μ_R (x, y) = [μ_RL (x, y) , μ_RU (x, y)] , ν_R (x, y) = [ν_RL (x, y) , ν_RU (x, y)] satisfying 0 ≤ μ_RU (x, y) + ν_RU (x, y) ≤1 for all (x, y) ∈ U × W.

Definition 8. Let R be an IVIF relation from U to W,

(U, W, R) be an IVIF approximation space. For all A∈

IVIF(W), we define the upper and lower IVIF rough approximations of A w.r.t. (U, W, R), denoted by $\bar{R} (A)$ and $\underline{R} (A)$ respectively, as follows: ∀ x ∈ U, $\begin{matrix} \bar{R} (A) (x) & = & (μ_{\bar{R} (A)} (x), & ν_{\bar{R} (A)} (x)), \\ \underline{R} (A) (x) & = & (μ_{\underline{R} (A)} (x), & ν_{\underline{R} (A)} (x)), \end{matrix}$ where $\begin{matrix} μ_{\bar{R} (A)} (x) & = & ⋁_{y \in W} [μ_{R} (x, y) \land μ_{A} (y)], \\ ν_{\bar{R} (A)} (x) & = & ⋀_{y \in W} [ν_{R} (x, y) \lor ν_{A} (y)]; \\ μ_{\underline{R} (A)} (x) & = & ⋀_{y \in W} [ν_{R} (x, y) \lor μ_{A} (y)], \\ ν_{\underline{R} (A)} (x) & = & ⋁_{y \in W} [μ_{R} (x, y) \land ν_{A} (y)] . \end{matrix}$

The pair $(\underline{R} (A), \bar{R} (A))$ is referred to as the IVIF rough set of A w.r.t. (U, W, R).

3 The knowledge reduction of the interval-valued intuitionistic fuzzy information system

Suppose U is a finite universe of discourse, let S = (U, C ∪ D) be an IVIF information system with decision attributes, where U = {x₁, x₂, ⋯ , x_s} , C = {c₁, c₂, ⋯ , c_n} are represented by IVIF numbers and D = {d₁, d₂, ⋯ , d_m} are crisp or IVIF numbers.

We present an example to illustrate the IVIF information table. Table 1 is an IVIF information table, where U = {x₁, x₂, ⋯ , x₅} is a set of objects and C is an IVIF condition attribute set that includes three attribute c₁, c₂, c₃, each with corresponding linguistic terms, e.g., c₁ has terms c₁₁, c₁₂ and c₁₃, and c₁₁, c₁₂, c₁₃ are three IVIF sets. The decision attribute D has only one attribute d which is separated into three linguistic terms, d₁, d₂, d₃, and d₁, d₂, d₃ are three IVIF sets.

From Table 1, we have U/C = {c₁₁, c₁₂, c₁₃, c₂₁, c₂₂, c₃₁, c₃₂, c₃₃, c₃₄}, where U/{c₁} = {c₁₁, c₁₂, c₁₃} , etc . Meanwhile, U/D = U/{d} = {d₁, d₂, d₃}.

Definition 9. Let U = {x₁, x₂, ⋯ , x_s} is a universe of discourse, , are two IVIF sets in U. We define the distance measure d : IVIF (U) × IVIF (U) → [0, 1] if it satisfies the following:

$\begin{matrix} (1) & 0 \leq d (A_{1}, A_{2}) \leq 1; \\ (2) & d (A_{1}, A_{2}) = 0 if and only if A_{1} = A_{2}; \\ (3) & d (A_{1}, A_{2}) = d (A_{2}, A_{1}); \\ (4) & \forall A_{1}, A_{2}, A_{3} \in IVIF (U), if A_{1} \subseteq A_{2} \subseteq A_{3}, then \\ d (A_{1}, A_{3}) \geq d (A_{1}, A_{2}), d (A_{1}, A_{3}) \geq d (A_{2}, A_{3}) . \end{matrix}$

In [21], Xu has introduced the IVIF Hamming distance, the IVIF Euclidean distance, their normalized forms and weighted forms of the above distances between IVIF sets.

