The approximation set of a vague set in rough approximation space

Information Sciences 300 (2015) 1–19

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The approximation set of a vague set in rough approximation space Qinghua Zhang a,b,⇑, Jin Wang a, Guoyin Wang a, Hong Yu a a b

The Chongqing Key Laboratory of Computational Intelligence, Chongqing University of Posts and Telecommunications, Chongqing 400065, China College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

a r t i c l e

i n f o

Article history: Received 10 November 2013 Received in revised form 8 December 2014 Accepted 9 December 2014 Available online 18 December 2014 Keywords: Vague set Approximation set Rough set Granular computing Similarity Fuzzy set

a b s t r a c t Vague set is a further generalization of fuzzy set. In rough set theory, a target concept may be a defined set, fuzzy set or vague set. That the target concept is a defined set or fuzzy set was analyzed in detail in our other papers respectively. In general, we can only get two boundaries of an uncertain concept when we use rough set to deal with the uncertain problems and can not get a useable approximation defined set which is a union set with many granules in Pawlak’s approximation space. In order to overcome above shortcoming, we mainly discuss the approximation set of a vague set in Pawlak’s approximation space in the paper. Firstly, many preliminary concepts or definitions related to the vague set and the rough set are reviewed briefly. And then, many new definitions, such as 0.5-crisp set, step-vague set and average-step-vague set, are defined one by one. The Euclidean similarity degrees between a vague set and its 0.5-crisp set, step-vague set and averagestep-vague set are analyzed in detail respectively. And then, the conclusion that the Euclidean similarity degree between a vague set and its 0.5-crisp set is better than the Euclidean similarity degree between the vague set and the other defined set in the approximation space ðU; RÞ is drawn. Afterward, it is proved that average-step-vague set is an optimal step-vague set because the Euclidean similarity degree between a vague set and its average-step-vague set in the approximation space ðU; RÞ can reach the maximum value. Finally, the change rules of the Euclidean similarity degree with the different knowledge granularities are discussed, and these rules are in accord with human cognitive mechanism in a multi-granularity knowledge space. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Since the fuzzy set theory was proposed by Zadeh in 1965 [46], it has been successfully applied to process many uncertain problems, such as fuzzy image segmentation, fuzzy clustering analysis, fuzzy control and so on. The uncertainty of an uncertain concept is described by a membership degree in the fuzzy set theory. In other words, a fuzzy set interprets the uncertain concepts with some membership function as well as a fundamental tool for revealing and analyzing uncertain problems, and it has been frequently used in real-life world applications. In fact, the fuzzy set theory is an extension of the classical Canter’s set theory. A fuzzy set A is a class of objects which satisfies a certain property and each object x has ⇑ Corresponding author at: The Chongqing Key Laboratory of Computational Intelligence, Chongqing University of Posts and Telecommunications, Chongqing 400065, China. Tel./fax: +86 023 62471793. E-mail address: [email protected] (Q. Zhang). http://dx.doi.org/10.1016/j.ins.2014.12.023 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

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a membership degree of A, denoted as lA ðxÞ. The membership function has the following characteristics: the single-value membership degree contains the evidences for both supporting and opposing x, and the value lA ðxÞ almost is a subjective value which is probably different due to the different researchers. Atanassov [2] proposed the intuitionistic fuzzy set as a generalization of the fuzzy set in 1986, and Gau and Buehrer [10] put forward the concept of the vague set in 1993. Afterward, Bustince and Burillo [5] drew a conclusion that the essence of the vague set was the same as that of the intuitionistic fuzzy set in 1996. Lu and Ng [14] analyzed the relationship between the vague set and the intuitionistic fuzzy set in detail, and they pointed out that the vague set was more natural than the intuitionistic fuzzy set to characterize fuzzy objects and it was not isomorphic to the intuitionistic fuzzy set. So the terms of both the vague set and the intuitionistic fuzzy set exist in different literatures or research papers. In this paper, the intuitionistic fuzzy set and the vague set are uniformly called the vague set due to the fact that many references are related to the vague set. Gau and Buehrer [10] pointed out that the drawback of the single membership value in the fuzzy set theory is that the evidence for xi 2 U and the evidence against xi 2 U are in fact mined together. They also stated that the single value revealed nothing about its accuracy. To tackle this problem, Gau and Buehrer [10] proposed the notion of vague set, which allowed to use interval-based membership instead of using point-based membership as the fuzzy set. They employed a truth-membership function t V ðxi Þ and a false-membership f V ðxi Þ to characterize the lower bounds on lV ðxi Þ. These bounds are used to create a subinterval on ½0; 1, namely, tV ðxi Þ 6 lV ðxi Þ 6 1  f V ðxi Þ. The vague set as well as the fuzzy set and the rough set [18], is not only an important tool to process uncertain problems [41] but also a typical soft computing method in data mining, especially, in the dynamic data mining field, knowledge has a greater uncertainty in the process of knowledge acquisition [52]. Vagueness, as well as fuzziness and roughness, has attracted the attentions of many researchers since the vague concept was proposed. And vagueness is current subject of vigorous debate in the philosophy of logic and language. Vague terms, such as ‘tall’, ‘red’, ‘bald’ and ‘tadpole’ have borderline cases (arguably, someone may be neither tall nor not tall), and they lack well-defined extensions (there is no sharp boundary between tall people and the rest). The phenomenon of vagueness poses a fundamental challenge to classical logic and semantics, which assumes that propositions are either true or false and that extensions are determinate [11,12,16,28]. Ronzitti [29] analyzed the vagueness and metaphysics which covers important questions concerning vagueness that arise in connection with the deployment of certain key metaphysical notions. Based on rough set theory, there are many papers discussing the relationships between the rough set and the vague set. Rough set theory seems to be well suited as a mathematical model of vagueness and uncertainty, and vagueness is a property of set (concept) and is strictly related to the existence of the boundary region of a set, whereas uncertainty is a property of elements of the set and is related to rough membership function [19,20]. Orowska [17] gave a formal framework to what was considered to be different from rough set theory ways of making vague concepts precise and described semantics in this framework. Skowron, Bazan and Swiniarski [3,26,27,30–37] proposed a rough set approach to search the approximate concept (or set) of vague concept within the adaptive learning framework, and the boundary regions of approximate concepts within the adaptive learning framework are satisfying the higher order vagueness condition, i.e., the boundary regions of vague concepts are not defined. Vagueness has been equated with the idea that, for several different reasons, an object can not always be considered as satisfying either A or not A as prescribed by classical logic [8]. The uncertainty of a vague concept has been an important issue since the vague set was proposed, and the fuzziness, roughness and vagueness are three main uncertainty metrics of the vague concept. In the Pawlak’s approximation space, the uncertainty of vague concept has been studied, and the change rules of uncertainty have been analyzed in a multi-granularity knowledge space [3,17,20,31,35]. There are three types of knowledge that can be specified according to the rough set theory, and three corresponding types of algebraic structures appear in rough set theory [39]. Wolski [40] thought that rough set theory can be viewed as a modified semantics, and in order to be more precise, rough set provided a modal version of this semantics. Then Bonikowski [4] proposed a new formal approach to vagueness, and many important conditions concerning the membership relation for vague sets, in connection to multi-sets and fuzzy sets, were established as well. Granular computing is an umbrella term to cover any theories, methodologies, techniques, and tools that make use of information granules in complex problem solving [24,44]. And it should focus on formalizing information granules and unifying them to create a coherent methodological and developmental environment for intelligent system design and analysis, and the granular fuzzy decision support systems were built successfully [1,21,23,25]. Multi-granularity knowledge discovery and management has attracted a great deal of researchers attention, and a knowledge management and semantic modeling based on information granularity is proposed and they capture semantics, facilitate comprehension, and support communication mechanisms [22]. The uncertainty of the decision information systems is an important parameter for obtaining a good decision making. Vagueness mainly reflects the boundary’s uncertainty of a vague concept, and it has been studied by many researchers. However, it is very difficult to describe a vague concept with an approximation defined set in Pawlak’s knowledge space. The vague set, as well as the rough set, focuses on finding the boundaries of a vague concept and neglects how to construct a defined set with many granules in current knowledge space as an approximation concept of the vague concept. Fuzzy set is a very successful approach to vagueness. In this approach, sets are defined by partial membership in contrast to defined membership used in the classical definition of a set. Rough set expresses vagueness not by means of membership but by employing the boundary region of a set. If the boundary region of a set is empty, it means that a particular set is defined, otherwise the set is rough. The non-empty boundary region of the set means that our knowledge about the set is not sufficient to define the set precisely [17,20,31]. In real-life world, any vague concept defined by philosophers [11,12] has the boundary region which is not definable by using fuzzy set or rough set. And this problem cannot be eliminated by ‘static’

