Attribute Reduction in Intuitionistic Fuzzy Concept Lattices

Abstract

As an effective tool for knowledge discovery, concept lattice has been successfully applied to various fields. And one of the key problems of knowledge discovery is attribute reduction. In order to understand the problems better, the attribute reduction is necessary to perfect the theory as well as expand application of concept lattice. This paper introduces the intuitionistic fuzzy theory into the concept lattice theory and proposes a kind of intuitionistic fuzzy concept lattice. Then, an approach to attribute reduction based on the discernibility matrix is proposed and investigated, which makes the discovery of implicit knowledge easier and the representation simpler in data; furthermore, the theory of concept lattice is perfected. The theory of intuitionistic fuzzy concept lattice is useful and meaningful in view of the complexity and fuzziness of information in real world, and the potential value of dealing with information is expected in the future.

1. Introduction

Concept lattice stems from the so-called formal concept analysis proposed by Wille in 1982 [1], which can be depicted by a Hasse diagram, where each node expresses a formal concept. A concept lattice is an ordered hierarchical structure of formal concepts that are defined by a binary relation between a set of objects and a set of attributes. Concretely, each formal concept is the pair (objects, attributes), which consists of two parts: the extension (objects covered by the concept) and intension (attributes describing the concept). As an effective tool for data analysis and knowledge processing, concept lattice has been applied to various fields, such as data mining, information retrieval, and software engineering [2–4].

The intuitionistic fuzzy (IF, for short) set theory initiated by Atanassov [5, 6] is also an important mathematical structure to cope with imprecise information. IF set, as an extension of Zadeh’s fuzzy set [7], considers both membership degree and nonmembership degree which are functions valued in [0,1], while a fuzzy set gives a membership degree only. The membership and nonmembership values induce an indeterminacy index, which models the hesitancy degree of how an object satisfies a particular property. So IF set theory can present vague information better. Recently, IF set theory has been successfully applied in decision analysis and pattern recognition [8–11].

In recent years, many new achievements on these topics have been achieved on theories such as construction of concept lattice [1, 2, 12–15] and acquisition of rules [16, 17]. For example, the paper [15] shows that an approach of creating fuzzy concept lattices proposed by Popescu was equivalent to the approach of Krajči called generalized concept lattices in some way by comparing this approach with several other approaches and give a straightforward generalization of Popescu’s approach to nonhomogeneous cases, and Li et al. [17] investigate the issue of rule acquisition in incomplete decision context. And although rough set theory and formal context analysis are different theories, they have much in common in terms of both goals and methodologies, and a formal context in formal concept analysis corresponds to an information system in rough set theory [18, 19]. Formal concept analysis abstracts the knowledge from a formal context through formal concepts, while rough set theory discovers the knowledge via lower and upper approximations, positive boundary, and negative regions from an information system [18, 20, 21]. In fact, there are strong connections between formal concept analysis and rough set theory, and some researchers have been devoted to comparing and combining these two useful theories [18, 22–24], based on which we can study the concept lattice in the similar way to the rough set. Similarly, the key to attribute reduction is to find the minimal subsets of attributes sets in concept lattice, which can determine a concept lattice isomorphic to the one determined by all attributes while the object set remains unchanged. It makes the discovery of implicit knowledge easier and the representation simpler in data and extends the theory of concept lattice; knowledge reduction in formal concept analysis has attracted much attention [16, 17, 25–40]. For instance, Ganter and Wille [27] develop a reduction method to remove the reducible attributes and objects of a formal context via some predefined arrow relations. The paper [25] proposed a method to reduce the size of the concept lattice of a formal context using 𝐾-means clustering. Reduction approaches were presented to avoid the redundancy in the attributes from the perspectives of extension equivalence in the paper [39]. The paper [30] investigated the issue of developing efficient knowledge reduction methods for real decision formal contexts and developed a corresponding heuristic algorithm to search for a minimal reduction.

Similarly, attribute reduction in IF concept lattice is to find the minimum attribute set which can assign the same concepts and hierarchy based on IF formal context keeping the same objects. Reduction approaches in both the papers [23, 36] are based on the equivalence relation between the objects and attributes. In most situations, however, it is fuzzy or intuitionistic fuzzy. At present, there are some achievements on knowledge reduction based on fuzzy formal context [15, 25, 28, 34], where Lifeng Li introduced and investigated the attribute reduction in fuzzy concept lattices based on the kind of transitive implication operator. However, the study on attribute reduction for intuitionistic fuzzy formal context has not been investigated perfectly, although the discernibility matrix was proposed by using of the cuts of IF sets and then established the method of the attribute reduction of IF concept lattice in paper [24]. Because it leaves out some useful information about transforming the concept lattice based on the IF the formal context to a classical one only by using the cuts of the IF sets. Thus, for the requirement of knowledge-handling systems, combining IF set theory and formal concept analysis theory directly can result in a new hybrid mathematical structure by establishing two appropriate operators from other views.

Actually, the IF relation is an important type of data tables in formal concept analysis in real life. The paper combines the IF theory with the formal concept analysis, the main purpose of which is to study attribute reduction in IF concept lattices by introducing a pair of implication operators, and we establish approaches and theories of attribute reduction based on IF formal context, which is also suitable for classical formal context and fuzzy formal context.

The paper is organized as follows. Section 2 reviews basic definitions in formal concept analysis. We give the definitions and propositions in concept lattice with IF attributes in Section 3. In the next section, we discuss the corresponding definitions of attribute reduction in concept lattices, and then we divide the attributes into four types and investigate some related propositions and establish some propositions to determine the type of an attribute. In Section 5, the discernibility matrix and discernibility function in concept lattice are introduced, and then we discuss the approach to reduction as well as the corresponding characteristics. Finally, a simple conclusion is given in the paper.

2. Preliminaries

To make this paper self-contained, the IF set theory and involved notions of formal concept analysis are introduced briefly. Detailed description of them can be found in corresponding references.

Definition 1 (Ganter and Wille [27]). A triple (𝑈,𝐴,𝐼) is called a formal context, if 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛} is an object set, where 𝑥_𝑖(𝑖≤𝑛) is called an object; 𝐴={𝑎₁,𝑎₂,…,𝑎_𝑚} is an attribute set, where 𝑎_𝑗(𝑗≤𝑚) is called an attribute; and 𝐼⊆𝑈×𝐴 is a binary relation between 𝑈 and 𝐴.
In a formal context (𝑈,𝐴,𝐼), if (𝑥,𝑎)∈𝐼, that is, 𝑥𝐼𝑎, we say that the object 𝑥 has the attribute 𝑎, or that 𝑎 is fulfilled by 𝑥. For convenience, we use “1” and “0” to represent (𝑥,𝑎)∈𝐼 and (𝑥,𝑎)∉𝐼, respectively. Thus, a formal context can be represented by a table only with 0 and 1.
For a formal context (𝑈,𝐴,𝐼), a pair of dual operators for 𝑋⊆𝑈 and 𝐵⊆𝐴 is defined as follows: 𝑋^∗={()},𝐵𝑎∈𝐴∣𝑥,𝑎∈𝐼,∀𝑥∈𝑋_∗={𝑥∈𝑈∣(𝑥,𝑎)∈𝐼,∀𝑎∈𝐵}. (1)
In fact, 𝑋^∗ is the set of all the attributes shared by all the objects in 𝑋, and 𝐵^∗ is the set of all the objects that fulfill all the attributes in 𝐵.
Meanwhile, the complement sets of 𝑋^∗ and 𝐵^∗ are denoted by ∼𝑋^∗ and ∼𝐵^∗, where ∼𝑋^∗={𝑎∈𝐴∣(𝑥,𝑎)∉𝐼,∃𝑥∈𝑋} and ∼𝐵^∗={𝑥∈𝑈∣(𝑥,𝑎)∉𝐼,∃𝑎∈𝐵}.

Proposition 2 (Ganter and Wille [27]). Let (𝑈,𝐴,𝐼) be a formal context, 𝑋₁,𝑋₂,𝑋⊆𝑈 and 𝐵₁,𝐵₂,𝐵⊆𝐴; the following properties hold:(1)𝑋₁⊆𝑋₂⇒𝑋^∗₂⊆𝑋^∗₁,𝐵₁⊆𝐵₂⇒𝐵^∗₂⊆𝐵^∗₁.(2)𝑋⊆𝑋^∗∗,𝐵⊆𝐵^∗∗.(3)𝑋^∗=𝑋^∗∗∗;𝐵^∗=𝐵^∗∗∗.(4)𝑋⊆𝐵^∗⇔𝐵⊆𝑋^∗.(5)(𝑋₁∪𝑋₂)^∗=𝑋^∗₁∩𝑋^∗₂,(𝐵₁∪𝐵₂)^∗=𝐵^∗₁∩𝐵^∗₂.(6)(𝑋₁∩𝑋₂)^∗⊇𝑋^∗₁∪𝑋^∗₂,(𝐵₁∩𝐵₂)^∗⊇𝐵^∗₁∪𝐵^∗₂.