Suppose U = {x₁, x₂, ⋯ , x_s}, ∀x_i ∈ U, A₁ (x_i) = ([μ_{A
₁
L} (x_i) , μ_{A
₁
U} (x_i)] , [ν_{A
₁
L} (x_i) , ν_{A
₁
U} (x_i)]), A₂ (x_i) = ([μ_{A
₂
L} (x_i) , μ_{A
₂
U} (x_i)] , [ν_{A
₂
L} (x_i) , ν_{A
₂
U} (x_i)]) are two IVIF sets in U. In this paper, we use the normalized Euclidean distance $\begin{matrix} \begin{matrix} d (A_{1}, A_{2}) = [\frac{1}{4 s} \sum_{i = 1}^{s} ((μ_{A_{1} L} (x_{i}) - μ_{A_{2} L} (x_{i}))^{2} \\ + (μ_{A_{1} U} (x_{i}) - μ_{A_{2} U} (x_{i}))^{2} + (ν_{A_{1} L} (x_{i}) - ν_{A_{2} L} (x_{i}))^{2} \\ + (ν_{A_{1} U} (x_{i}) - ν_{A_{2} U} (x_{i}))^{2})]^{1 / 2} . \end{matrix} \end{matrix}$ Especially, if A₁ = ([μ_{A
₁
L}, μ_{A
₁
U}] , [ν_{A
₁
L}, ν_{A
₁
U}]) and A₂ = ([μ_{A
₂
L}, μ_{A
₂
U}] , [ν_{A
₂
L}, ν_{A
₂
U}]) are two IVIF numbers, then $\begin{matrix} d (A_{1}, A_{2}) & = [\frac{1}{4} ({(μ_{A_{1} L} - μ_{A_{2} L})}^{2} + {(μ_{A_{1} U} - μ_{A_{2} U})}^{2} \\ {+ {(ν_{A_{1} L} - ν_{A_{2} L})}^{2} + {(ν_{A_{1} U} - ν_{A_{2} U})}^{2})]}^{1 / 2} . \end{matrix}$

Definition 10. Suppose $R_{m} = (r_{ij}^{(m)})_{n \times n} (m = 1, 2)$

be two IVIF relations in U, where $r_{ij}^{(m)} = ([a_{ij}^{(m)}, b_{ij}^{(m)}],$

$[c_{ij}^{(m)}, d_{ij}^{(m)}]),$ i, j = 1, 2, ⋯ , n . If R = R₁ ∘ R₂ = (r_ij) _n×n, we then call R a composition matrix of R₁ and R₂, where $\begin{matrix} r_{ij} & = & ⋃_{k = 1}^{n} (r_{ik}^{(1)} \cap r_{kj}^{(2)}) \\ = & ([⋁_{k = 1}^{n} (a_{ik}^{(1)} \land a_{kj}^{(2)}), ⋁_{k = 1}^{n} (b_{ik}^{(1)} \land b_{kj}^{(2)})], \\ [⋀_{k = 1}^{n} (c_{ik}^{(1)} \lor c_{kj}^{(2)}), ⋀_{k = 1}^{n} (d_{ik}^{(1)} \lor d_{kj}^{(2)})]) . \end{matrix}$

We also see that the composition of finite IVIF relations is also an IVIF relation.

∀ x_i, x_j ∈ U, every IVIF attribute c_k (k = 1, 2, ⋯ ,

n) can define an IVIF relation R_{c
_k} as

$\begin{matrix} R_{c_{k}} (x_{i}, x_{j}) = \\ {\begin{matrix} \min {c_{k} (x_{i}), c_{k} (x_{j})}, & c_{k} (x_{i}) \neq c_{k} (x_{j}), \\ 1_{L}, & c_{k} (x_{i}) = c_{k} (x_{j}) . \end{matrix} \end{matrix}$ (1)If the threshold value λ ∈ [0, 1] is given, then Equations. 1 can be modified as:

$\begin{matrix} R_{c_{k}} (x_{i}, x_{j}) = \\ {\begin{matrix} \min {c_{k} (x_{i}), c_{k} (x_{j})}, & d (c_{k} (x_{i}), c_{k} (x_{j})) \geq λ, \\ 1_{L}, & d (c_{k} (x_{i}), c_{k} (x_{j})) < λ . \end{matrix} \end{matrix}$ (2)

Table 2 illustrates the relation R_{c
₁₁} generated by the attribute c₁₁ when λ = 0.05 in Table 1.

∀ B ⊆ C, the relation generated by B is R_B = ∘ {R_b|b ∈ B}. For example, if B = {c₁₁, c₁₂, c₂₁}, λ = 0.05 in Table 1, then the relation generated by B is computed in Table 3.

Furthermore, if B = C, λ = 0.05 in Table 1, we obtain the relation R_C in Table 4.

Theorem 1. Let R be an IVIF relation, then after finite times of compositions: $\begin{matrix} R \to R^{2} \to R^{4} \to \dots \to R^{2^{k}} \to \dots, \end{matrix}$ there must exist a positive integer k such that R^{2^k} = R^{2^k+1}, and R^{2^k+1} is also an IVIF similarity relation which is denoted by sim(R).

Table 5 gives the IVIF similarity relation sim(R_C) obtained from the IVIF relation R_C by Theorem 1.

Definition 11. Let S = (U, C ∪ D) be an IVIF information system, U/D = { d₁, d₂, ⋯ , d_m }. ∀x ∈ U, B ⊆ C, R_B is the IVIF relation generated by B, if the relation $\begin{matrix} \underline{R_{B}} (d_{i}) (x) > \underline{R_{B}} (d_{j}) (x) \\ \Leftrightarrow \underline{R_{C}} (d_{i}) (x) > \underline{R_{C}} (d_{j}) (x), \forall i \neq j \end{matrix}$ hold, then B is called the consistent set of C. If B is the minimum consistent set of C in the inclusion set, then B is called the reduction of the IVIF information system S.