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definitions such as fuzzy set or rough set. We reviewed a lot of literatures [11,12,16,17,19,20,28,29], and found an important approach to solve above problem is to use some adaptive strategies to make it possible to adaptively change approximations of concepts when domain information knowledge is changing. Many well-known researchers, such as Skowron, Swiniarski, Peters, Nguyen, Bazan and Polkowski, have done many researches in this area [3,26,27,30–37]. And there are important consequences on adaptive approximation of vague concepts and reasoning about approximated concepts. And these researches realize a step toward approximate reasoning in multi-agent systems, intelligent systems, complex dynamic systems [3] and so on. However, from another perspective, we can obtain an approximation set of a vague concept in Pawlak’s knowledge space if there is no additional information. So, in this paper we will focus on constructing the approximation set of a vague set, such as 0.5-crisp set, approximation set, step-vague set and average-step-vague set. In other words, we hope to obtain many approximation sets in Pawlak’s knowledge space when the information is ‘static’. When the Pawlak’s knowledge space is changing with the changing condition attributes, boundary region of vague concept will dynamically change. In this paper, the change rules of the Euclidean similarity degree with the different knowledge granularities are discussed, and these rules will be in accord with human cognitive mechanism in multi-granularity knowledge space. These consequences reflect human cognitive process for a new thing, especially with the increasing awareness of information, the boundary region becomes smaller and smaller, and the similarity degree between a target concept (vague concept) and its approximation set becomes bigger and bigger. In this paper we will present several kinds of approximation sets of a target concept from a new viewpoint of multi-granularity. And these approximation sets will be important to solve vague boundary problems which may lead to conflict or war between two countries or regions for disputed borders, such as the disputed borders and territories between South Sudan and Sudan. There is another example for explaining the applications of the approximation sets of a vague set. Supposing there are 100 papers submitted to a conference, we only need to accept 40 excellent papers. After these papers are reviewed, there are three kinds of results: (1) there are 30 papers which are certainly accepted; (2) there are 20 papers which need to be further reviewed; (3) there are 50 papers which are rejected. Here, the boundary of this vague concept is clear, but we can not obtain the final review result (40 papers) if there is not any additional information because those papers (20 papers) which need to be further reviewed can not be accurately classified. This problem can be solved for a decision maker if an approximation set of all accepted paper (40 papers) can be constructed in current knowledge space. As a well-known fact, it is one of the basic characteristics in human being problem solving that people has a kind of ability to conceptualize the world with different granularity and translate from one abstraction level to others easily. This is a powerful ability of human being to deal with complex problems [54]. The rough set proposed by Pawlak [18] is an important tool to describe the boundary’s uncertainty of a set, and it describes the uncertainty of a concept X with two defined boundaries named upper-approximation set and lower-approximation set [18,38]. But, rough set theory does not give out the method for precisely or approximately characterizing the uncertain set X (or concept) with existing knowledge granules (which are defined subsets of the universe of discourse) in Pawlak’s knowledge space [15,50]. In our other research work [47,48,50,51],  the disadvantages of using upper-approximation set RðXÞ or lower-approximation set RðXÞ as an approximation set of the uncertain set X (or concept) were analyzed, and a defined approximation set R0:5 ðXÞ of the uncertain set X was proposed, and the change rules of similarity degree between R0:5 ðXÞ and X with the changing knowledge granularities in Pawlak’s space were discussed in detail. In fact, the defined approximation set R0:5 ðXÞ is a union set of many defined granules in Pawlak’s space. We also proposed the definition of 0.5-crisp set which is a defined approximation set of the fuzzy set X. A kind of similarity degree between a fuzzy set X and its 0.5-crisp set was discussed in detail, and the change rules of the similarity degree with the changing knowledge granularities in Pawlak’s approximation space were analyzed too. When the target concept is a vague set V in Pawlak’s approximation space, how can we obtain a defined approximation set of V? It is very important to obtain an approximation set which is a defined set composed of many granules. In this paper, the definition of similarity degree between two vague sets is presented, and many concepts, such as 0.5-crisp set, step-vague set and average-step-vague set of a vague set V will be defined one by one. The Euclidean similarity degree formula is presented because this kind of similarity degree is used by many researchers, such as Chen [6,7], Li [13]. And the Euclidean similarity degree between vague set V and its 0.5-crisp set or average-step-vague set are discussed in detail respectively, and the change regularities of the similarity degree between a vague set V and its 0.5-crisp set with the changing knowledge granularities in Pawlak’s space will be analyzed. Many relevant preliminary concepts are reviewed briefly and many new definitions about the vague set are presented in Section 2. In Section 3, the Euclidean similarity degrees between a vague set and its approximation sets are proposed, and the related theorems and properties are proposed and proved successfully. In Section 4, the change rules of the similarity degree between a vague set and its approximation sets with different knowledge granularities are found and proved. Finally, the paper is concluded in Section 5.

2. Preliminaries In this section, the basic concepts or terms related to the vague set and the rough set will be reviewed briefly and many new definitions in Pawlak’s approximation space, such as 0.5-crisp set, approximation set of the vague set, step-vague set and average-step-vague set, will be defined one by one.