Definition 3 (Ganter and Wille [27]). Let (𝑈,𝐴,𝐼) be a formal context. A pair (𝑋,𝐵) is called a formal concept (in brief a concept) if 𝑋^∗=𝐵 and 𝑋=𝐵^∗ for 𝑋⊆𝑈,𝐵⊆𝐴. Furthermore, 𝑋 and 𝐵 are called the extension and the intension of (𝑋,𝐵), respectively.
From the above discussions, it is clear that both (𝑋^∗∗,𝑋^∗) and (𝐵^∗,𝐵^∗∗) are concepts.
According to [23, 27], we have the corresponding account as follows.
For convenience, all concepts of a formal context (𝑈,𝐴,𝐼) are denoted by 𝐿(𝑈,𝐴,𝐼), and they are ordered by 𝑋₁,𝐵₁≤𝑋₂,𝐵₂⟺𝑋₁⊆𝑋₂⟺𝐵₁⊇𝐵₂, (2) where (𝑋₁,𝐵₁) and (𝑋₂,𝐵₂) are concepts. Moreover, (𝑋₁,𝐵₁) is called a subconcept of (𝑋₂,𝐵₂), and (𝑋₂,𝐵₂) is called a superconcept of (𝑋₁,𝐵₁). And (𝑋₁,𝐵₁)<(𝑋₂,𝐵₂) means that (𝑋₁,𝐵₁)≤(𝑋₂,𝐵₂) and (𝑋₁,𝐵₁)≠(𝑋₂,𝐵₂) hold at the same time. If (𝑋₁,𝐵₁)<(𝑋₂,𝐵₂) and there does not exist a concept (𝑌,𝐶) such that (𝑋₁,𝐵₁)<(𝑌,𝐶)<(𝑋₂,𝐵₂), then (𝑋₁,𝐵₁) is called a child concept (immediate subconcept) of (𝑋₂,𝐵₂) and (𝑋₂,𝐵₂) is called a parent concept (immediate superconcept) of (𝑋₁,𝐵₁), and this is denoted by (𝑋₁,𝐵₁)≺(𝑋₂,𝐵₂).
For any two concepts (𝑋₁,𝐵₁) and (𝑋₂,𝐵₂) of a formal context (𝑈,𝐴,𝐼), it is easy to prove that both (𝑋₁∩𝑋₂,(𝐵₁∪𝐵₂)^∗∗) and ((𝑋₁∪𝑋₂)^∗∗,𝐵₁∩𝐵₂) are also concepts. Hence, if the meet and join are given by (Ganter and Wille [27]): 𝑋₁,𝐵₁∧𝑋₂,𝐵₂=𝑋₁∩𝑋₂,𝐵₁∪𝐵₂^∗∗,𝑋₁,𝐵₁∨𝑋₂,𝐵₂=𝑋₁∪𝑋₂^∗∗,𝐵₁∩𝐵₂, (3) then the concept lattice 𝐿(𝑈,𝐴,𝐼) is complete lattice.

Definition 4 (Atanassov [5]). Let 𝑈 be a finite and non-empty set called universe. An IF set 𝐴 of 𝑈 has the following form: 𝐴(𝑥)𝐴𝐴=𝑥,𝜇,𝛾(𝑥)∣𝑥∈𝑈, (4) where 𝜇𝐴∶𝑈→[0,1] and 𝛾𝐴∶𝑈→[0,1] and 𝜇𝐴(𝑥) and 𝛾𝐴(𝑥) are, respectively, called the membership degree and nonmembership degree to 𝐴 of the object 𝑥∈𝑈. Furthermore, they satisfy 𝐴𝐴0≤𝜇(𝑥)+𝛾(𝑥)≤1 for any 𝑥∈𝑈. In general, we use ℐℱ(𝑈) to denote all IF sets in the universe 𝑈.

Definition 5 (Atanassov [6]). Let 𝐴,𝐵∈ℐℱ(𝑈), 𝐴𝐵𝐴⊆𝐵⇔𝜇(𝑥)≤𝜇(𝑥), and 𝛾𝐴𝐵(𝑥)≥𝛾(𝑥) for any 𝑥∈𝑈. If both 𝐴⊆𝐵 and 𝐵⊆𝐴, then we say 𝐴 is equal to 𝐵, denoted by 𝐴=𝐵. The universe set and empty set are special IF set, where 𝑈={⟨𝑥,1,0⟩∣𝑥∈𝑈} and ∅={⟨𝑥,0,1⟩∣𝑥∈𝑈}.
Let us denote intersection and union of 𝐴 and 𝐵 by 𝐴∩𝐵 and 𝐴∪𝐵, respectively. Moreover, we denote complement of 𝐴 by ∼𝐴.

Definition 6 (Atanassov [6]). Let 𝐴,𝐵∈ℐℱ(𝑈); then 𝜇𝐴(𝑥)𝐵(𝑥𝛾𝐴(𝑥)𝐵(𝑥,𝜇𝐴(𝑥)𝐵(𝑥𝛾𝐴(𝑥)𝐵(𝑥,∼𝐴(𝑥)𝐴(𝑥.𝐴∩𝐵=𝑥,∧,𝜇),∨,𝛾)∣𝑥∈𝑈𝐴∪𝐵=𝑥,∨,𝜇),∧,𝛾)∣𝑥∈𝑈𝐴=𝑥,𝛾,𝜇)∣𝑥∈𝑈 (5)
Many properties of these operators in IF set theory are similar to fuzzy set theory. Detailed description can be found easily in the corresponding references.

3. IF Concept Lattice

The definition of concept lattice with IF attributes is introduced, and some important properties are discussed in this section.

Definition 7. A triple (𝑈,𝐴,𝐼)̃ is called an IF formal context, if 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛} is an object set, where 𝑥_𝑖(𝑖≤𝑛) is called an object; 𝐴={𝑎₁,𝑎₂,…,𝑎_𝑚} is an attribute set, where 𝑎_𝑗(𝑗≤𝑚) is called an attribute; and 𝐼̃ is an IF set of 𝑈×𝐴, where ̃̃𝐼={⟨(𝑥,𝑎),𝜇𝐼(𝑥,𝑎),𝛾𝐼(̃𝑥,𝑎)⟩∣(𝑥,𝑎)∈𝑈×𝐴}, 𝜇𝐼̃∶𝑈×𝐴→[0,1], and 𝛾𝐼̃∶𝑈×𝐴→[0,1].
The complement of 𝐼̃ is denoted by ∼𝐼̃={⟨(𝑥,𝑎),𝛾𝐼(̃𝑥,𝑎),𝜇𝐼(̃𝑥,𝑎)⟩∣(𝑥,𝑎)∈𝑈×𝐴}.
We denote ̃̃𝐼(𝑥,𝑎)=⟨𝜇𝐼(𝑥,𝑎),𝛾𝐼(̃𝑥,𝑎)⟩; then the set of 𝐼(̃𝑥,𝑎)(𝑥∈𝑈,𝑎∈𝐴) is denoted by 𝑉=𝐼(̃𝑥,𝑎)∣𝑥∈𝑈,𝑎∈𝐴. (6)
Let 𝐼(̃𝑥,𝑎), 𝐼(̃𝑦,𝑎)∈𝑉; then 𝐼(̃)≥𝐼̃̃𝐼()𝐼̃𝐼(̃)𝐼̃.𝑥,𝑎𝑦,𝑎⟺𝜇𝑥,𝑎≥𝜇𝑦,𝑎∧𝛾𝑥,𝑎≤𝛾𝑦,𝑎 (7)
With respect to an IF formal context (𝑈,𝐴,𝐼)̃, for 𝑋⊆𝑈, 𝐵⊆𝐴 and 𝐴, 𝐵∈ℐℱ(𝑈), where ∀𝐵(𝑏),𝐴(𝑏)∈{𝐼(̃𝑥,𝑏)∣∀𝑥∈𝑈}.
A pair of operators is defined by 𝑋^∗=𝐴=𝑎,𝜇_𝑋∗(𝑎),𝛾_𝑋∗(𝑎)∣𝑎∈𝐴, (8) where 𝐴(𝑎)=⟨∧_{∀𝑥∈𝑋}𝜇𝐼(̃𝑥,𝑎),∨_{∀𝑥∈𝑋}𝛾𝐼(̃𝑥,𝑎)⟩(𝑎∈𝐴) and denote ∅^∗=𝐴={⟨𝑎,1,0⟩∣𝑎∈𝐴}. 𝐵^∗=𝑥∈𝑈∣𝐼(̃𝑥,𝑏)≥𝐵(𝑏),∀𝑏∈𝐵, (9) where 𝐼(̃𝑥,𝑏)∈𝑉 and denote 𝐵(𝑏)=⟨0,1⟩, if 𝑏∉𝐵.
Similarly, ∀𝑥∈𝑈, we use 𝑥^∗ and 𝑎^∗ instead of {𝑥}^∗ and {𝑎}^∗, respectively, and for any 𝐵⊆𝐴 denote 𝑈^𝐵=𝐵∣𝐵(𝑏)=𝐼(̃𝑥,𝑏),𝑥∈𝑈,𝑏∈𝐵. (10)

Example 8. An IF formal context is shown as in Table 1.
In this context, let 𝑋={𝑥₁,𝑥₂,𝑥₄} and 𝐵={(𝑎,0.7,0.3),(𝑏,0.8,0.2),(𝑐,0.7,0.1),(𝑑,0.1,0.8)}∈ℐℱ(𝐵), where 𝐵={𝑎,𝑏,𝑐,𝑑}⊂𝐴; then from the definition we can obtain that 𝑋^∗=),(),(),(),(,𝐵𝐴={(𝑎,0.7,0.2𝑏,0.7,0.2𝑐,0.2,0.5𝑑,0.2,0.6𝑒,0.2,0.8)}_∗=𝑥₂,𝑥₄. (11)

Table 1 

Proposition 9. Let (𝑈,𝐴,𝐼)̃ be an IF formal context, 𝑋₁,𝑋₂,𝑋⊆𝑈, 𝐵₁,𝐵₂,𝐵⊆𝐴; then the above operators have the following properties:(1)𝑋₁⊆𝑋₂⇒𝑋^∗₂⊆𝑋^∗₁,𝐵₁⊆𝐵₂⇒𝐵^∗₂⊆𝐵^∗₁.(2)𝑋⊆𝑋^∗∗,𝐵⊆𝐵^∗∗.(3)𝑋^∗=𝑋^∗∗∗,𝐵^∗=𝐵^∗∗∗.(4)𝑋⊆𝐵^∗⇔𝐵⊆𝑋^∗.(5)(𝑋₁∪𝑋₂)^∗=𝑋^∗₁∩𝑋^∗₂,(𝐵₁∪𝐵₂)^∗=𝐵^∗₁∩𝐵^∗₂.(6)(𝑋₁∩𝑋₂)^∗⊇𝑋^∗₁∪𝑋^∗₂,(𝐵₁∩𝐵₂)^∗⊇𝐵^∗₁∪𝐵^∗₂.