Definition 12. Let S = (U, C ∪ D) be an IVIF information system, U/D = { d₁, d₂, ⋯ , d_m }. ∀x ∈ U, B ⊆ C, R_B is the IVIF relation generated by B, we define $\begin{matrix} \underline{R_{B}} (D) (x) \\ = (\underline{R_{B}} (d_{1}) (x), \underline{R_{B}} (d_{2}) (x), \dots, \underline{R_{B}} (d_{m}) (x)) . \end{matrix}$

The positive region of D with regard to B is defined as follows:

μ_{pos_B(D)} (x)

$\begin{matrix} = & ⋁_{d \in U / D} \underline{R_{B}} (d) (x) = ⋁_{d \in U / D} (μ_{\underline{R_{B}} (d)} (x), ν_{\underline{R_{B}} (d)} (x)) \\ = & (⋁_{d \in U / D} μ_{\underline{R_{B}} (d)} (x), ⋀_{d \in U / D} ν_{\underline{R_{B}} (d)} (x)) \\ = & ([(⋁_{d \in U / D} μ_{\underline{R_{B}} (d)} (x))^{-}, (⋁_{d \in U / D} μ_{\underline{R_{B}} (d)} (x))^{+}], \\ [(⋀_{d \in U / D} ν_{\underline{R_{B}} (d)} (x))^{-}, (⋀_{d \in U / D} ν_{\underline{R_{B}} (d)} (x))^{+}]), \end{matrix}$

where

$\begin{array}{l} \underset{d \in U / D}{⋁} μ_{\underline{R_{B}} (d)} (x) \\ = [(\underset{d \in U / D}{⋁} μ_{\underline{R_{B}} (d)} (x))^{-}, (\underset{d \in U / D}{⋁} μ_{\underline{R_{B}} (d)} (x))^{+}], \\ \underset{d \in U / D}{⋀} ν_{\underline{R_{B}} (d)} (x) \\ = [(\underset{d \in U / D}{⋀} ν_{\underline{R_{B}} (d)} (x))^{-}, (\underset{d \in U / D}{⋀} ν_{\underline{R_{B}} (d)} (x))^{+}] \end{array}$

The dependency degree γ_B (D) of B with regard to D is defined as follows: $\begin{matrix} γ_{B} (D) = \frac{1}{4 | U |} (\sum_{x \in U} (⋁_{d \in U / D} μ_{\underline{R_{B}} (d)} (x))^{-} \\ + \sum_{x \in U} (⋁_{d \in U / D} μ_{\underline{R_{B}} (d)} (x))^{+} + \sum_{x \in U} (⋀_{d \in U / D} ν_{\underline{R_{B}} (d)} (x))^{-} \\ + \sum_{x \in U} (⋀_{d \in U / D} ν_{\underline{R_{B}} (d)} (x))^{+}) . \end{matrix}$

In Table 1, we can compute the positive region of

D with regard to C as follows: $\begin{matrix} μ_{{pos}_{C} (D)} (x_{1}) & = & ([0.15, 0.2], [0.6, 0.75]), \\ μ_{{pos}_{C} (D)} (x_{2}) & = & ([0.2, 0.25], [0.5, 0.6]), \\ μ_{{pos}_{C} (D)} (x_{3}) & = & ([0.15, 0.2], [0.6, 0.75]), \\ μ_{{pos}_{C} (D)} (x_{4}) & = & ([0.15, 0.2], [0.55, 0.7]), \\ μ_{{pos}_{C} (D)} (x_{5}) & = & ([0.15, 0.2], [0.6, 0.75]) . \end{matrix}$

Given the threshold value δ ∈ [0, 1], we define the following discernibility matrix.

Definition 13. Let S = (U, C ∪ D) be an IVIF information system, U = {x₁, x₂, ⋯ , x_n} , ∀ x_i, x_j ∈ U, we denote a n × n matrix (m_ij), called the discernibility matrix of S denoted by M_D (U, C) = (m_ij) if $\begin{matrix} m_{ij} = {\begin{matrix} \bar{c_{ij}}, & μ_{posC} (D) (x_{i}) \neq μ_{posC} (D) (x_{j}), \\ C, & μ_{posC} (D) (x_{i}) = μ_{posC} (D) (x_{j}), \end{matrix} \end{matrix}$ where $\bar{c_{ij}} ≐ {c_{k} \in C | d (c_{k} (x_{i}), c_{k} (x_{j})) > δ}$ .

Remark 1. In Definition 13, given an appropriate threshold value η, the condition μ_posC (D) (x_i)≠μ_posC (D) (x_j) can be replaced by $d (μ_{posC} (D) (x_{i}), μ_{posC} (D) (x_{j})) > η .$

Theorem 2. Let S = (U, C ∪ D) be an IVIF information system. If there exists a subset B ⊆ C such that B∩ M_D (U, C) ≠ ∅, then B is the consistent set of C.