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Let U ¼ fx1 ; x2 ; . . . ; xn g be a classical Cantor set of objects, called the universe of discourse, and let R be an equivalence relation on U which is called indiscernibility relation. Equivalence classes of the relation R are called elements set (or called partition-block of U). Any union of elementary set is called a composed set. The pair S ¼ ðU; RÞ will be called Pawlak’s approx  imation space, and U=R ¼ ½xR jx 2 U ¼ fX 1 ; X 2 ; . . . ; X m g is said to be a partition of U induced by the equivalence relation R. Definition 1 (Rough Set [18,38]). Let X # U be a subset of U, the lower-approximation set and the upper-approximation set of     T X are defined and denoted by RðX Þ and RðX Þ respectively as follows, RðX Þ ¼ [ ½xR j½xR # X , and RðX Þ ¼ [ ½xR j½xR X – / . If RðX Þ ¼ RðX Þ in S, the set X is called defined set, and if RðX Þ – RðX Þ, the set X is called Rough set where ½xR denotes the equivalence class of R and contains the element x. BR ðX Þ ¼ RðX Þ  RðX Þ is called the Boundary of X in S. N R ðX Þ ¼ U  RðX Þ is called the Negative region of X in S. PR ðX Þ ¼ RðX Þ is called the Positive region of X in S. Obviously, if X is a defined set, BR ðX Þ ¼ RðX Þ  RðX Þ ¼ /, and if X is a rough set, BR ðX Þ ¼ RðX Þ  RðX Þ – /.   Definition 2 (Fuzzy Set [46]). A fuzzy set A ¼ < x; lA ðxÞ > jx 2 U in a universe of discourse named U is characterized by a membership function lA which is defined as follows, lA : U ! ½0; 1. For any point x of U, its value lA ðxÞ shows the degree that x belongs to the fuzzy set A. Definition 3 (Vague Set [10]). A vague set V in a universe of discourse U is characterized by a truth-membership function tV ðxÞ and a false-membership function f V ðxÞ; t V ðxÞ is a lower boundary on the grade of membership of x derived from the evidence for x, and f V ðxÞ is a lower boundary on the negation of x derived from the evidence against x. Both t V ðxÞ and f V ðxÞ are associated with a real number in the interval ½0; 1 with each point in U, where tV ðxÞ + f V ðxÞ 6 1. That is, tV : U ! ½0; 1 and f V : U ! ½0; 1. R When U is continuous, a vague set V can be represented by V ¼ U ½tV ðxÞ; 1  f V ðxÞ=xdx. There is a vague set shown by 2 2 Fig. 1, where tV ðxÞ ¼ epx and f V ðxÞ ¼ ð1  epx Þ=2. When U is discrete, a vague set V can be represented by

V¼

n X ½t V ðxi Þ; 1  f V ðxi Þ=xi : i¼1

Here, ½tV ðxi Þ; 1  f V ðxi Þ denotes a vague value of xi where t V ðxi Þ 6 1  f V ðxi Þ; 1 6 i 6 n. Actually, the fuzzy set is a special vague set (that is, tV ðxi Þ ¼ 1  f V ðxi Þ), i.e., if a vague value interval [t V ðxi Þ; 1  f V ðxi Þ] becomes a single point set, 1 6 i 6 n, the vague set will degenerate into a fuzzy set. Definition 4 (Containment [10]). A vague set V 1 is contained in another vague set V 2 , i.e., V 1 # V 2 , if and only if tV 1 ðxÞ 6 tV 2 ðxÞ and 1  f V 1 ðxÞ 6 1  f V 2 ðxÞ for any point x in U.

Fig. 1. A vague set.

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 and is defined as follow, for any point x in Definition 5 (Complement [10]). The complement of a vague set V is denoted by V a universe of discourse U; t V ðxÞ ¼ f V ðxÞ and f V ðxÞ ¼ t V ðxÞ. According to a large number of documents, we find that the different researchers have different interpretation to the similarity degree between two vague sets [48]. In their research process, different researchers maybe define the different formulas of similarity degree with their own understandings. In order to better measure the similarity degree between two vague sets, afterwards, we present the definition of similarity degree between two vague sets based on the definition of the similarity degree between two fuzzy sets referred to [47]. Definition 6 (Similarity Degree between two Vague Sets). Let RðUÞ denote a set of all vague sets on U, the mapping S : RðUÞ  RðUÞ ! ½0; 1; SðV 1 ; V 2 Þ is said to be the similarity degree between V 1 2 RðUÞ and V 2 2 RðUÞ if and only if SðV 1 ; V 2 Þ satisfies the following properties, (1) (2) (3) (4) (5)

0 6 SðV 1 ; V 2 Þ 6 1 (Boundedness); SðV 1 ; V 2 Þ ¼ SðV 2 ; V 1 Þ (Symmetry); If V 1 # V 2 # V 3 2 RðUÞ, then SðV 1 ; V 3 Þ 6 SðV 1 ; V 2 Þ and SðV 1 ; V 3 Þ 6 SðV 2 ; V 3 Þ; SðV 1 ; V 2 Þ ¼ 1 () V 1 ¼ V 2 (Reflexivity);  2 . Where PðUÞ denotes the power set of U(Contrariety). SðV 1 ; V 2 Þ ¼ 0 if and only if V 1 2 PðUÞ; V 2 2 PðUÞ and V 1 ¼ V

According to the definition of similarity between two vague sets, a typical similarity degree formula SðV 1 ; V 2 Þ with Euclidean distance based on literatures [6,7,13] between V 1 and V 2 can be constructed as follow, n h i X 1 2 SðV 1 ; V 2 Þ ¼ 1  pffiffiffiffiffiffi ðtV 1 ðxi Þ  t V 2 ðxi ÞÞ2 þ ðf V 1 ðxi Þ  f V 2 ðxi ÞÞ 2n i¼1

!12 :

ð1Þ

SðV 1 ; V 2 Þ is called Euclidean similarity degree in this paper. And let n h i X 1 2 dðV 1 ; V 2 Þ ¼ pffiffiffiffiffiffi ðtV 1 ðxi Þ  tV 2 ðxi ÞÞ2 þ ðf V 1 ðxi Þ  f V 2 ðxi ÞÞ 2n i¼1

!12 :

ð2Þ

The dðV 1 ; V 2 Þ denotes the distance between V 1 and V 2 . In this paper, we mainly discuss the discrete domain U. In general, a vague set on discrete domain U ¼ fx1 ; x2 ; . . . ; xn g is denoted by

V¼

½t V ðx1 Þ; 1  f V ðx1 Þ ½t V ðx2 Þ; 1  f V ðx2 Þ ½t V ðxn Þ; 1  f V ðxn Þ þ þ  þ : x1 x2 xn

In the approximation space ðU; RÞ, how to obtain a defined approximation set of a vague set with existing granules is a very important issue. It is very useful to solve the problem of disputed borders, such as the disputed borders and territories between South Sudan and Sudan. In other words, obtaining a defined set as an approximation set of a vague set is a better way than only getting the boundaries of a vague set, such as the upper-approximation set and lower-approximation set in rough set theory. The definition of the 0.5-crisp set of a vague set in the approximation space ðU; RÞ will be proposed as follows. Definition 7 (0.5-crisp Set). Let U ¼ fx1 ; x2 ; . . . ; xn g; R be an equivalence relation on U, and V be a vague set on U. V R0:5 is called 0.5-crisp set of vague set in the approximation space ðU; RÞ if and only if for any x 2 U satisfies the following condition,

V R0:5 ðxÞ

¼

8 ½1; 1; > > < > > : ½0; 0;

X

tV ðyÞ P

y2½xR

X

y2½xR

X

f V ðyÞ;

y2½xR

tV ðyÞ <

X

f V ðyÞ:

y2½xR

Obviously, V R0:5 is a subset of U, and it is a defined set which is a union set of many equivalence classes in U=R. There is a vague set shown as Fig. 2, in which 0.5-crisp set V R0:5 is denoted by special lines composed of hollow circles. Definition 8 (The Approximation Set of Vague Set). Let V be a vague set on U; R be an equivalence relation on U and U=R ¼ fX 1 ; X 2 ; . . . ; X m g. Let,

RðVÞ ¼ [fX k jX k 2 U=R ^ ð8x 2 X k ! ½t V ðxÞ; 1  f V ðxÞ ¼ ½1; 1Þg;  RðVÞ ¼ [fX k jX k 2 U=R ^ ð9x 2 X k ^ ½t V ðxÞ; 1  f V ðxÞ – ½0; 0Þg:  are called Lower-approximation defined set and Upper-approximation defined set of V in the approximaRðV Þ and RðVÞ  tion space ðU; RÞ respectively. BR ðVÞ ¼ RðVÞR ðV Þ is called the Boundary of V.