Proof.  (1) Denote that 𝑋^∗₁=𝐴₁ and 𝑋^∗₂=𝐴₂, so it can be known for any 𝑎∈𝐴𝐴₁(𝑎)=∧_{∀𝑥∈𝑋1}𝜇𝐼(̃𝑥,𝑎),∨_{∀𝑥∈𝑋1}𝛾𝐼(̃,𝐴𝑥,𝑎)₂(𝑎)=∧_{∀𝑥∈𝑋2}𝜇𝐼(̃𝑥,𝑎),∨_{∀𝑥∈𝑋2}𝛾𝐼(̃𝑥,𝑎), (12) Since 𝑋₁⊆𝑋₂, it is true that 𝐴₁(𝑎)≥𝐴₂(𝑎). It follows that 𝐴₂⊆𝐴₁; that is, 𝑋^∗₂⊆𝑋^∗₁. In addition, from the above definitions, we can obtain that if 𝑎∈𝐵_𝑖(𝑖=1,2), 𝐵_𝑖(𝑎)=𝐴(𝑎), otherwise 𝐵_𝑖(𝑎)=⟨0,1⟩. So 𝐵^∗₁=𝑥∈𝑈∣𝐼(̃𝑥,𝑏)≥𝐵₁(𝑏)𝐼(̃)𝐵,∀𝑏∈𝐴,where𝑥,𝑏∈𝑉,_∗2=𝑥∈𝑈∣𝐼(̃𝑥,𝑏)≥𝐵₂(𝑏),∀𝑏∈𝐴,where𝐼(̃𝑥,𝑏)∈𝑉. (13) Since 𝐵₁⊆𝐵₂, we can obtain that 𝐵₁(𝑏)≤𝐵₂(𝑏). It follows that 𝐼(̃𝑥,𝑏)≥𝐵₂(𝑏) implies 𝐼(̃𝑥,𝑏)≥𝐵₁(𝑏) for any 𝐼(̃𝑥,𝑏)∈𝑉. Hence, if 𝑥∈𝐵^∗₂, then 𝑥∈𝐵^∗₁; that is, 𝐵^∗₂⊆𝐵^∗₁.(2) On one hand, assume that 𝑋^∗=𝐴; then 𝑋^∗∗=𝐴^∗={𝑥∈𝑈∣𝐼(̃𝑥,𝑎)≥𝐴(𝑎),∀𝑎∈𝐴}, where ̃̃𝐴(𝑎)=∧{𝐼(𝑥,𝑎)∀𝑥∈𝑋}=⟨∧𝜇𝐼(𝑥,𝑎),∨𝛾𝐼(̃𝑥,𝑎)⟩,∀𝑥∈𝑋, according to Definition 7. If 𝑥∈𝑋, then ̃̃𝐼(𝑥,𝑎)≥⟨∧𝜇𝐼(𝑥,𝑎),∨𝛾𝐼(̃𝑥,𝑎)⟩,∀𝑥∈𝑋=𝐴(𝑎). Thus, 𝑥∈𝑋^∗∗; that is, 𝑋⊆𝑋^∗∗. On the other hand, assume that 𝐵⊆𝑈^𝐵; then we can denote 𝐵={⟨𝜇𝐼(̃𝑥,𝑏),𝛾𝐼(̃𝑥,𝑏)⟩∣𝑏∈𝐴}, where 𝐵(𝑏)=⟨0,1⟩, if 𝑏∈𝐴−𝐵. And we can obtain that 𝐵^∗={𝑥∈𝑈∣𝐼(̃𝑥,𝑏)≥𝐵(𝑏),∀𝑏∈𝐴}, which follows that for any 𝑥∈𝐵^∗ and 𝑏∈𝐵, 𝐼(̃𝑥,𝑏)≥𝐵(𝑏) holds. So, we can obtain that 𝐵^∗∗(𝑏)=⟨∧𝜇𝐼(̃𝑥,𝑏),∨𝛾𝐼(̃𝑥,𝑏)⟩,∀𝑥∈𝐵^∗≥𝐵(𝑏). Hence, 𝐵⊆𝐵^∗∗.(3) It is obvious from (1) and (2).(4) From (1) we can have 𝑋⊆𝐵^∗⇒𝐵^∗∗⊆𝑋^∗, and from (2) we conclude 𝐵⊆𝐵^∗∗⇒𝐵⊆𝑋^∗.(5) It is obvious that (𝐵₁∪𝐵₂)^∗⊆𝐵^∗₁∩𝐵^∗₂. Furthermore, ∀𝑥∈𝐵₁^∗∩𝐵₂^∗⇔𝑥∈𝐵₁^∗∧𝑥∈𝐵₂^∗⇔𝐼(̃𝑥,𝑏)≥𝐵(𝑏),∀𝑏∈𝐵₁ and 𝐼(̃𝑥,𝑏)≥𝐵(𝑏),∀𝑏∈𝐵₂⇔𝐼(̃𝑥,𝑏)≥𝐵(𝑏), ∀𝑏∈𝐵₁∪𝐵₂⇔𝑥∈(𝐵₁∪𝐵₂)^∗⇒𝐵^∗₁∩𝐵^∗₂⊆(𝐵₁∪𝐵₂)^∗. Hence, (𝐵₁∪𝐵₂)^∗=𝐵^∗₁∩𝐵^∗₂. (𝑋₁∪𝑋₂)^∗=𝑋^∗₁∩𝑋^∗₂ can be obtained similarly.(6) It can be easily proved from (1).

Definition 10. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. A pair (𝑋,𝐵) is called an IF formal concept (in brief a concept) if 𝑋^∗=𝐵 and 𝑋=𝐵^∗ for 𝑋⊆𝑈, 𝐵⊆𝐴. 𝑋 and 𝐵 are called the extension and the intension of (𝑋,𝐵), respectively.
From the above, it is clear that both (𝑋^∗∗,𝑋^∗) and (𝐵^∗,𝐵^∗∗) are concepts.
IF concept lattice 𝐿(𝑈,𝐴,𝐼)̃ is referred to as all concepts of an IF formal context (𝑈,𝐴,𝐼)̃, and they are ordered by 𝑋₁,𝐵₁≤𝑋₂,𝐵₂⟺𝑋₁⊆𝑋₂⟺𝐵₁⊇𝐵₂, (14) where (𝑋₁,𝐵₁) and (𝑋₂,𝐵₂) are concepts. (𝑋₁,𝐵₁) is called a subconcept of (𝑋₂,𝐵₂), and (𝑋₂,𝐵₂) is called a superconcept of (𝑋₁,𝐵₁).
And we denote the family of all IF concept lattices by ℒ={𝐿(𝑈,𝐴,𝐼)̃∣(𝑈,𝐴,𝐼)̃isanIFformalcontext}.

Proposition 11. If (𝑋₁,𝐵₁) and (𝑋₂,𝐵₂) are two concepts of an IF formal context (𝑈,𝐴,𝐼)̃, then both (𝑋₁∩𝑋₂,(𝐵₁∪𝐵₂)^∗∗) and ((𝑋₁∪𝑋₂)^∗∗,𝐵₁∩𝐵₂) are also concepts.

Proof. It is straight from the definition and Proposition 9.

Hence, from the above, if the meet and join are given by 𝑋₁,𝐵₁∧𝑋₂,𝐵₂=𝑋₁∩𝑋₂,𝐵₁∪𝐵₂^∗∗,𝑋₁,𝐵₁∨𝑋₂,𝐵₂=𝑋₁∪𝑋₂^∗∗,𝐵₁∩𝐵₂, (15) then the IF concept lattice 𝐿(𝑈,𝐴,𝐼)̃ is complete lattice.