Proof. If μ_posC (D) (x_i) ≠ μ_posC (D) (x_j), then there exists k ∈ {1, 2, ⋯ , m} such that $\underline{R_{C}} (d_{k}) (x_{i}) \neq$ $\underline{R_{C}} (d_{k}) (x_{j})$ . If B∩ M_D (U, C) ≠ ∅, then there is a ∈ B such that d (a (x_i) , a (x_j)) >δ. That is, d_k (x_i) and d_k (x_j) can be discerned by B.□

In Table 1, given the threshold value λ = 0.05, δ = 0.2, η = 0, we obtain the discernibility matrix M_D (U, C) of S in Table 6.

Let B = {c₃₁}, it is easy to verify that B satisfies Theorem 1, i.e., B is the consistent set of C. Moreover, it can be easily tested that B is a minimum consistent set. Therefore, B is one of the reductions.

Similar to Definition 12, we have:

Definition 14.Let S = (U, C ∪ D) be an IVIF information system, U / D = {d₁, d₂, ⋯ , d_m}. ∀x ∈ U, B ⊆ C, R_B is the IVIF relation generated by B, we define $\begin{matrix} \bar{R_{B}} (D) (x) \\ = (\bar{R_{B}} (d_{1}) (x), \bar{R_{B}} (d_{2}) (x), \dots, \bar{R_{B}} (d_{m}) (x)), \\ {\underline{R_{B}}}^{(m)} (D) (x) = {d_{j} | \underline{R_{B}} (d_{j}) (x) = ⋁_{k \leq m} \underline{R_{B}} (d_{k}) (x)}, \\ {\bar{R_{B}}}^{(+)} (D) (x) = {d_{j} | \bar{R_{B}} (d_{j}) (x) \geq 0_{L}} . \end{matrix}$

If $\underline{R_{B}} (x) = \underline{R_{C}} (x)$ or $\bar{R_{B}} (x) = \bar{R_{C}} (x)$ for any x ∈ U, then B is called the lower (or upper) approximation consistent set of C.

If ${\underline{R_{B}}}^{(m)} (D) (x) = {\underline{R_{C}}}^{(m)} (D) (x)$ for any x ∈ U, then B is called the maximum lower approximation consistent set of C.

If ${\bar{R_{B}}}^{(+)} (D) (x) = {\bar{R_{C}}}^{(+)} (D) (x)$ for any x ∈ U, then B is called the non-negative upper approximation consistent set of C.

If B ⊆ C is the lower (or upper) approximation consistent set of C, and any subset of B are not the lower (or upper) approximation consistent set of C, then B is called the lower (or upper) approximation reduction of C.

It is similar to define the maximum lower approximation reduction and the non-negative upper approximation reduction.

Similar to Definition 13, given the threshold value δ ∈ [0, 1], we define the following discernibility matrix.

Definition 15. Let S = (U, C ∪ D) be an IVIF information system, U = {x₁, x₂, ⋯ , x_n} , ∀ x_i, x_j ∈ U, we denote the n × n matrix ${\underline{M}}_{D} (U, C) = ({\underline{m}}_{ij})$ , ${\bar{M}}_{D} (U, C) = ({\bar{m}}_{ij})$ , ${\underline{M}}_{D}^{(m)} (U, C) = ({\underline{m}}_{ij}^{(m)})$ and ${\bar{M}}_{D}^{(+)} (U, C) = ({\bar{m}}_{ij}^{(+)})$ respectively by $\begin{matrix} {\underline{m}}_{ij} & = & {\begin{matrix} \bar{c_{ij}}, & {\underline{R}}_{C} (D) (x_{i}) \neq {\underline{R}}_{C} (D) (x_{j}), \\ C, & {\underline{R}}_{C} (D) (x_{i}) = {\underline{R}}_{C} (D) (x_{j}), \end{matrix} \\ {\bar{m}}_{ij} & = & {\begin{matrix} \bar{c_{ij}}, & {\bar{R}}_{C} (D) (x_{i}) \neq {\bar{R}}_{C} (D) (x_{j}), \\ C, & {\bar{R}}_{C} (D) (x_{i}) = {\bar{R}}_{C} (D) (x_{j}), \end{matrix} \\ {\underline{m}}_{ij}^{(m)} & = & {\begin{matrix} \bar{c_{ij}}, & {\underline{R_{C}}}^{(m)} (D) (x_{i}) \neq {\underline{R_{C}}}^{(m)} (D) (x_{j}), \\ C, & {\underline{R_{C}}}^{(m)} (D) (x_{i}) = {\underline{R_{C}}}^{(m)} (D) (x_{j}), \end{matrix} \\ {\bar{m}}_{ij}^{(+)} & = & {\begin{matrix} \bar{c_{ij}}, & {\bar{R_{C}}}^{(+)} (D) (x_{i}) \neq {\bar{R_{C}}}^{(+)} (D) (x_{j}), \\ C, & {\bar{R_{C}}}^{(+)} (D) (x_{i}) = {\bar{R_{C}}}^{(+)} (D) (x_{j}), \end{matrix} \end{matrix}$ where $\bar{c_{ij}} ≐ {c_{k} \in C | d (c_{k} (x_{i}), c_{k} (x_{j})) > δ}$ , ${\underline{M}}_{D}$ (U, C), ${\bar{M}}_{D} (U, C)$ , ${\underline{M}}_{D}^{(m)} (U, C)$ and ${\bar{M}}_{D}^{(+)} (U, C)$ are called the discernibility matrix of the lower approximation, upper approximation, maximum lower approximation and non-negative upper approximation attribute set in the IVIF information system S.