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Fig. 2. A vague set and its 0.5-crisp set.

There is a trapezoid vague set shown as Fig. 3, and its approximation sets can be obtained as follows.

RðVÞ ¼ [f½0:2; 0; ½0; 0:2g ¼ ½0:2; 0 [ ½0; 0:2 ¼ ½0:2; 0:2;  RðVÞ ¼ [f½0:8; 0:6; ½0:6; 0:4; ½0:4; 0:2; ½0:2; 0; ½0; 0:2; ½0:2; 0:4; ½0:4; 0:6; ½0:6; 0:8g ¼ ½0:8; 0:8; BR ðVÞ ¼ [f½0:8; 0:6; ½0:6; 0:4; ½0:4; 0:2; ½0:2; 0:4; ½0:4; 0:6; ½0:6; 0:8g ¼ [f½0:8; 0:2; ½0:2; 0:8g ¼ ½0:8; 0:2 [ ½0:2; 0:8: Definition 9 (Step-vague Set). Let V be a vague set on U; R be an equivalence relation on U and U=R ¼ fX 1 ; X 2 ; . . . ; X m g. If for any x 2 X 1 ; ½t V ðxÞ; 1  f V ðxÞ ¼ ½t 1 ; 1  f 1  is always satisfied, and for any x 2 X 2 ; ½tV ðxÞ; 1  f V ðxÞ ¼ ½t2 ; 1  f 2  always is satisfied, . . ., and for any x 2 X m ; ½tV ðxÞ; 1  f V ðxÞ ¼ ½tm ; 1  f m  is held also, then the vague set V is called Step-vague set on U=R, and denoted as V J , where 0 6 t i 6 1; 0 6 f i 6 1, and ti þ f i 6 1ði ¼ 1; 2; . . . ; mÞ.

Definition 10 (Average-step-vague Set). Let V be a vague set on U; R be an equivalence relation on U and U=R ¼ fX 1 ; X 2 ; . . . ; X m g. For any x 2 U, let

Fig. 3. Trapezoid vague set.

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

 J ðxÞ ¼ V

"P

y2½xR t V ðyÞ

j½xR j

P ;1 

7

#

y2½xR f V ðyÞ

j½xR j

;

 J is called Average-step-vague set in the approximation space ðU; RÞ. then the vague set V A vague set and its average-step-vague set in the approximation space ðU; RÞ are shown as Fig. 4. For example, let U ¼ fx1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 ; x8 ; x9 ; x10 g,

U=R ¼ ffx1 ; x2 g; fx3 ; x4 ; x5 g; fx6 ; x7 g; fx8 ; x9 ; x10 gg; ½0:1; 0:3 ½0:3; 0:5 ½0:5; 0:6 ½0:6; 0:8 ½0:7; 1 ½1; 1 ½1; 1 ½0; 0 ½0; 0 ½0; 0 þ þ þ þ þ þ þ þ þ : V¼ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10  ¼ fx1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 g; BR ðVÞ ¼ fx1 ; x2 ; x3 ; x4 ; x5 g; N R ðVÞ ¼ fx8 ; x9 ; x10 g, and P R ðVÞ ¼ fx6 ; x7 g, Then, RðVÞ ¼ fx6 ; x7 g; RðVÞ

½0; 0 ½0; 0 ½1; 1 ½1; 1 ½1; 1 ½1; 1 ½1; 1 ½0; 0 ½0; 0 ½0; 0 þ þ þ þ þ þ þ þ þ ; x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 ½0:2; 0:4 ½0:2; 0:4 ½0:6; 0:8 ½0:6; 0:8 ½0:6; 0:8 ½1; 1 ½1; 1 ½0; 0 ½0; 0 ½0; 0 þ þ þ þ þ þ þ þ þ : V J ¼ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

V R0:5 ¼

Obviously, V R0:5 is a defined set, i.e., V R0:5 ¼ fx3 ; x4 ; x5 ; x6 ; x7 g. Next, we will prove that V R0:5 is the best defined approximation set of V in all defined subsets of U, that is to say, V R0:5 has the maximum Euclidean similarity degree with V in all defined subsets of U. 3. The similarity degree between a vague set and its approximation sets Obviously, in the approximation space ðU; RÞ, for any x 2 U, (1) If x 2 PR ðVÞ, then ½t V ðxÞ; 1  f V ðxÞ ¼ ½1; 1. (2) If x 2 N R ðVÞ, then ½tV ðxÞ; 1  f V ðxÞ ¼ ½0; 0. (3) If x 2 BR ðVÞ, then 0 < t V ðxÞ < 1 or 0 < f V ðxÞ < 1. In this section, we will discuss the Euclidean similarity degree between a vague set and its approximation sets in detail, such as 0.5-crisp set, lower-approximation set, upper-approximation set, and average-step-vague set. Let V 1 and V 2 be two vague sets on U, as follows,

½t V 1 ðx1 Þ; 1  f V 1 ðx1 Þ ½t V 1 ðx2 Þ; 1  f V 1 ðx2 Þ ½tV ðxn Þ; 1  f V 1 ðxn Þ þ þ  þ 1 ; x1 x2 xn ½t V ðx1 Þ; 1  f V 2 ðx1 Þ ½t V 2 ðx2 Þ; 1  f V 2 ðx2 Þ ½tV ðxn Þ; 1  f V 2 ðxn Þ V2 ¼ 2 þ þ  þ 2 : x1 x2 xn

V1 ¼

Theorem 1. Let V be a vague set on U; R be an equivalence relation on U and U=R ¼ fX 1 ; X 2 ; . . . ; X m g, then SðV; V R0:5 Þ P SðV; RðV ÞÞ, where S is the Euclidean similarity degree.

Fig. 4. A vague set and its average-step-vague set.