Example 12. In Example 8, we can find all concepts of the IF formal context by the definition, which are (1,𝐴₁), (2,𝐴₂), (4,𝐴₃), (12,𝐴₄), (13,𝐴₅), (24,𝐴₆), (123,𝐴₇), (124,𝐴₈), (𝑈,𝐴₉), (∅,𝐴), respectively, and we denote objects set {𝑥_𝑖,𝑥_𝑗} by 𝑖𝑗(𝑖,𝑗=1,2,3,4) which is same to others, where 𝐴₁=),(),(),(),(,𝐴{(𝑎,0.9,0.0𝑏,0.7,0.2𝑐,0.2,0.5𝑑,0.9,0.1𝑒,0.8,0.1)}₂=),(),(),(),(,𝐴{(𝑎,0.8,0.1𝑏,0.8,0.2𝑐,0.8,0.1𝑑,0.3,0.5𝑒,0.2,0.7)}₃=),(),(),(),(,𝐴{(𝑎,0.7,0.2𝑏,0.8,0.1𝑐,0.7,0.1𝑑,0.2,0.6𝑒,0.2,0.8)}₄=),(),(),(),(,𝐴{(𝑎,0.8,0.1𝑏,0.7,0.2𝑐,0.2,0.5𝑑,0.3,0.5𝑒,0.2,0.7)}₅=),(),(),(),(,𝐴{(𝑎,0.1,0.8𝑏,0.2,0.6𝑐,0.1,0.9𝑑,0.8,0.2𝑒,0.2,0.6)}₆=),(),(),(),(,𝐴{(𝑎,0.7,0.2𝑏,0.8,0.2𝑐,0.7,0.1𝑑,0.2,0.6𝑒,0.2,0.8)}₇=),(),(),(),(,𝐴{(𝑎,0.1,0.8𝑏,0.2,0.6𝑐,0.1,0.9𝑑,0.3,0.5𝑒,0.2,0.7)}₈=),(),(),(),(,𝐴{(𝑎,0.7,0.2𝑏,0.7,0.2𝑐,0.2,0.5𝑑,0.2,0.6𝑒,0.2,0.8)}₉=),(),(),(),(,),(),(),(),(.{(𝑎,0.1,0.8𝑏,0.2,0.6𝑐,0.1,0.9𝑑,0.2,0.6𝑒,0.2,0.8)}𝐴={(𝑎,1,0𝑏,1,0𝑐,1,0𝑑,1,0𝑒,1,0)} (16)
Furthermore, we can obtain the following IF concept lattice of the IF context (Figure 1).

Figure 1 

Concept lattice of the IF formal context (𝑈,𝐴,𝐼)̃.

4. Attribute Reduction in IF Concept Lattices

In this section, we discuss attribute reduction in IF concept lattices and obtain some propositions in a similar way to Zhang et al. [40] and Li and Zhang [28].

Definition 13. Let 𝐿(𝑈,𝐴₁,𝐼₁̃) and 𝐿(𝑈,𝐴₂,𝐼₂̃) be two IF concept lattices. If for any (𝑋,𝐵)∈𝐿(𝑈,𝐴₂,𝐼₂̃), there exists (𝑋^′,𝐵^′)∈𝐿(𝑈,𝐴₁,𝐼₁̃) such that 𝑋=𝑋^′, then we say that 𝐿(𝑈,𝐴₂,𝐼₂̃) is coarser than 𝐿(𝑈,𝐴₁,𝐼₁̃) or 𝐿(𝑈,𝐴₁,𝐼₁̃) is thinner than 𝐿(𝑈,𝐴₂,𝐼₂̃), denoted by 𝐿𝑈,𝐴₁,𝐼₁̃≤𝐿𝑈,𝐴₂,𝐼₂̃. (17)
If 𝐿(𝑈,𝐴₁,𝐼₁̃)≤𝐿(𝑈,𝐴₂,𝐼₂̃) and 𝐿(𝑈,𝐴₂,𝐼₂̃)≤𝐿(𝑈,𝐴₁,𝐼₁̃), we say that 𝐿(𝑈,𝐴₁,𝐼₁̃) and 𝐿(𝑈,𝐴₂,𝐼₂̃) are isomorphic with each other and denoted by 𝐿(𝑈,𝐴₁,𝐼₁̃)≅𝐿(𝑈,𝐴₂,𝐼₂̃).

Definition 14. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. The set of all extensions of 𝐿(𝑈,𝐴,𝐼)̃ is defined to be 𝐿_𝑈𝑈,𝐴,𝐼̃=𝑋∣𝑋,𝐵∈𝐿𝑈,𝐴,𝐼̃. (18)
Let (𝑈,𝐴,𝐼)̃ be an IF formal context and 𝐷⊆𝐴. We denote 𝐼_𝐷̃=𝐼̃∩𝐼^′̃, where 𝐼^′̃ is an IF set of 𝑈×𝐷; that is, 𝐼_𝐷̃̃={⟨(𝑥,𝑎),𝜇𝐼_𝐷(𝑥,𝑎),𝛾𝐼_𝐷̃(𝑥,𝑎)⟩∣(𝑥,𝑎)∈𝑈×𝐷}. Obviously, (𝑈,𝐷,𝐼_𝐷̃) is also an IF formal context, we denote all concepts of (𝑈,𝐷,𝐼_𝐷̃) by 𝐿(𝑈,𝐷,𝐼_𝐷̃) similarly. For any (𝑋,𝐵)∈𝐿(𝑈,𝐷,𝐼_𝐷̃), it satisfies that if 𝑎∈𝐷, 𝑋^∗𝐷(𝑎)=𝑋^∗(𝑎), otherwise 𝑋^∗𝐷(𝑎)=⟨0,1⟩, and 𝐵^∗𝐷={𝑥∈𝑈∣𝐼(̃𝑥,𝑏)≥𝐵(𝑏),∀𝑏∈𝐷}.

Proposition 15. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. If 𝑋⊆𝑈, 𝐷⊆𝐴 and 𝐵∈ℐℱ(𝑈), then 𝑋^∗𝐷⊆𝑋^∗and 𝐵^∗⊆𝐵^∗𝐷 hold.

Proof. They can be easily obtained from Proposition 9 and Definition 14.

Proposition 16. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. If 𝐷⊆𝐴 and 𝐷≠∅, then there must exist the following relation: 𝐿𝑈,𝐴,𝐼̃≤𝐿𝑈,𝐷,𝐼_𝐷̃. (19)

Proof. For any (𝑋,𝐵)∈𝐿(𝑈,𝐷,𝐼_𝐷̃), 𝑋^∗𝐷=𝐵 and 𝐵^∗𝐷=𝑋. From the above discussions, we know that (𝑋^∗∗,𝑋^∗)∈𝐿(𝑈,𝐴,𝐼)̃ is concept; thus we only need to verify 𝑋^∗∗=𝑋.
According to Proposition 9 (2), we can obtain that 𝑋^∗∗⊇𝑋. In addition to Proposition 15, 𝐵=𝑋^∗𝐷⊆𝑋^∗⇒𝑋^∗∗⊆𝐵^∗⊆𝐵^∗𝐷=𝑋. Therefore, 𝐿(𝑈,𝐴,𝐼)̃≤𝐿(𝑈,𝐷,𝐼_𝐷̃).

Corollary 17. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. If 𝐷⊆𝐴, 𝐷≠∅, then 𝐿_𝑈(𝑈,𝐷,𝐼_𝐷̃)⊆𝐿_𝑈(𝑈,𝐴,𝐼)̃.

Definition 18. Let (𝑈,𝐴,𝐼)̃ be an IF formal context, 𝐷⊆𝐴. We say that 𝐷 is a consistent set of 𝐿(𝑈,𝐴,𝐼)̃, if 𝐿(𝑈,𝐷,𝐼_𝐷̃)≅𝐿(𝑈,𝐴,𝐼)̃. If 𝐷 is a consistent set, and for any 𝑑∈𝐷, there exists 𝐿(𝑈,𝐷−𝑑,𝐼̃_𝐷−𝑑)≇𝐿(𝑈,𝐴,𝐼)̃, then 𝐷 is called an attribute reduction of 𝐿(𝑈,𝐴,𝐼)̃. The intersection set of all reductions is called the core of 𝐿(𝑈,𝐴,𝐼)̃.
Obviously, we can obtain the following propositions by the above definition.

Proposition 19. For any 𝐿(𝑈,𝐴,𝐼)̃, there must exist a reduction of it.

Proof. This proposition is immediately obtained in a similar way to the 𝐿(𝑈,𝐴,𝐼).

Proposition 20. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. If 𝐷⊆𝐴 and 𝐷≠∅, then 𝐷isaconsistentset⟺𝐿𝑈,𝐷,𝐼_𝐷̃≤𝐿𝑈,𝐴,𝐼̃. (20)

Proof. According to Proposition 15 and Definition 18, the conclusion can be obtained easily.

Definition 21. Let (𝑈,𝐴,𝐼)̃ be an IF formal context and suppose that 𝜏 is an index set and all the reductions are denoted by {𝐷^𝑖∣𝐷^𝑖isareduction,𝑖∈𝜏}. Then, attributes can be divided into four types as follows:(1)absolutely necessary attribute (core attribute) 𝑏∶𝑏∈∩_𝑖∈𝜏𝐷^𝑖;(2)relatively necessary attribute 𝑐∶𝑐∈∪_𝑖∈𝜏𝐷^𝑖−∩_𝑖∈𝜏𝐷^𝑖;(3)absolutely unnecessary attribute 𝑑∶𝑑∈𝐴−∪_𝑖∈𝜏𝐷^𝑖;(4)unnecessary attribute 𝑒∶𝑒∈𝐴−∩_𝑖∈𝜏𝐷^𝑖.
In general, the reduction of (𝑈,𝐴,𝐼)̃ is not unique. An example will be used to illustrate the above discussions as follows.