Remark 2. In Definition 15, given an appropriate threshold value τ, the inequations in the definitions of ${\underline{m}}_{ij}, {\bar{m}}_{ij}, {\underline{m}}_{ij}^{(m)}$ and ${\bar{m}}_{ij}^{(+)}$ can be replaced by $\begin{matrix} d (\underline{R_{C}} (D) (x_{i}), \underline{R_{C}} (D) (x_{j})) > τ, \\ d (\bar{R_{C}} (D) (x_{i}), \bar{R_{C}} (D) (x_{j})) > τ, \\ d ({\underline{R_{C}}}^{(m)} (D) (x_{i}), {\underline{R_{C}}}^{(m)} (D) (x_{j})) > τ, \\ d ({\bar{R_{C}}}^{(+)} (D) (x_{i}), {\bar{R_{C}}}^{(+)} (D) (x_{j})) > τ . \end{matrix}$

Similar to Theorem 2, we can easily obtain the following:

Theorem 3. Let (U, C ∪ D) be the IVIF information system. ∀ B ⊆ C, then we have that

B is the lower approximation consistent set of C iff $B \cap {\underline{M}}_{D} (U, C) \neq \emptyset$ ,

B is the upper approximation consistent set of C iff $B \cap {\bar{M}}_{D} (U, C) \neq \emptyset$ ,

B is the maximum lower approximation consistent set of C iff $B \cap {\underline{M}}_{D}^{(m)} (U, C) \neq \emptyset$ ,

B is the non-negative upper approximation consistent set of C iff $B \cap {\bar{M}}_{D}^{(+)} (U, C) \neq \emptyset$ .

In Table 1, given the threshold value λ = 0.05, δ = 0.2, τ = 0.2, we obtain the lower discernibility matrix ${\underline{M}}_{D} (U, C)$ of S in Table 7.

From Table 6, we can see that B₁ = {c₁₁, c₂₁} is not a lower approximation consistent set of C in this sense, while B₂ = {c₁₁, c₂₁, c₃₄} is a lower approximation consistent set of C. Furthermore it is also one of the reductions.

4 Concept lattices associated with interval-valued intuitionistic fuzzy contexts

4.1 IVIF formal concept and Galois connection

Let L be a complete lattice, U a universe of discourse, the set of all L-fuzzy sets in U is denoted by L^U. $\forall {\tilde{X}}_{1}, {\tilde{X}}_{2} \in L^{U}$ , we define $\tilde{X_{1}} \subseteq {\tilde{X}}_{2} \Leftrightarrow$ $\forall x \in U, {\tilde{X}}_{1} (x) \leq {\tilde{X}}_{2} (x),$ so (L^U, ⊆) is a poset.

The following definitions and theorems are the relevant facts about Galois connections and complete residuated lattices.

Definition 16. Let $(L_{1}^{U}, \subseteq)$ and $(L_{2}^{A}, \subseteq)$ be two posets. The pair (f, g) of mappings $f : L_{1}^{U} \to L_{2}^{A}$ and $g : L_{2}^{A} \to L_{1}^{U}$ is called a Galois connection between $L_{1}^{U}$ and $L_{2}^{A}$ if and only if $\begin{matrix} X \subseteq g (B) \Leftrightarrow B \subseteq f (X) \end{matrix}$ for all $X \in L_{1}^{U}$ and $B \in L_{2}^{A}$ (both X and B can be either a crisp set or a L-fuzzy set).

Theorem 4. Let $(L_{1}^{U}, \subseteq)$ and $(L_{2}^{A}, \subseteq)$ be two posets and $f : L_{1}^{U} \to L_{2}^{A}$ and $g : L_{2}^{A} \to L_{1}^{U}$ be two mappings, then the following statements are equivalent:

(f, g) is a Galois connection between $L_{1}^{U}$ and $L_{2}^{A}$ ;

$\forall X_{1}, X_{2}, X \in L_{1}^{U}$ , $B_{1}, B_{2}, B \in L_{2}^{A}$ , $\begin{matrix} \begin{matrix} X_{1} \subseteq X_{2} & \Rightarrow & f (X_{2}) & \subseteq & f (X_{1}), \\ B_{1} \subseteq B_{2} & \Rightarrow & g (B_{2}) & \subseteq & g (B_{1}), \end{matrix} \\ X \subseteq g \circ f (X), B \subseteq f \circ g (B); \end{matrix}$

$\forall X_{1}, X_{2}, X \in L_{1}^{U}$ , $B_{1}, B_{2}, B \in L_{2}^{A}$ , $\begin{matrix} \begin{matrix} f (X_{1} \cup X_{2}) & = & f (X_{1}) & \cap & f (X_{2}), \\ g (B_{1} \cup B_{2}) & = & g (B_{1}) & \cap & g (B_{2}), \end{matrix} \\ X \subseteq g \circ f (X), B \subseteq f \circ g (B) . \end{matrix}$

Definition 17. Let $(L_{1}^{U}, \subseteq)$ and $(L_{2}^{A}, \subseteq)$ be two posets and $f : L_{1}^{U} \to L_{2}^{A}$ and $g : L_{2}^{A} \to L_{1}^{U}$ be two mappings. A pair $(X, B), X \in L_{1}^{U}, B \in L_{2}^{A}$ , is called a formal concept if f (X) = B, g (B) = X, and X and B are respectively referred to as the extension and the intension of the formal concept (X, B). The set of all formal concepts is denoted by L (U, A, f, g) .