8

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

  RðVÞ ¼ [fX i1 ; X i2 ; . . . ; X it g, where X i1 2 U=R; X i2 2 U=R; . . ., and X it 2 U=R. According to the definition Proof. Let BR ðVÞ ¼ RðVÞ of the Euclidean similarity degree between two vague sets and formula (2), the distance between a vague set V and its lower-approximation set RðV Þ is denoted as dðV; RðV ÞÞ. For convenience, we only need to compute the square of dðV; RðV ÞÞ, that is,  2

d ðV; RðVÞÞ ¼

n h i 1 X 1 2 ðt V ðxi Þ  tRðVÞ ðxi ÞÞ2 þ ðf V ðxi Þ  f RðVÞ ðxi ÞÞ ¼ 2n i¼1 2n

þ

X h

2

ðt V ðxÞ  tRðVÞ ðxÞÞ þ ðf V ðxÞ  f RðVÞ ðxÞÞ

2

i

þ

x2NR ðVÞ

(

X h

2

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ

x2PR ðVÞ

X h

2

2

ðt V ðxÞ  t RðVÞ ðxÞÞ þ ðf V ðxÞ  f RðVÞ ðxÞÞ

i

i

) :

x2BR ðVÞ

2

In the formula d ðV; RðVÞÞ, if x 2 P R ðVÞ, then tV ðxÞ ¼ t RðV Þ ðxÞ ¼ t V R ðxÞ ¼ 1, 0:5

f V ðxÞ ¼ f RðVÞ ðxÞ ¼ f V R ðxÞ ¼ 0; 0:5

so,

X h

2

ðt V ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ

i

X h

¼

x2P R ðVÞ

2

ðt V ðxÞ  tV R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ 0:5

x2PR ðVÞ

i

0:5

¼ 0:

If x 2 N R ðVÞ, then t V ðxÞ ¼ tRðVÞ ðxÞ ¼ t V R ðxÞ ¼ 0, and f V ðxÞ ¼ f RðVÞ ðxÞ ¼ f V R ðxÞ ¼ 1, so, 0:5

0:5

i i X h X h 2 2 ðt V ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ ¼ ðtV ðxÞ  tV R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ ¼ 0:

x2N R ðVÞ

0:5

x2NR ðVÞ

0:5

Therefore,

8 i i 1 X h 1
i2

it

P P (1) If there is at least a block, supposed to be X i1 , which satisfies the inequality x2X i t V ðxÞ P x2X i f V ðxÞ, and the P P P P 1 1 other blocks satisfy t ðxÞ < f ðxÞ; . . ., and t ðxÞ < f ðxÞ. And then, for any x 2 X i1 , we have V V V V x2X i x2X i x2X i x2X i 2

t

2

t

V R0:5 ðxÞ ¼ ½1; 1 according to the definition of 0.5-crisp set, and for any x 2 X i2 ; . . ., or x 2 X it , we get V R0:5 ðxÞ ¼ ½0; 0. Let X i1 ¼ fx11 ; x12 ; . . . ; x1s1 g, then

Xh

2

ðt V ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ

i

x2X i

1

¼

s1 h X

2

ðt V ðx1i Þ  tRðVÞ ðx1i ÞÞ2 þ ðf V ðx1i Þ  f RðVÞ ðx1i ÞÞ

i

i¼1 2

2

¼ ðt V ðx11 Þ  0Þ2 þ ðf V ðx11 Þ  1Þ þ ðtV ðx12 Þ  0Þ2 þ ðf V ðx12 Þ  1Þ þ    þ ðt V ðx1s1 Þ  0Þ2 þ ðf V ðx1s1 Þ  1Þ 2

2

¼ ðt V ðx11 ÞÞ2 þ ðt V ðx12 ÞÞ2 þ    þ ðt V ðx1s1 ÞÞ2 þ ðf V ðx11 Þ  1Þ þ ðf V ðx12 Þ  1Þ þ    þ ðf V ðx1s1 Þ  1Þ 2

2

¼ ðt V ðx11 ÞÞ2 þ ðt V ðx12 ÞÞ2 þ    þ ðt V ðx1s1 ÞÞ2 þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1s1 ÞÞ

2

2

2

 2ðf V ðx11 Þ þ f V ðx12 Þ þ    þ f V ðx1s1 ÞÞ þ s1 2 2 2 P ðt V ðx11 ÞÞ2 þ ðt V ðx12 ÞÞ2 þ    þ ðt V ðx1s1 ÞÞ2 þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1s1 ÞÞ  2ðt V ðx11 Þ þ t V ðx12 Þ þ    þ tV ðx1s1 ÞÞ þ s1 2

2

¼ ðtV ðx11 Þ  1Þ2 þ ðt V ðx12 Þ  1Þ2 þ    þ ðtV ðx1s1 Þ  1Þ2 þ ðf V ðx11 Þ  0Þ þ ðf V ðx12 Þ  0Þ þ    þ ðf V ðx1s1 Þ  0Þ ¼ ðtV ðx11 Þ  tV R ðx11 ÞÞ2 þ ðt V ðx12 Þ  tV R ðx12 ÞÞ2 þ    þ ðt V ðx1s1 Þ  t V R ðx1s1 ÞÞ2 0:5

0:5

2

0:5

2

2

þ ðf V ðx11 Þ  f V R ðx11 ÞÞ þ ðf V ðx12 Þ  f V R ðx12 ÞÞ þ    þ ðf V ðx1s1 Þ  f V R ðx1s1 ÞÞ 0:5 0:5 0:5 i Xh 2 2 ¼ ðtV ðxÞ  tV R ðxÞÞ þ ðf V ðxÞ  f V R ðxÞÞ : x2X i

0:5

0:5

1

And then, let X i2 ¼ fx21 ; x22 ; . . . ; x2s2 g, for any x 2 X i2 , we can get V R0:5 ðxÞ ¼ ½0; 0 due to

X

x2X i

2

tV ðxÞ <

X

x2X i

2

f V ðxÞ:

2

9

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

Therefore,

Xh x2X i

2

ðt V ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ

i

s2 h X

¼

2

ðt V ðx2i Þ  tRðVÞ ðx2i ÞÞ2 þ ðf V ðx2i Þ  f RðVÞ ðx2i ÞÞ

i

i¼1

2

2

¼ ðt V ðx21 Þ  0Þ2 þ ðf V ðx21 Þ  1Þ þ ðtV ðx22 Þ  0Þ2 2

þ ðf V ðx22 Þ  1Þ þ    þ ðt V ðx2s2 Þ  0Þ2 þ ðf V ðx2s2 Þ  1Þ i Xh 2 ¼ ðt V ðxÞ  t V R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ : 0:5

x2X i

2

0:5

2

In the same way, for any x 2 X it , we have

Xh

ðt V ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ

2

i

Xh

¼

x2X it

i 2 ðt V ðxÞ  t V R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ : 0:5

x2X it 2

0:5

2

Therefore, d ðV; RðVÞÞ P d ðV; V R0:5 Þ is held. (2) If all the blocks in approximation space ðU; RÞ satisfy the following inequalities, i.e.,

X x2X i

X

t V ðxÞ <

x2X i

1

X

f V ðxÞ;

x2X i

1

t V ðxÞ <

X x2X i

2

f V ðxÞ; . . . ;

X

tV ðxÞ <

x2X it

2

X

f V ðxÞ:

x2X it 2

2

In a similar way of above (1), we can easily obtain the equation d ðV; RðVÞÞ ¼ d ðV; V R0:5 Þ. According to (1) and (2), we draw a conclusion that dðV; RðVÞÞ P dðV; V R0:5 Þ, and so, the inequality SðV; RðVÞÞ 6 SðV; V R0:5 Þ is held.  Theorem 2. Let V be a vague set on U; R be an equivalence relation on U and U=R ¼ fX 1 ; X 2 ; . . . ; X m g, then SðV; V R0:5 Þ P SðV; RðVÞÞ, where S is the Euclidean similarity degree. Proof. According to the definition of Euclidean similarity degree between two vague sets and the formula (2), and similar to the proof of Theorem 1 we have 2