Example 22. For the IF formal context in Table 1, if we take out {𝑎,𝑐} from the attributes set 𝐴, then we can obtain a new IF formal context (𝑈,𝐷¹,𝐼̃_𝐷1), where 𝐷¹=𝐴−{𝑎,𝑐,𝑑}. And we can get all concepts of (𝑈,𝐷¹,𝐼̃_𝐷1), they are (1,𝐷¹₁), (2,𝐷¹₂), (4,𝐷¹₃), (12,𝐷¹₄), (13,𝐷¹₅), (24,𝐷¹₆), (123,𝐷¹₇), (124,𝐷¹₈), (𝑈,𝐷¹₉), and(∅,𝐷¹) marked by IFC1′, IFC2′, FC3′, IFC4′, IFC5′, IFC6′, IFC7′, IFC8′, IFC9′, and IFC10′, respectively, where 𝐷¹₁=),(,𝐷{(𝑏,0.7,0.2𝑒,0.8,0.1)}₁₂=),(,𝐷{(𝑏,0.8,0.2𝑒,0.2,0.7)}₁₃=),(,𝐷{(𝑏,0.8,0.1𝑒,0.2,0.8)}₁₄=),(,𝐷{(𝑏,0.7,0.2𝑒,0.2,0.7)}₁₅=),(,𝐷{(𝑏,0.2,0.6𝑒,0.2,0.6)}₁₆=),(,𝐷{(𝑏,0.7,0.2𝑒,0.2,0.8)}₁₇=),(,𝐷{(𝑏,0.8,0.2𝑒,0.2,0.8)}₁₈=),(,𝐷{(𝑏,0.2,0.6𝑒,0.2,0.7)}₁₉=),(,𝐷{(𝑏,0.7,0.2𝑒,0.2,0.8)}₁={(𝑏,1,0),(𝑒,1,0)}. (21)
In addition, we can obtain concept lattice of context 𝐿(𝑈,𝐷¹,𝐼̃_𝐷¹), as shown in Figure 2.
From Figures 1 and 2, we can find easily that 𝐿(𝑈,𝐷¹,𝐼̃_𝐷1) and 𝐿(𝑈,𝐴,𝐼)̃ are isomorphic. So, 𝐷¹ is a consistent set of (𝑈,𝐴,𝐼)̃. In fact, we can find that 𝐿(𝑈,𝐷¹−𝑏,𝐼̃_𝐷1−𝑏)≇𝐿(𝑈,𝐴,𝐼)̃, 𝐿(𝑈,𝐷¹−𝑒,𝐼̃_𝐷1−𝑒)≇𝐿(𝑈,𝐴,𝐼)̃ by calculation. Hence, 𝐷¹ is a reduction of (𝑈,𝐴,𝐼)̃.
If we take out {𝑎,𝑐,𝑒} from the attributes set 𝐴, then we can obtain a new IF formal context (𝑈,𝐷²,𝐼̃_𝐷2), where 𝐷²=𝐴−{𝑎,𝑐,𝑒}. And we can get all concepts of (𝑈,𝐷²,𝐼̃_𝐷2), which are (1,𝐷²₁), (2,𝐷²₂), (4,𝐷²₃), (12,𝐷²₄), (13,𝐷²₅), (24,𝐷²₆), (123,𝐷²₇), (124,𝐷²₈), (𝑈,𝐷²₉), and (∅,𝐷²), respectively, where 𝐷²₁=),(,𝐷{(𝑏,0.7,0.2𝑑,0.9,0.1)}₂₂=),(,𝐷{(𝑏,0.8,0.2𝑑,0.3,0.5)}₂₃=),(,𝐷{(𝑏,0.8,0.1𝑑,0.2,0.6)}₂₄=),(,𝐷{(𝑏,0.7,0.2𝑑,0.3,0.5)}₂₅=),(,𝐷{(𝑏,0.2,0.6𝑑,0.8,0.2)}₂₆=),(,𝐷{(𝑏,0.8,0.2𝑑,0.2,0.6)}₂₇=),(,𝐷{(𝑏,0.2,0.6𝑑,0.3,0.5)}₂₈=),(,𝐷{(𝑏,0.7,0.2𝑑,0.2,0.6)}₂₉=),(,𝐷{(𝑏,0.2,0.6𝑑,0.2,0.6)}₂={(𝑏,1,0),(𝑑,1,0)}. (22)
Obviously, 𝐿(𝑈,𝐷²,𝐼̃_𝐷2) is isomorphic with 𝐿(𝑈,𝐴,𝐼)̃.

Figure 2 

Concept lattice of the IF formal context (𝑈,𝐴,𝐼)̃.

Corollary 23. The core is the reduction ⇔ The reduction is only one.

Proof. ⇐Obviously.
⇒Assume that the core is the reduction, and the reduction is not unique; that is, there are two reductions: 𝐷^𝑖≠𝐷^𝑗 at least. Hence, the core of the reductions ∩𝐷^𝑡⊆𝐷^𝑖∩𝐷^𝑗⊂𝐷^𝑖. For 𝐷^𝑖 is the reduction, the proper subset of it (where it is the core of the reductions) must not be the reduction. This clearly contradicts the known conditions. So, if the core is the reduction, the reduction is only one.

Obviously, the following corollaries can be obtained by the above definitions and propositions.

Corollary 24. Let (𝑈,𝐴,𝐼)̃ be an IF formal context; 𝑎∈𝐴 is a core attribute ⇔𝐴−{𝑎} is not a consistent set.

Corollary 25. Let (𝑈,𝐴,𝐼)̃ be an IF formal context, 𝑎∈𝐴 is an unnecessary attribute ⇔𝐴−{𝑎} is a consistent set.
Since the reduction 𝐷 of an IF formal context satisfies the following conditions: (1) 𝐷⊆𝐴 a consistent set; (2) ∀𝑑∈𝐷,𝐷⧵{𝑑} is not a consistent set; it is helpful to give the necessary and sufficient conditions of consistent sets in order to get reductions more easily.

Proposition 26. Let (𝑈,𝐴,𝐼)̃ be an IF formal context, 𝐷⊆𝐴,𝐷≠∅. 𝐼̃⟺𝐵𝑇ℎ𝑒𝑛𝐷𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡𝑠𝑒𝑡𝑜𝑓𝑈,𝐴,_{∗∗𝐷∗𝐷}=𝐵^∗𝑓𝑜𝑟𝑎𝑛𝑦𝐵∈𝑈^𝐴. (23)

Proof. Assume that 𝐷 is a consistent set; then we have 𝐿(𝑈,𝐷,𝐼_𝐷̃)≤𝐿(𝑈,𝐴,𝐼)̃ according to Proposition 20. For any 𝐵∈𝑈^𝐴, it is easy to see that (𝐵^∗,𝐵^∗∗)∈𝐿(𝑈,𝐴,𝐼)̃. Thus, by Definition 13, there exists 𝐵^′∈𝑈^𝐷 such that (𝐵^∗,𝐵^′)∈𝐿(𝑈,𝐷,𝐼_𝐷̃). Hence, 𝐵^′=𝐵^∗∗𝐷 and 𝐵^∗=𝐵^′∗𝐷, which concludes 𝐵^{∗∗𝐷∗𝐷}=𝐵^∗.
Conversely, suppose that 𝐵^{∗∗𝐷∗𝐷}=𝐵^∗ for any 𝐵∈𝑈^𝐴. To prove that 𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃, then it suffices to show that for any (𝑋,𝐵)∈𝐿(𝑈,𝐴,𝐼)̃, there exists 𝐵^′∈𝑈^𝐷 such that (𝑋,𝐵^′)∈𝐿(𝑈,𝐷,𝐼_𝐷̃). Thus, suppose that (𝑋,𝐵)∈𝐿(𝑈,𝐴,𝐼)̃; then we can get that 𝑋^∗=𝐵 and 𝑋=𝐵^∗. Taking 𝐵^′=𝐵^∗∗𝐷, then we can obtain that 𝐵^′∗𝐷=𝐵^{∗∗𝐷∗𝐷}=𝐵^∗=𝑋, and so (𝑋,𝐵^′)∈𝐿(𝑈,𝐷,𝐼_𝐷̃).

Corollary 27. Let (𝑈,𝐴,𝐼)̃ be an IF formal context, 𝐷⊆𝐴, and   𝐷≠∅. Then 𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃⇔∀𝐵∈𝑈^𝐴, ∃𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵^∗.

Proof. It can be certified easily by Proposition 20.

Proposition 28. Let (𝑈,𝐴,𝐼)̃ be an IF formal context, 𝐷⊆𝐴,𝐷≠∅, and 𝐸=𝐴−𝐷. Then, 𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃ if and only if  ∀𝐵∈𝑈^𝐸, ∃𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵^∗𝐸.