Theorem 5. Let $(L_{1}^{U}, \subseteq)$ and $(L_{2}^{A}, \subseteq)$ be two posets and (f, g) a Galois connection between $L_{1}^{U}$ and $L_{2}^{A}$ , then L (U, A, f, g) is a complete lattice. Let I be an index set, ∀ (X_i, B_i) ∈ L (U, A, f, g) , i ∈ I, the meet and join are defined as follows: $\begin{matrix} \land_{i \in I} (X_{i}, B_{i}) = (\cap_{i \in I} X_{i}, f \circ g (\cup_{i \in I} B_{i})), \\ \lor_{i \in I} (X_{i}, B_{i}) = (g \circ f (\cup_{i \in I} X_{i}), \cap_{i \in I} B_{i}) . \end{matrix}$

Remark 3. Let R be a binary relation in Definition 16, then there is a pair of mappings f : 2^U → 2^A and g : 2^A → 2^U, which form a Galois connection between power sets $\begin{matrix} f (X) & = & {y \in A : (x, y) \in R, \forall x \in X}, \\ g (B) & = & {x \in U : (x, y) \in R, \forall y \in B}, \end{matrix}$ where the corresponding concept lattice generated by (f, g) is denoted by L(U,A,R).

4.2. Variable threshold IVIF concept lattices

Definition 18. A residuated lattice is a structure (L, ∧ , ∨ , ⊗ , →0, 1) in which ∧, ∨ , ⊗ and → are binary operators on the set L which satisfies:

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

⊗ is commutative and associative, with 1 as neutral element,

x ⊗ y ≤ z iff x ≤ y → z for all x, y and z in L (residuation principle).

Definition 19. A L-fuzzy formal context is a tuple (L,U,A,R) where U≠ ∅ and A≠ ∅ are the object and attribute sets respectively, and R ∈ L^U×A is a L-fuzzy relation. We call the L-fuzzy formal context (L,U,A,R) an interval-valued intuitionistic fuzzy formal context (IVIF formal context for short) if L∈IVIF(U).

Now, we give four types of IVIF formal contexts based on the residual implicators.

Definition 20. Let (L, U, A, R) be an IVIF formal context, $\forall \tilde{X} \in L^{U}, \tilde{B} \in L^{A}$ , two operators are defined as follows: $\begin{matrix} {\tilde{X}}^{♯} (a) = ⋀_{x \in U} (\tilde{X} (x) \to R (x, a)), a \in A, \\ {\tilde{B}}^{♯} (x) = ⋀_{a \in A} (\tilde{B} (a) \to R (x, a)), x \in U . \end{matrix}$

Theorem 6. The pair (♯ , ♯) in Definition 20 is a Galois connection between L^U and L^A.

Let X be a subset of U, then Definition 20 generates to the following definition.

Definition 21. Let (L, U, A, R) be an IVIF formal context, given a confidence threshold (α, β) ∈ L, $\forall X \in P (U), \tilde{B} \in L^{A}$ , two operators are defined as follows: $\begin{matrix} X^{□} (a) = (α, β) \to ⋀_{x \in X} R (x, a), a \in A, \\ {\tilde{B}}^{□} = {x \in U | ⋀_{a \in A} (\tilde{B} (a) \to R (x, a)) \geq (α, β)} . \end{matrix}$

Theorem 7. The pair (□ , □) in Definition 21 is a Galois connection between $P$ (U) and L^A.

Let B be a subset of A, then Definition 20 generates to the following definition.

Definition 22. Let (L, U, A, R) be an IVIF formal context, given a confidence threshold (α, β) ∈L, $\forall \tilde{X} \in L^{U}, B \in P$ (A), two operators are defined as follows: $\begin{matrix} {\tilde{X}}^{◊} = {a \in A | ⋀_{x \in U} (\tilde{X} (x) \to R (x, a)) \geq (α, β)}, \\ B^{◊} (x) = (α, β) \to ⋀_{a \in B} R (x, a), x \in U . \end{matrix}$

Theorem 8. The pair (◊ , ◊) in Definition 22 is a Galois connection between $P$ (A) and L^U.

Let X be a subset of U and B be a subset of A, then Definition 20 generates to the following definition.