 ¼ d ðV; RðVÞÞ

¼

n h i 1 X 2 ðtV ðxi Þ  t RðVÞ ðxi ÞÞ2 þ ðf V ðxi Þ  f RðVÞ ðxi ÞÞ   2n i¼1

1 2n

(

i X h 2 ðt V ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

x2P R ðVÞ

þ

X h

2

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

i

x2NR ðVÞ

þ

X h

2

ðtV ðxÞ  t RðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

i

) :

x2BR ðVÞ 2

 If x 2 P R ðVÞ, then t V ðxÞ ¼ tRðVÞ In the formula d ðV; RðVÞÞ, ðxÞ ¼ t V R ðxÞ ¼ 1,  0:5

f V ðxÞ ¼ f RðVÞ ðxÞ ¼ f V R ðxÞ ¼ 0;  0:5

so,

X h

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

2

i

x2PR ðVÞ

¼

X h

2

ðt V ðxÞ  t V R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ

x2PR ðVÞ

0:5

i

0:5

¼ 0:

If x 2 N R ðVÞ, then t V ðxÞ ¼ tRðVÞ ðxÞ ¼ t V R ðxÞ ¼ 0; f V ðxÞ ¼ f RðV   Þ ðxÞ ¼ f V R ðxÞ ¼ 1, so, we can have

X h

0:5

2

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

0:5

i

x2NR ðVÞ

¼

X h

x2NR ðVÞ

2

ðtV ðxÞ  tV R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ 0:5

0:5

i

¼ 0:

10

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

  RðVÞ ¼ [fX i1 ; X i2 ; . . . ; X it g, then we have Let BR ðVÞ ¼ RðVÞ

8
i2

it

P P (1) If there is at least a block, supposed to be X i1 , which satisfies the inequality x2X i tV ðxÞ < x2X i f V ðxÞ, and the other 1 1 P P P P blocks satisfy the following inequalities, x2X i t V ðxÞ P x2X i f V ðxÞ; . . ., and x2X i t V ðxÞ P x2X i f V ðxÞ, then for any x 2 X i1 , 2

t

2

t

we have V R0:5 ðxÞ ¼ ½0; 0. On the contrary, for any x 2 X i2 ; . . . ; x 2 X it , we have V R0:5 ðxÞ ¼ ½1; 1. Let X i1 ¼ fx11 ; x12 ; . . . ; x1s1 g, then

Xh

x2X i

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

2

i

1

s1 h X

¼

2

ðtV ðx1i Þ  t RðVÞ ðx1i ÞÞ2 þ ðf V ðx1i Þ  f RðVÞ ðx1i ÞÞ  

i

i¼1 2

2

2

¼ ðtV ðx11 Þ  1Þ2 þ ðf V ðx11 Þ  0Þ þ ðt V ðx12 Þ  1Þ2 þ ðf V ðx12 Þ  0Þ þ    þ ðt V ðx1s1 Þ  1Þ2 þ ðf V ðx1s1 Þ  0Þ 2

2

¼ ðtV ðx11 Þ  1Þ2 þ ðt V ðx12 Þ  1Þ2 þ    þ ðtV ðx1s1 Þ  1Þ2 þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1s1 ÞÞ 2

2

2

2

¼ ðtV ðx11 ÞÞ2 þ ðtV ðx12 ÞÞ2 þ    þ ðt V ðx1s1 ÞÞ2 þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1s1 ÞÞ  2ðt V ðx11 Þ þ tV ðx12 Þ þ    þ t V ðx1s1 ÞÞ þ s1 2

2

2

P ðtV ðx11 ÞÞ2 þ ðtV ðx12 ÞÞ2 þ    þ ðtV ðx1s1 ÞÞ2 þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1s1 ÞÞ  2ðf V ðx11 Þ þ f V ðx12 Þ þ    þ f V ðx1s1 ÞÞ þ s1 2

2

¼ ðtV ðx11 ÞÞ2 þ ðtV ðx12 ÞÞ2 þ    þ ðt V ðx1s1 ÞÞ2 þ ðf V ðx11 Þ  1Þ þ ðf V ðx12 Þ  1Þ þ    þ ðf V ðx1s1 Þ  1Þ

2

¼ ðtV ðx11 Þ  t V R ðx11 ÞÞ2 þ ðtV ðx12 Þ  tV R ðx12 ÞÞ2 þ    þ ðtV ðx1s1 Þ  tV R ðx1s1 ÞÞ2 0:5

0:5

0:5

2

2

2

þ ðf V ðx11 Þ  f V R ðx11 ÞÞ þ ðf V ðx12 Þ  f V R ðx12 ÞÞ þ    þ ðf V ðx1s1 Þ  f V R ðx1s1 ÞÞ 0:5 0:5 0:5 i Xh 2 2 ¼ ðt V ðxÞ  t V R ðxÞÞ þ ðf V ðxÞ  f V R ðxÞÞ : x2X i

0:5

0:5

1

And as well, let X i2 ¼ fx21 ; x22 ; . . . ; x2s2 g, for any x 2 X i2 , we have V R0:5 ðxÞ ¼ ½1; 1 due to have

Xh x2X i

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

2

P

x2X i

2

t V ðxÞ P

P

x2X i

2

f V ðxÞ. Therefore, we

i

2

¼

s2 h X

2

ðtV ðx2i Þ  t RðVÞ ðx2i ÞÞ2 þ ðf V ðx2i Þ  f RðVÞ ðx2i ÞÞ  

i

i¼1 2

2

2

¼ ðtV ðx21 Þ  1Þ2 þ ðf V ðx21 Þ  0Þ þ ðt V ðx22 Þ  1Þ2 þ ðf V ðx22 Þ  0Þ þ    þ ðt V ðx2s2 Þ  1Þ2 þ ðf V ðx2s2 Þ  0Þ i Xh 2 ¼ ðt V ðxÞ  t V R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ : x2X i

0:5

0:5

2

In a similar way, if x 2 X it , we have

Xh

2

ðtV ðxÞ  tRðVÞ ðxÞÞ2 þ ðf V ðxÞ  f RðVÞ ðxÞÞ  

i

¼

x2X it

i Xh 2 ðt V ðxÞ  t V R ðxÞÞ2 þ ðf V ðxÞ  f V R ðxÞÞ : x2X it

0:5

0:5

2 2  P d ðV; V R0:5 Þ is held. Consequently, d ðV; RðVÞÞ P P (2) If all the blocks in approximation space ðU; RÞ satisfy the following inequalities, x2X i1 t V ðxÞ P x2X i1 f V ðxÞ; P P P P t ðxÞ P f ðxÞ; . . . ; t ðxÞ P f ðxÞ. Similarly, through the proof of (1), we easily obtain the equation x2X i V x2X i V x2X i V x2X i V 2

2

2

t

t

2

  ¼ d ðV; V R0:5 Þ. According to above (1) and (2), the inequality dðV; RðVÞÞ P dðV; V R0:5 Þ is always held, which means d ðV; RðVÞÞ  6 SðV; V R0:5 Þ is proved successfully.  the conclusion SðV; RðVÞÞ Theorem 1 shows that the Euclidean similarity degree between a vague set and its lower-approximation defined set is lower than the Euclidean similarity degree between the vague set and its 0.5-crisp set, and Theorem 2 shows that the

11

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

Euclidean similarity degree between a vague set and its upper-approximation defined set is lower than the Euclidean similarity degree between the vague set and its 0.5-crisp set. These conclusions can be shown by Fig. 5. In Fig. 5, the star line stands for SðV; V R0:5 Þ, the line composed of round dots stands for SðV; RðVÞÞ and the plus-line stands for SðV; RðVÞÞ, where