Proof. Suppose that 𝐷 is a consistent set; then 𝐵₁^∗=𝐵₁^∗𝐷 for any 𝐵₁∈𝑈^𝐴. For any 𝐵∈𝑈^𝐸, let 𝐵₂=𝐵∪𝐵₁, with 𝐵₁∈𝑈^𝐷, 𝐵₁(𝑎)=⟨∧_{∀𝑥∈𝑈}𝜇𝐼(̃𝑥,𝑎),∨_{∀𝑥∈𝑈}𝛾𝐼(̃𝑥,𝑎)⟩, for 𝑎∈𝐷; then 𝐵₁^∗=𝐵^∗𝐸. Thus, there exists 𝐵^′∈𝑈^𝐷 such that then 𝐵^′∗𝐷=𝐵^∗𝐸 by Corollary 25.
Suppose that there exists 𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵^∗𝐸 for any 𝐵∈𝑈^𝐸. Let (𝑋,𝐵₂)∈𝐿(𝑈,𝐴,𝐼)̃; then 𝑋^∗=𝐵₂ and 𝑋=𝐵₂^∗. Let 𝐵₂=𝐵∪𝐵₁ with 𝐵∈𝑈^𝐸 and 𝐵₁∈𝑈^𝐷, then 𝐵₂^∗=(𝐵∪𝐵₁)^∗=𝐵^∗∩𝐵₁^∗=𝐵^∗𝐷∩𝐵₁^∗𝐸. Hence, there exists 𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵₁^∗𝐸, so 𝐵^∗𝐷∩𝐵₁^∗𝐸=𝐵₁^∗𝐷∩𝐵^′∗𝐷=(𝐵₁∪𝐵^′)^∗𝐷=𝑋. Now it follows from Proposition 9 (3) and Definition 10 that ((𝐵₁∪𝐵^′)^∗𝐷,(𝐵₁∪𝐵^′)^{∗𝐷∗𝐷})∈𝐿(𝑈,𝐷,𝐼_𝐷̃), and so 𝐿(𝑈,𝐷,𝐼_𝐷̃)≤𝐿(𝑈,𝐴,𝐼)̃, which means that 𝐷 is a consistent set.

The functions of attributes, which are closely related to consistent sets, vary from one to another. So, sufficient conditions to determine the type of attributes are useful in attribute reduction.

Proposition 29. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. Then, 𝑎 is an absolutely necessary attribute if there exists 𝑥_𝑖,𝑥_𝑗∈𝑈 such that 𝐼(̃𝑥_𝑖,𝑎)>𝐼(̃𝑥_𝑗,𝑎), and for any 𝑏≠𝑎  𝐼(̃𝑥_𝑖,𝑏)≤𝐼(̃𝑥_𝑗,𝑏).

Proof. Suppose that 𝑎 is an unnecessary attribute, then 𝐷=𝐴−{𝑎} is a consistent set; that is, 𝐿(𝑈,𝐷,𝐼_𝐷̃)≅𝐿(𝑈,𝐴,𝐼)̃. Let 𝐵=𝑥^∗_𝑖, then (𝐵^∗,𝐵^∗∗)∈𝐿(𝑈,𝐴,𝐼)̃. Since 𝐼(̃𝑥_𝑖,𝑎)>𝐼(̃𝑥_𝑗,𝑎), we know that 𝑥_𝑖∈𝐵^∗ and 𝑥_𝑗∉𝐵^∗. Assume that (𝑋,𝐵^′)∈𝐿(𝑈,𝐷,𝐼_𝐷̃) and 𝑥_𝑖∈𝑋, then 𝑋^∗𝐷=𝐵^′, 𝐵^′∗𝐷=𝑋. From 𝑥_𝑖∈𝑋, we know that 𝐵^′=𝑋^∗𝐷⊆𝑥^∗𝐷_𝑖. But since for any 𝑏≠𝑎, 𝐼(̃𝑥_𝑖,𝑏)≤𝐼(̃𝑥_𝑗,𝑏). It follows that 𝐵^′(𝑏)≤𝐼(̃𝑥_𝑗,𝑏) and so 𝑥_𝑗∈𝑋. That is to say, in IF formal context (𝑈,𝐷,𝐼_𝐷̃), every concept which contains 𝑥_𝑖 also contains 𝑥_𝑗. But 𝐼(̃𝑥_𝑖,𝑎)>𝐼(̃𝑥_𝑗,𝑎). Consistently, there is no concept in 𝐿(𝑈,𝐷,𝐼_𝐷̃) whose extent is equal to 𝐵^∗. Thus, 𝐷 is not a consistent set, which gets a contradiction. Therefore, 𝑎 is an absolutely necessary attribute.

Proposition 30. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. Then, 𝑎 is an unnecessary attribute if the following conditions hold: for any 𝑥_𝑖,𝑥_𝑗∈𝑈, if 𝐼(̃𝑥_𝑖,𝑎)>𝐼(̃𝑥_𝑗,𝑎), then there exists 𝑏≠𝑎 such that 𝐼(̃𝑥_𝑖,𝑏)>𝐼(̃𝑥_𝑗,𝑏). Moreover, if there exists 𝑥_𝑘∈𝐴 such that 𝐼(̃𝑥_𝑘,𝑎)≥𝐼(̃𝑥_𝑖,𝑎), then 𝐼(̃𝑥_𝑘,𝑏)≥𝐼(̃𝑥_𝑗,𝑏).

Proof. Suppose that 𝑎=𝑎_𝑙, 𝐷=𝐴−{𝑎_𝑙}. It suffices to prove that 𝐷 is consistent set. By Corollary 27, it remains to prove that for any 𝐵∈𝑈^𝐴, there exists 𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵^∗. So suppose that 𝐵={𝐼(̃𝑥_𝑠1,𝑎₁),𝐼(̃𝑥_𝑠2,𝑎₂),…,𝐼(̃𝑥_𝑠𝑚,𝑎_𝑚)}, where 𝑥_𝑠𝑡∈𝑈𝑎_𝑡∈𝐴, 1≤𝑠≤|𝑈|, 1≤𝑡≤𝑚.
If for any 𝑥∈𝑈,  𝐼(̃𝑥,𝑎_𝑙)≥𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙), then let 𝐵^′={𝐼(̃𝑥_𝑠1,𝑎₁), 𝐼(̃𝑥_𝑠2,𝑎₂),…, 𝐼(̃𝑥_𝑠𝑙−1,𝑎_𝑙−1), 𝐼(̃𝑥_𝑠𝑙+1,𝑎_𝑙+1),…,𝐼(̃𝑥_𝑠𝑚,𝑎_𝑚)}, and so we can get that 𝐵^′∗𝐷=𝐵^∗.
Otherwise, assume that there are {𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝛼}⊆𝑈, such that 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙)>𝐼(̃𝑥_𝑡𝛽,𝑎_𝑙)(1≤𝛽≤𝛼); then there exist, according the condition, 𝑎_𝑞1,𝑎_𝑞2,…,𝑎_𝑞𝜂∈𝐴−{𝑎_𝑙} such that 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑞𝜎)>𝐼(̃𝑥_𝑡𝛽,𝑎_𝑞𝜎)(1≤𝜎≤𝜂). Moreover, if there exists 𝑥_𝑘 such that 𝐼(̃𝑥_𝑘,𝑎_𝑙)≥𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙), then 𝐼(̃𝑥_𝑘,𝑎_𝑞𝜎)≥𝐼(̃𝑥_𝑠𝑙,𝑎_𝑞𝜎). Let 𝐵^′=𝐵₁∪𝐵₂, where 𝐵₁=𝐼̃𝑥_𝑠1,𝑎₁,𝐼̃𝑥_𝑠2,𝑎₂,…,𝐼̃𝑥_𝑠𝑙−1,𝑎_𝑙−1,𝐼̃𝑥_𝑠𝑙+1,𝑎_𝑙+1,…,𝐼̃𝑥_𝑠𝑚,𝑎_𝑚,𝐵₂=𝐼̃𝑥_𝑠𝑙,𝑎_𝑞1,𝐼̃𝑥_𝑠𝑙,𝑎_𝑞2,…,𝐼̃𝑥_𝑠𝑙,𝑎_𝑞𝜂; (24) then it follows that 𝐵^′∗𝐷=𝐵₁^∗𝐷∩𝐵₂^∗𝐷. So we know that if 𝑥∈𝐵^∗, that is, 𝐼(̃𝑥,𝑎_𝑡)≥𝐼(̃𝑥_𝑠𝑡,𝑎_𝑡), then 𝐼(̃𝑥,𝑎_𝑞𝜎)≥𝐼(̃𝑥_𝑠𝑙,𝑎_𝑞𝜎). Then, 𝑥∈𝐵₁^∗ and 𝑥∈𝐵₂^∗. It follows that 𝑥∈𝐵^′∗𝐷 and so 𝐵^∗⊆𝐵^′∗𝐷. If 𝑥∉𝐵^∗, that is, there exists 𝑎_𝑡0∈𝐴 such that 𝐼(̃𝑥_𝑠𝑡,𝑎_𝑡0)>𝐼(̃𝑥,𝑎_𝑡0). If 𝑎_𝑡0=𝑎_𝑙, that is, 𝑥∈{𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝛼}, then 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑞𝜎)>𝐼(̃𝑥,𝑎_𝑞𝜎); that is, 𝑥∉𝐵₂^∗𝐷, and so 𝑥∉𝐵^′∗𝐷. If 𝑎_𝑡0≠𝑎_𝑙, that is, then 𝐼(̃𝑥_𝑠𝑡0,𝑎_𝑡0)>𝐼(̃𝑥,𝑎_𝑡0); that is, 𝑥∉𝐵₁^∗𝐷, and so 𝑥∉𝐵^′∗𝐷. Hence, we conclude that for any 𝐵∈𝑈^𝐴 there exists 𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵^∗.