Definition 23. Let (L, U, A, R) be an IVIF formal context, given a confidence threshold (α, β) ∈ L, $\forall X \in P (U)$ , $B \in P (A)$ , two operators are defined as follows: $\begin{matrix} X^{*} & = & (α, β) \to ⋀_{x \in X} R (x, a), a \in A, \\ B^{*} & = & (α, β) \to ⋀_{a \in B} R (x, a), x \in U . \end{matrix}$

Theorem 9. The pair (∗ , ∗) in Definition 23 is a Galois connection between $P$ (A) and $P$ (U).

Now, we give the following IVIF formal concepts respectively by $\begin{matrix} L (\tilde{U}, \tilde{A}, R) = {(\tilde{X}, \tilde{B}) | {\tilde{X}}^{♯} = \tilde{B}, \tilde{B} = {\tilde{X}}^{♯}}, \\ \begin{matrix} L_{(α, β)} (U, \tilde{A}, R) & = & {(X, \tilde{B}) | X^{□} = \tilde{B}, X = {\tilde{B}}^{□}}, \\ L_{(α, β)} (\tilde{U}, A, R) & = & {(\tilde{X}, B) | {\tilde{X}}^{◊} = B, \tilde{X} = B^{◊}}, \\ L_{(α, β)} (U, A, R) & = & {(X, B) | X = B^{*}, B = X^{*}} . \end{matrix} \end{matrix}$

From Theorem 5 and Theorem 6–9, we can prove that all of them are IVIF formal concepts. We call L_(α,β) (U, A, R) (respectively, $L_{(α, β)} (U, \tilde{A}, R)$ , L_(α,β)

$(\tilde{U}, A, R)$ ) a crisp-crisp (respectively, a crisp-fuzzy, fuzzy-crisp) variable threshold concept lattice.

Corresponding to above IVIF formal concept lattices, we denote the sets of all extensions by Ext_(α,β)

(U, A, R), ${Ext}_{(α, β)} (U, \tilde{A}, R)$ , ${Ext}_{(α, β)} (\tilde{U}, A, R)$ and $Ext (\tilde{U}, \tilde{A}, R)$ , and the sets of all intensions by Int_(α,β)

$(U, A, R), {Int}_{(α, β)} (U, \tilde{A}, R), {Int}_{(α, β)} (\tilde{U}, A, R)$ and $Int (\tilde{U}, \tilde{A}, R)$ respectively.

We introduce another IVIF formal concept based on the (α, β) - cut set.

Definition 24. Let (L, U, A, R) be an IVIF formal context, given a confidence threshold (α, β) ∈ L,∀X ⊆ U, B ⊆ A, we define two mappings $↑_{(α, β)} : P (U) \to P (A)$ and $↓_{(α, β)} : P (A) \to P (U)$ as follows: $\begin{matrix} ↑_{(α, β)} (X) & = & {a \in A : \forall x \in X, R (x, a) \geq (α, β)}, \\ ↓_{(α, β)} (B) & = & {x \in X : \forall a \in B, R (x, a) \geq (α, β)}, \end{matrix}$ where ↑_(α,β) (X) represents the set of attribute which all the elements of the subset of objective X have based on the (α, β) - cut set, ↓_(α,β) (B) represents the set of objects which have the same attributes B based on the (α, β) - cut set.

Definition 25. Given a confidence threshold (α, β) ∈ L, a (α, β)-IVIF formal concept of an IVIF formal context (L, U, A, R) is a pair (X, B) where X ⊆ U, B ⊆ A, ↑ _(α,β) (X) = B and ↓_(α,β) (B) = X. X and B are the (α, β)-extent and the (α, β)-intent of the IVIF formal concept, respectively.

Remark 4. The (α, β)-IVIF formal concept is acturally the concept of the classical formal context (U, A, R_(α,β)), where R_(α,β) = {(x, a) ∈ U × A : R (x, a) ≥ (α, β)}.

Moreover, we introduce the one-sided formal concept lattice which independently described by Kraj $\overset{ˇ}{c}$ i [11] and Been Yahia, Jaoua [7], cf. also [6].

Let U be a universe of discourse, ∀A ∈ IVIF (U), if U = {x₁, x₂, ⋯ , x_n} is finite, the IVIF set A can be denoted by $\frac{A (x_{1})}{x_{1}} + \frac{A (x_{2})}{x_{2}} + \dots + \frac{A (x_{n})}{x_{n}}$ .

Definition 26. Given an IVIF formal context (L, U, A,

R), we define two mappings $\tilde{f} : P$ (U) → L^A and $\tilde{g} : L^{A} \to P$ (U) as follows: $\begin{matrix} \tilde{f} (X) & = & {\frac{(α, β)}{d} | (α, β) = ⋀_{g \in X} R (g, d) for all d \in A}, \\ \tilde{g} (\tilde{B}) & = & {g \in U | R (g, d) \geq \tilde{B} (d) for all d \in A} . \end{matrix}$

Definition 27. An IVIF one-sided formal concept of an IVIF formal context (L, U, A, R) is a pair $(X, \tilde{B})$ where $X \in P$ $(U), \tilde{B} \in L^{A}, \tilde{f} (X) = \tilde{B}$ and $\tilde{g} (\tilde{B}) = X$ . X and $\tilde{B}$ are the extent and the intent of the IVIF formal concept, respectively.