8 0; > > > > > > 5 4 > > >3x þ 3; > < t V ðxÞ ¼ 1; > > > > 5 >  3 x þ 43 ; > > > > > : 0;

8 1; > > > > > > 5 > xð0:8; 0:2 > >  2 x  1; > < x 2 ð0:2; 0:2 ; f V ðxÞ ¼ 0; > > > >5 > x 2 ð0:2; 0:8 x  1; > > 2 > > > : x 2 ð0:8; 1 1;

x 2 ½1; 0:8

x 2 ½1; 0:8 x 2 ð0:8; 0:4 x 2 ð0:4; 0:4 : x 2 ð0:4; 0:8 x 2 ð0:8; 1

Theorem 3. Let V be a vague set on U; R be an equivalence relation on U and U=R ¼ fX 1 ; X 2 ; . . . ; X m g. For any step-vague set V J in  J Þ P SðV; V J Þ, where S is the Euclidean similarity degree. the approximation space ðU; RÞ; SðV; V Proof. Let V J be a step-vague set in the approximation space ðU; RÞ, for any x 2 U,

V J ðxÞ ¼

8 ½t 1 ; 1  f 1  > > > > > > > < ½t 2 ; 1  f 2 

x 2 X1 x 2 X2

.. > > > . > > > > : ½t m ; 1  f m  x 2 X m

:



The distance between V and V J is denoted as dðV; V J Þ based on formula (2), and the square of dðV; V J Þ is shown as follows,

1 d ðV; V J Þ ¼ 2n

(

2

Xh

Xh

2

ðt V ðxÞ  t V J ðxÞÞ2 þ ðf V ðxÞ  f V J ðxÞÞ

x2X 1

i

þ

Xh x2X 2

2

ðtV ðxÞ  tV J ðxÞÞ2 þ ðf V ðxÞ  f V J ðxÞÞ

i

) :

x2X m

Let X 1 ¼ fx11 ; x12 ; . . . ; x1n1 g; X 2 ¼ fx21 ; x22 ; . . . ; x2n2 g; . . ., and X m ¼ fxm1 ; xm2 ; . . . ; xmnm g. And as well, let

y1 ¼

X x2X 1

ðt V ðxÞ  t V J ðxÞÞ2 ;

z1 ¼

X

2

ðtV ðxÞ  tV J ðxÞÞ2 þ ðf V ðxÞ  f V J ðxÞÞ

2

ðf V ðxÞ  f V J ðxÞÞ ;

x2X 1

Fig. 5. The Euclidean similarity degree between a vague set and its approximation sets.

i

þ þ

12

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

then

y1 ¼ ðtV ðx11 Þ  t V J ðx11 ÞÞ2 þ ðt V ðx12 Þ  tV J ðx12 ÞÞ2 þ    þ ðt V ðx1n1 Þ  tV J ðx1n1 ÞÞ2 ¼ ðtV ðx11 Þ  t 1 Þ2 þ ðtV ðx12 Þ  t 1 Þ2 þ    þ ðtV ðx1n1 Þ  t1 Þ2 : 2

2

2

In a similar way, z1 ¼ ðf V ðx11 Þ  f 1 Þ þ ðf V ðx12 Þ  f 1 Þ þ    þ ðf V ðx1n1 Þ  f 1 Þ . We compute the derivation of y1 and the derivation of z1 respectively as follows,

dy1 ¼ 2ðt V ðx11 Þ  t 1 Þ  2ðt V ðx12 Þ  t 1 Þ      2ðt V ðx1n1 Þ  t 1 Þ ¼ 0; dt 1

ð3Þ

dz1 ¼ 2ðf V ðx11 Þ  f 1 Þ  2ðf V ðx12 Þ  f 1 Þ      2ðf V ðx1n1 Þ  f 1 Þ ¼ 0: df1

ð4Þ

The solution of Eq. (3) is

t1 ¼

1 ðtV ðx11 Þ þ tV ðx12 Þ þ    þ t V ðx1n1 ÞÞ; n1

and the solution of Eq. (4) is

f1 ¼

1 ðf ðx11 Þ þ f V ðx12 Þ þ    þ f V ðx1n1 ÞÞ: n1 V

Because their second derivatives satisfy 2

d y1 2 dt 1

2

d z1

> 0;

2

df 1

> 0;

when 1 1X t V ðx1i Þ; n1 i¼1

n

t1 ¼

1 1X f ðx1i Þ; n1 i¼1 V

n

f1 ¼

both y1 and z1 reach their minimum values respectively. In a similar way, let

yi ¼

X ðtV ðxÞ  tV J ðxÞÞ2 ; x2X i

zi ¼

X 2 ðf V ðxÞ  f V J ðxÞÞ ; x2X i

Pni Pni we have the similar conclusions that when t i ¼ n1 j¼1 t V ðxij Þ and f i ¼ n1 j¼1 f V ðxij Þ, both yi and zi reach their minimum values i i ði ¼ 1; 2; . . . ; mÞ. Pni Pni Therefore, when t i ¼ n1 j¼1 t V ðxij Þ and f i ¼ n1 j¼1 f V ðxij Þ ði ¼ 1; 2; . . . ; mÞ, i

i

2

d ðV; V J Þ reaches its minimum value, that is 2

d ðV; V J Þ ¼

P

n h i 1 X 2 ðtV ðxi Þ  t V J ðxi ÞÞ2 þ ðf V ðxi Þ  f V J ðxi ÞÞ 2n i¼1

1 2n

(

" P P 2  2 # X  x2X 1 t V ðxÞ x2X 1 f V ðxÞ tV ðxÞ  þ f V ðxÞ  jX 1 j jX 1 j x2X 1

" " P P P 2  2 #  X  X  t V ðxÞ 2 x2X 2 t V ðxÞ x2X 2 f V ðxÞ þ  þ þ t V ðxÞ  þ f V ðxÞ  t V ðxÞ  x2X m þ jX 2 j jX 2 j jX m j x2X x2X m 2

P  2 #) f V ðxÞ 2 ¼ d ðV; V J Þ: þ f V ðxÞ  x2X m jX m j  J Þ, that is to say, the inequality SðV; V  J Þ P SðV; V J Þis proved successfully. So dðV; V J Þ P dðV; V  J ofVis an optimal approximation vague set of V in all step-vague sets Theorem 3 shows that the average-step-vague set V in the approximation space ðU; RÞ. For example, supposing a step-vague set V J is defined as formula (5), the Euclidean similarity degree between a vague set and this step-vague set V J and the Euclidean similarity degree between the vague set and  J are shown by Fig. 6. In Fig. 6, the star-line stands for SðV; V  J Þ and the line composed of round its average-step-vague set V dots stands for SðV; V J Þ, where V J ðxÞ is formula (5).

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

13

Fig. 6. The Euclidean similarity degree between a vague set and its step-vague set.