Proposition 31. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. Then 𝑎 is an unnecessary attribute if there exists 𝑏∈𝐴 such that for any 𝑥_𝑖,𝑥_𝑗∈𝑈, 𝐼(̃𝑥_𝑖,𝑎)>𝐼(̃𝑥_𝑗,𝑎) implies that 𝐼(̃𝑥_𝑖,𝑏)>𝐼(̃𝑥_𝑗,𝑏). Moreover, if 𝑏 is an absolutely necessary attribute, then 𝑎 is an absolutely unnecessary attribute.

Proof. We denote 𝑎=𝑎_𝑙, 𝑏=𝑎_𝑙−1, and 𝐷=𝐴−{𝑎_𝑙}. It suffices to prove that 𝐷 is consistent set. By Corollary 27, it remains to prove that for any 𝐵∈𝑈^𝐴, there exists 𝐵^′∈𝑈^𝐷, such that 𝐵^′∗𝐷=𝐵^∗. So suppose that 𝐵={𝐼(̃𝑥_𝑠1,𝑎₁),𝐼(̃𝑥_𝑠2,𝑎₂),…,𝐼(̃𝑥_𝑠𝑚,𝑎_𝑚)}, where 𝑥_𝑠𝑡∈𝑈, 𝑎_𝑡∈𝐴, 1≤𝑠≤|𝑈|, 1≤𝑡≤𝑚.
If for any 𝑥∈𝑈, 𝐼(̃𝑥,𝑎_𝑙)≥𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙), then let 𝐵^′={𝐼(̃𝑥_𝑠1,𝑎₁),𝐼(̃𝑥_𝑠2,𝑎₂),…,𝐼(̃𝑥_𝑠𝑙−1,𝑎_𝑙−1), 𝐼(̃𝑥_𝑠𝑙+1,𝑎_𝑙+1),…,𝐼(̃𝑥_𝑠𝑚,𝑎_𝑚)}, and it follows that 𝐵^′∗𝐷=𝐵^∗.
Otherwise, there exists 𝑥_𝑘 such that 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙)>𝐼(̃𝑥_𝑘,𝑎_𝑙); then 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙−1)>𝐼(̃𝑥_𝑘,𝑎_𝑙−1). Denote {𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝛼}⊆𝑈 to be the set whose elements satisfy the condition that 𝐼(̃𝑥_𝑡𝛽,𝑎_𝑙)≥𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙)(1≤𝛽≤𝛼). Then, 𝐼(̃𝑥_𝑡𝛽,𝑎_𝑙)>𝐼(̃𝑥_𝑘,𝑎_𝑙) and thus 𝐼(̃𝑥_𝑡𝛽,𝑎_𝑙−1)>𝐼(̃𝑥_𝑘,𝑎_𝑙−1). Let 𝐵^′=𝐵₁∪𝐵₂, where 𝐵₁=𝐼̃𝑥_𝑠1,𝑎₁,𝐼̃𝑥_𝑠2,𝑎₂,…,𝐼̃𝑥_𝑠𝑙−1,𝑎_𝑙−1,𝐼̃𝑥_𝑠𝑙+1,𝑎_𝑙+1,…,𝐼̃𝑥_𝑠𝑚,𝑎_𝑚,𝐵₂=𝐼̃𝑥_𝑠1,𝑎₁,𝐼̃𝑥_𝑠2,𝑎₂,…,𝐼̃𝑥_𝑠𝑙−2,𝑎_𝑙−2,_1≤∧_𝛽≤𝛼𝐼̃𝑥_𝑡𝛽,𝑎_𝑙−1,𝐼̃𝑥_𝑠𝑙+1,𝑎_𝑙+1,…,𝐼̃𝑥_𝑠𝑚,𝑎_𝑚, (25) then it follows that 𝐵^′∗𝐷=𝐵₁^∗𝐷∩𝐵₂^∗𝐷. So we know that if 𝑥∈𝐵^∗, that is, for all 𝑎_𝑡∈𝐴,𝐼(̃𝑥,𝑎_𝑡)≥𝐼(̃𝑥_𝑠𝑡,𝑎_𝑡), then 𝑥∈𝐵^∗𝐷₁ and 𝑥∈{𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝛼}. Then, 𝑥∈𝐵₁^∗ and 𝑥∈𝐵₂^∗. It follows that 𝑥∈𝐵^′∗𝐷 and so 𝐵^∗⊆𝐵^′∗𝐷. If 𝑥∉𝐵^∗, then 𝑥∉𝐵^∗𝐷₁ or 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙)>𝐼(̃𝑥,𝑎_𝑙). If 𝑥∉𝐵^∗𝐷₁, then 𝑥∈𝐵^′∗𝐷. If 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑙)>𝐼(̃𝑥,𝑎_𝑙), then 𝐼(̃𝑥_𝑡𝛽,𝑎_𝑙−1)>𝐼(̃𝑥,𝑎_𝑙−1), and so, there exists 𝑎_𝑡0∈𝐴 such that 𝐼(̃𝑥_𝑠𝑡,𝑎_𝑡0)>𝐼(̃𝑥,𝑎_𝑡0). If 𝑎_𝑡0=𝑎_𝑙, that is, 𝑥∈{𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝛼}, then 𝐼(̃𝑥_𝑠𝑙,𝑎_𝑞𝜎)>𝐼(̃𝑥,𝑎_𝑞𝜎); that is, 𝑥∉𝐵₂^∗𝐷, and so ∧_{1≤𝛽≤𝛼}𝐼(̃𝑥_𝑡𝛽,𝑎_𝑙−1)>𝐼(̃𝑥,𝑎_𝑙−1) that is, 𝑥∉𝐵^∗𝐷₂ and so 𝑥∉𝐵^′∗𝐷. Thus, we conclude that for any 𝐵∈𝑈^𝐴 there exists 𝐵^′∈𝑈^𝐷 such that 𝐵^′∗𝐷=𝐵^∗. Therefore 𝑎_𝑙 is an unnecessary attribute.
Moreover, suppose that 𝑎_𝑙−1 is an absolutely necessary attribute and 𝐷 is a consistent set which contains 𝑎_𝑙−1. Since 𝑎_𝑙−1 is an absolutely necessary attribute, we have 𝑎_𝑙−1∈𝐷; thus 𝐷−{𝑎_𝑙} is also a consistent set, that is, 𝐷 is not a reduction. Therefore, 𝑎_𝑙 is an absolutely unnecessary attribute.

Corollary 32. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. Then, 𝑎∈𝐴 is an absolutely unnecessary attribute if for any 𝑥_𝑖,𝑥_𝑗∈𝑈, 𝐼(̃𝑥_𝑗,𝑎)≥𝐼(̃𝑥_𝑖,𝑎).

5. Approach to Reduction

In this section, discernibility matrix and discernibility function [18, 41] are introduced to compute all reductions for an IF formal context based on the conclusions discussed in Section 4, and we discuss the approach to reduction as well as the corresponding characteristics. Furthermore, we also show corresponding reduction algorithm.

Definition 33. Let (𝑈,𝐴,𝐼)̃ be an IF formal context and 𝑋_𝑖,𝑋_𝑗∈𝐿_𝑈(𝑈,𝐴,𝐼)̃∪{{𝑥_𝑗}∣𝑥_𝑗∈𝑈}, we define 𝒟^∗𝑋_𝑖,𝑋_𝑗=𝑎∈𝐴∣𝜇_𝑋∗𝑖(𝑎)>𝜇_𝑋∗𝑗(𝑎)or𝛾_𝑋∗𝑖(𝑎)<𝛾_𝑋∗𝑗(𝑎)𝑋_𝑖∈𝐿_𝑈𝐼̃,𝑋𝑈,𝐴,_𝑗=𝑥_𝑗⊄𝑋_𝑖𝒟∅otherwise,_∗=𝒟^∗𝑋_𝑖,𝑋_𝑗∣𝑋_𝑖,𝑋_𝑗∈𝐿_𝑈𝑈,𝐴,𝐼̃∪𝑥_𝑗∣𝑥_𝑗=𝒟∈𝑈_∗𝑋_𝑖,𝑋_{𝑗|𝐿𝑈(𝑈,𝐴,}̃𝐼^)∪{{𝑥_𝑗^}∣𝑥_𝑗^∈𝑈}|2. (26)
Then, 𝒟^∗(𝑋_𝑖,𝑋_𝑗) is called discernibility attributes set between 𝑋_𝑖 and 𝑋_𝑗. And 𝒟^∗ is referred to as discernibility matrix of the IF formal context (𝑈,𝐴,𝐼)̃.

Proposition 34. Let (𝑈,𝐴,𝐼)̃ be an IF formal context and 𝐷⊆𝐴,𝐷≠∅. Then, the following two propositions are equivalent.(1)𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃.(2)If 𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅, then 𝐷∩𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅, for all 𝒟^∗(𝑋_𝑖,𝑋_𝑗)∈𝒟^∗.