Example 1. Table 8 shows an example of an IVIF formal context with U = {o₁, o₂, o₃, o₄} and A = {a₁, a₂, a₃, a₄}. $\forall X \in P (U), \tilde{B} \in L^{A}$ , let the implicator in Definition 21 be the S-implicator, we have X^□ (a) = (α, β) → ⋀ _x∈XR (x, a) , a ∈ A and ${\tilde{B}}^{□} = {x \in U | ⋀_{a \in A} (\tilde{B} (a) \to R (x, a)) \geq (α, β)} .$

In this example, we will choose the threshold value (α, β) dynamically. For example, we will determine the threshold value of the subset X = {o₁, o₃} where $\begin{matrix} X^{□} & = {\frac{([0.52, 0.54], [0.4, 0.45])}{a_{1}} + \frac{([0.8, 0.9], [0.05, 0.1])}{a_{2}} \\ + \frac{([0.65, 0.75], [0.15, 0.23])}{a_{3}} + \frac{([0.22, 0.26], [0.5, 0.6])}{a_{4}}} \end{matrix}$ by computing in Table 10. It is reasonable that the object o₁ and o₃ should belong to the set $X^{□ □} = {x \in U | ⋀_{a \in A} (X^{□} (a) \to R (x, a)) \geq (α, β)} .$

It follows that $\begin{matrix} ⋀_{a \in A} (X^{□} (a) & \to & R (o_{1}, a)) \geq (α, β), \\ ⋀_{a \in A} (X^{□} (a) & \to & R (o_{3}, a)) \geq (α, β) . \end{matrix}$

Because the obtainment of X^□ (a) rely on the value of (α, β), we take R (o₁, a_i) → R (o₁, a_i)) ≥ (α, β) , R (o₃, a_i) → R (o₁, a_i)) ≥ (α, β) and R (o₁, a_i)

→ R (o₃, a_i)) ≥ (α, β) , R (o₃, a_i) → R (o₃, a_i)) ≥ (α, β) (∀ i = 1, 2, 3, 4) as a substitute. Table 9 gives the value of (α, β) on each attribute. For example, the values ([1.0, 1.0] , [0.0, 0.0]), ([0.9, 0.95] , [0.02, 0.03]), ([1.0, 1.0] , [0.0, 0.0]), ([0.5, 0.6] , [0.22, 0.26]) of (α, β) are chosen during the construction of the concept FC₂. Table 1 and Fig. 2 give the variable threshold IVIF formal concept of the IVIF formal context in Table 8 and the IVIF formal concept lattice.

The object set {o₁, o₄} is not a extent of the IVIF formal concept in Table 8, because we can check that

${o_{1}, o_{4}}^{□} = {\frac{([0.52, 0.54], [0.4, 0.45])}{a_{1}} + \frac{([0.8, 0.85], [0.1, 0.15])}{a_{2}}$

$+ \frac{([0.65, 0.75], [0.15, 0.23])}{a_{3}} + \frac{([0.2, 0.25], [0.6, 0.7])}{a_{4}}}$ , but we have {o₁, o₄} ^□□ = {o₁, o₃, o₄} which do not satisfy the condition of {o₁, o₄} ^□□ = {o₁, o₄}. It is easy to see that the subsets {o₁}, {o₄}, {o₁, o₂}, {o₁, o₄}, {o₁, o₂, o₃}, {o₁, o₂, o₄} are not extents of the IVIF formal concept of the IVIF formal context in Table 8.

Example 2. Table 11 and Fig. 1 give the one-sided IVIF concept of the IVIF formal context in Table 8 and the IVIF concept lattice.

It is not difficult to verify that the subsets {o₁} , {o₄} ,

{o₁, o₂} , {o₁, o₄} , {o₂, o₄} , {o₁, o₂ , o₄} are also not extents of the IVIF formal concept of the IVIF formal context in Table 8.

Taking above two examples into consideration, we can find that the concepts obtained by above two ways are different.

5 Further research and conclusion

In this paper, we have developed a general framework for the study of IVIF rough sets and IVIF formal concept lattices. Firstly, we introduced IVIF rough approximation operators with respect to an IVIF approximation space. Secondly, the knowledge reduction of the IVIF information system based on the positive region of decision attributes is discussed. Finally, we considered four types of variable threshold IVIF concept lattices and the one-sided IVIF concept lattice associated with the IVIF formal context. Similarly, relative properties are examined.

However, the knowledge reduction of the IVIF formal concept lattice is not investigated completely in this paper. The axiomatic characterization of the IVIF information system and the IVIF formal context are not studied. In further research, we will develop proposed approaches to those various aspects of interval-valued intuitionistic fuzzy rough set theory.

Acknowledgments

The authors would like to show their sincere thanks to the anonymous referees for their valuable suggestions. This work was supported by the Scientific Research Fund of Heilongjiang Provincial Education Department (No.12531033).

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