Proof. (1)⇒(2) We assume that property (2) does not hold. That is, ∃𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅∈𝒟^∗, 𝑋_𝑗={𝑥_𝑗}⊄𝑋_𝑖such that 𝐷∩𝒟^∗(𝑋_𝑖,𝑋_𝑗)=∅. That is to say for all 𝑎∈𝐷 such that 𝜇_𝑋∗𝑖(𝑎)≤𝜇_𝑋∗𝑗(𝑎) and  𝛾_𝑋∗𝑖(𝑎)≥𝛾_𝑋∗𝑗(𝑎); hence 𝑥_𝑗∈𝑋^{∗𝐷∗𝐷}_𝑖. In other words, (𝑋_𝑖,𝑋^∗𝐷_𝑖)∉𝐿(𝑈,𝐷,𝐼_𝐷̃). It is paradoxical that 𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃.
(2)⇒(1) If for all 𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅∈𝒟^∗, then for all 𝑎∈𝐷∩𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅ such that 𝜇_𝑋∗𝑖(𝑎)>𝜇_𝑋∗𝑗(𝑎) or 𝛾_𝑋^∗_𝑇𝑖(𝑎)<𝛾_𝑋^∗_𝑇𝑗(𝑎) and ⇒𝜇_𝑋^∗_𝐷𝑖(𝑎)>𝜇_𝑋^∗_𝐷𝑗(𝑎) or 𝛾_𝑋^∗_𝐷𝑖(𝑎)<𝛾_𝑋^∗_𝐷𝑗(𝑎). Hence, for all 𝑋_𝑖∈𝐿_𝑈(𝑈,𝐴,𝐼)̃, 𝑋_𝑗={𝑥_𝑗}⊄𝑋_𝑖 such that 𝑥_𝑗∉𝑋^{∗𝐷∗𝐷}_𝑖⇒𝑋_𝑖=𝑋^{∗𝐷∗𝐷}_𝑖. So, there exists (𝑋_𝑖,𝑋^∗𝐷_𝑖)∈𝐿(𝑈,𝐷,𝐼_𝐷̃). Thus, 𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃.

Definition 35. Let (𝑈,𝐴,𝐼)̃ be an IF formal context and 𝒟^∗ be discernibility matrix of (𝑈,𝐴,𝐼)̃. We define ℳ^∗=∧_{𝒟∗(𝑋𝑖,𝑋𝑗)∈𝒟∗}∨_{𝑎𝑘∈𝒟∗(𝑋𝑖,𝑋𝑗)}𝑎_𝑘,𝒟^∗𝑋_𝑖,𝑋_𝑗≠∅. (27)
Then, ℳ^∗ is called discernibility function of the IF formal context (𝑈,𝐴,𝐼)̃.

Proposition 36. Let (𝑈,𝐴,𝐼)̃ be an IF formal context. The minimal disjunctive normal form of discernibility function is defined as ℳ^∗=∨^𝑝∧𝑎_𝑘=1^𝑞𝑘_𝑠𝑠=1. (28) Denote 𝐵_𝑘={𝑎_𝑠∣𝑠=1,2,…,𝑞_𝑘}, then {𝐵_𝑘∣𝑘=1,2,…,𝑝} are all reductions of the IF formal context (𝑈,𝐴,𝐼)̃.

Proof. It can be easily verified by Proposition 34, Definition 35, and the definition of minimal disjunctive normal of discernibility function.

From above discussion, we know that to get the attribute reductions in the IF concept lattices is equal to find the minimum consistent set 𝐷 which satisfies 𝐷∩𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅ for any 𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅.

Similarly, for any 𝑋_𝑖∈𝐿_𝑈(𝑈,𝐴,𝐼)̃ and 𝒟^∗(𝑋_𝑖,𝑋_𝑗)≠∅, the discernibility function of 𝑋_𝑖 is defined as ℳ^∗𝑋_𝑖=∧_{𝑋𝑗={𝑥𝑗}⊄𝑋𝑖}∨_{𝑎𝑘∈𝒟∗(𝑋𝑖,𝑋𝑗)}𝑎_𝑘=∨^𝑝∧𝑎_𝑘=1^𝑞𝑘_𝑠𝑠=1. (29)

Corollary 37. 𝐷 is a consistent set of (𝑈,𝐴,𝐼)̃⇔𝑋^∗∗𝐷⊆𝑋^∗∗.

In the following, we investigate how to get IF attributes reductions from all attributes in concept lattices by illustrating an example.

Example 38 (renewal Example 22). All the reductions can be computed by discernibility matrix and discernibility function in Example 22.
By the definition of discernibility matrix, the results are presented in Table 2.
Hence, we can get that ℳ^∗=()∧()∧()∧()∧()∧()()=()∨().𝑎∨𝑑∨𝑒𝑎∨𝑏∨𝑐∨𝑑∨𝑒𝑏∨𝑐𝑎∨𝑏∨𝑐𝑎∨𝑐∨𝑑∨𝑒𝑑∨𝑒∧𝑏=𝑏∧𝑑∨𝑒𝑏∧𝑑𝑏∧𝑒 (30)
Through calculation and analysis, there are two reductions which are 𝐷¹={𝑏,𝑑} and 𝐷²={𝑏,𝑒}, for the IF formal context in Table 1. The core attribute of the IF formal context is 𝑏. 𝑑,𝑒 are relatively necessary attributes; 𝑎,𝑐 are absolutely unnecessary attributes.

Table 2 

Discernibility matrix in Example 22.

Algorithm 39. Discernibility functions are monotonic Boolean functions, and we obtain that the minimal disjunctive normal form of the discernibility function determine all the reductions. Algorithm of attribute reduction of concept based on the IF formal context is described as follows, and the flow chart of the algorithm is shown in Figure 3. Consider

Figure 3 

The flow chart of Algorithm 39.

Input: An IF formal context 𝐾=(𝑈,𝐴,𝐼)̃, where 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛}, 𝐴={𝑎₁,𝑎₂,…,𝑎_𝑚}.

Output: RED(𝐾) // All reductions of the IF formal context.

Step  1: Initialized setting. We denote the initialized information by 𝐾 and read data table.

Step  2: Compute 𝑋^∗_𝑖 for 𝑋_𝑖∈𝑃(𝑈).

Step  3: Compute 𝑋_𝑖^∗∗, the corresponding extension of 𝑋^∗_𝑖, according to Definitions 7 and 10. And remove duplicates.

Step  4: Compute the set of discernibility attributes 𝒟^∗(𝑋_𝑖^∗∗,𝑋_𝑗) for according to Definition 33.

Step  5: Compute ℳ^∗=∧_𝒟^∗_{(𝑋𝑖,𝑋𝑗)∈𝒟}^∗{∨{𝑎_𝑘∣𝑎_𝑘∈𝒟^∗(𝑋_𝑖^∗∗,𝑋_𝑗)}}.

Step  6: Compute ℳ^∗=∨^𝑝_𝑘=1(∧𝑎_𝑠^𝑞𝑘_𝑠=1).

Step  7: Let 𝐵_𝑘={𝑎_𝑠∣𝑠≤𝑞_𝑘} and RED(𝐾)={𝐵_𝑘∣𝑘≤𝑝}.

Step  8: Output 𝐿(𝑈,𝐴,𝐼)̃, 𝒟^∗ and RED(𝐾).

In the worst case, the time complexity of the proposed algorithm is 𝑂(2^|𝑈||𝑈||𝐴|), where |𝑈| is the number of objects and |𝐴| is the number of attributes. If the number of attributes has an upper bound, as usually happens, the cost of time grows geometrically with the number of objects.

Experimental computing program can be designed and carried out so as to apply the algorithm studied more directly and applicably. The main process of the program will be introduced by the flow chart. According to Algorithm 39, the process of the program can be designed and listed in Figure 3: the flow chart of the program.

Furthermore, the program has been employed to compute all concepts and reductions of the IF formal context in Example 8, which are consistent with the results obtained in Example 38. The test shows that the program is effective.

Example 40. Let𝐾=(𝑈,𝐴,𝐼)̃ be an IF formal context about some emerging viruses presented in Table 3, where 𝑈={𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇} is the set of emerging viruses and 𝐴={𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₅} is the set of some important characteristics about viruses. The interpretations of the attributes are listed as follows.(𝑎₁) The type of genome’s nucleic acids, where membership degree is referred to as DNA; nonmembership degree is referred to as RNA.(𝑎₂)Envelope. (𝑎₃) Strand: membership degree is referred to as single strand; nonmembership degree is referred to as double strand.(𝑎₄) The heredity of protein. (𝑎₅) Pathogenicity: membership degree is referred to as pathogenicit; nonmembership degree is referred to as opposite.
We can obtain one reduction RED(𝐾)={𝑎₁,𝑎₃,𝑎₄} and thirty concepts of 𝐾, of which detailed description are shown in Table 4.
The results obtained above presented the relations between these new viruses which may be useful to find their ancestors and evolution mechanism for the viruses’ researchers. In a way, the approach to construction and attribute reduction of IF concept lattice, presented in the paper, can be used to prune the redundancies and process the information.

Table 3 

The target IF formal context 𝐾.

Table 4 

The results of Example 40.

6. Conclusion

This paper introduced the intuitionistic fuzzy set into the concept lattice theory firstly. Then, we offered the related definitions and propositions of attribute reduction in concept lattices which were examined by some examples. What is more, the discernibility matrix and discernibility function in concept lattice were defined which presented an approach to attribute reduction of concept lattice based on the intuitionistic fuzzy context. The attribute reduction of IF concept lattice made the representation of implicit knowledge simpler in IF formal context. In fact, an approach to reduction was introduced based on the discernibility matrix of IF concept lattice. The results of this paper extended the theory of concept lattice, although there were some problems in the applications.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 61105041), Science and Technology Program of Board of Education of Chongqing (no. KJ120805), National Natural Science Foundation of CQ CSTC (no. cstc2013jcyjA40051), and Postgraduate Innovation Foundation of Chongqing University of Technology (no. YCX2012203).