V J ðxÞ ¼

8 ½0:4; 0:6 > > > > > > > ½0:5; 0:6 > > > > > > > ½0:6; 0:7 > > > > > > > > ½0:6; 0:8 > > > > > > < ½0:7; 0:8 > > ½0:4; 0:6 > > > > > > > ½0:5; 0:6 > > > > > > > ½0:6; 0:7 > > > > > > > ½0:6; 0:8 > > > > > : ½0:7; 0:8

x 2 ½1:0; 0:8 x 2 ð0:8; 0:6 x 2 ð0:6; 0:4 x 2 ð0:4; 0:2 x 2 ð0:2; 0:0 :

ð5Þ

x 2 ð0:0; 0:2 x 2 ð0:2; 0:4 x 2 ð0:4; 0:6 x 2 ð0:6; 0:8 x 2 ð0:8; 1:0

4. The change rules of similarity degree with different knowledge granularities In an approximation space ðU; RÞ, the knowledge granularity will change with the equivalence relation R. In the artificial intelligence field, the change rules of uncertainty of rough set with different knowledge granularities are very important to incremental knowledge acquisition from a hierarchical knowledge space. From the viewpoint of granular computing, we can acquire knowledge in different knowledge granularity levels. In 1990, Zhang [54,55] developed a structural definition of membership function, and found that for a fuzzy set (concept), it may probably be described by different types of membership functions, as long as the structures of these membership functions are the same, they characterize the fuzzy set(i.e., concept) with the same property. That is to say, although these membership functions are different in appearance, they are the same in essence. The structural description is more essential to a fuzzy concept than the membership function. This structure is called hierarchical quotient space structure in quotient space theory developed by Zhang, et al. [55]. As a well-known fact, it is one of basic characteristics that one person has a kind of ability to conceptualize the world with different granularities and translate from one abstraction level to others easily. Such is a powerful ability of human being to deal with complex problems [52,53]. So, in this section, we will discuss the change rules of Euclidean similarity degree between a vague set and its 0.5-crisp set in different knowledge granularity levels. Based on human cognitive mechanism, the finer the knowledge granule is, the greater the Euclidean similarity degree between a vague set and its approximation set is. Do the change rules of SðV; V R0:5 Þ with different knowledge granularity accord with the human cognitive mechanism? Theorem 4 will give out the answer. Definition 11 [49]. Let U ¼ fx1 ; x2 ; . . . ; xn g be a non-empty finite set, P0 ¼ fP 01 ; P02 ; . . . ; P 0l g and P 00 ¼ fP001 ; P002 ; . . . ; P 00m g are two partitions on U. If 8P0i 2P0 ð9P00j 2P00 ðP i 0 # P 00j ÞÞ, then P 0 is finer than P00 , denoted as P 0 P00 .

14

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

Definition 12 [49]. Let U ¼ fx1 ; x2 ; . . . ; xn g be a non-empty finite set, P 0 ¼ fP 01 ; P 02 ; . . . P 0l g and P00 ¼ fP001 ; P002 ; . . . ; P 00m g are two partitions on U. If P0 P 00 , and 9P0i 2P0 ð9P00j 2P00 ðPi 0  P00j ÞÞ, then P 0 is strictly finer than P00 , denoted by P 0  P00 . Lemma 1. Let R1 and R2 be two equivalence relations on U, R2 # R1 , then U=R2 U=R1 . 1 2 Theorem 4. Let V be a vague set on U, R1 and R2 be two equivalence relations on U, and R2 # R1 , then SðV; V R0:5 Þ 6 SðV; V R0:5 Þ, where S is the Euclidean similarity degree.

Proof. Let U=R1 ¼ fX 1 ; X 2 ; . . . ; X m g and U=R2 ¼ fY 1 ; Y 2 ; . . . ; Y k g. According to Lemma 1, we have U=R2 U=R1 due to R2 # R1 , that is to say, the partition U=R2 is finer than the partition U=R1 . In other words, for any block Y i 2 U=R2 , there must exist a block X j 2 U=R1 and they satisfy the inclusion relation, i.e., Y i # X j . In order to simplify the proof, let X 1 ¼ Y 1 [ Y 2 , 1 X 2 ¼ Y 3 , X 3 ¼ Y 4 , . . ., and X m ¼ Y k ðk ¼ m þ 1Þ. We have the square of distance between V and V R0:5 in approximation space ðU; R1 Þ as follows,

 n  1 X 2 ðtV ðxi Þ  t V R1 ðxi ÞÞ2 þ ðf V ðxi Þ  f V R1 ðxi ÞÞ 2n i¼1 0:5 0:5 (   1 X 2 ðt V ðxÞ  t V R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ ¼ 2n x2X 0:5 0:5

2

1 d ðV; V R0:5 Þ¼

1

þ

X

x2X 2

2

ðt V ðxÞ  t V R1 ðxÞÞ þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

0:5

2

 þ  þ

X x2X m

) ðt V ðxÞ  t V R1 ðxÞÞ þ ðf V ðxÞ  f V R1 ðxÞÞ :

2

2

2

0:5

ð6Þ

0:5

2

1 2 In the formula (6), if X 1 2 R1 ðVÞ, then Y 1 2 R2 ðVÞ and Y 2 2 R2 ðVÞ, d ðV; V R0:5 Þ ¼ d ðV; V R0:5 Þ is held. If X 1 2 N R1 ðVÞ, then

2

Y 1 2 N R2 ðVÞ and Y 2 2 N R2 ðVÞ, d

1 ðV; V R0:5 Þ

2

¼d

2 ðV; V R0:5 Þ

is held.

If X 1 2 BR1 ðVÞ, this case is very complex. The theorem will be proved in several cases as follows.  (1) If

X

t V ðxÞ P

x2X 1

X

f V ðxÞ;

x2X 1

that is, for any x 2 X 1 , we have t V R1 ðxÞ ¼ 1 and f V R1 ðxÞ ¼ 0. So we have

X

0:5

2

ðtV ðxÞ  tV R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

x2X 1

0:5



¼

0:5

Xh

i 2 ðt V ðxÞ  1Þ2 þ ðf V ðxÞÞ :

x2X 1

Let X 1 ¼ fx11 ; x12 ; . . . ; x1n1 g, Y 1 ¼ fx11 ; x12 ; . . . ; x1r1 g and Y 2 ¼ fx1;r1 þ1 ; x1;r1 þ2 ; . . . ; x1n1 g. Then we have the following 3 cases. r If

X

t V ðxÞ P

x2Y 1

X

f V ðxÞ and

x2Y 1

X

t V ðxÞ P

x2Y 2

X

f V ðxÞ;

x2Y 2

then

 X 2 ðt V ðxÞ  t V R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

x2X 1

Xh

¼

0:5

ðt V ðxÞ  1Þ2 þ ðf V ðxÞÞ

2

i

x2X 1

 X 2 ðt V ðxÞ  t V R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ

¼

0:5

x2Y 1

0:5

 X 2 þ ðt V ðxÞ  t V R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

x2Y 2

0:5

s If

X x2Y 1

t V ðxÞ P

X x2Y 1

f V ðxÞ;

X x2Y 2

tV ðxÞ <

X x2Y 2

f V ðxÞ;

15

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

then

X

2

ðt V ðxÞ  t V R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

x2X 1

Xh

¼

 ¼

0:5

2

2

ðtV ðxÞ  1Þ þ ðf V ðxÞÞ

i

þ

x2Y 1

Xh

Xh

2

ðt V ðxÞ  1Þ2 þ ðf V ðxÞÞ

i

x2X 1

i 2 ðt V ðxÞ  1Þ2 þ ðf V ðxÞÞ :

x2Y 2

Obviously,

 i X Xh 2 2 ðt V ðxÞ  1Þ2 þ ðf V ðxÞÞ ¼ ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ :

x2Y 1

0:5

x2Y 1

0:5

But,

Xh

2

ðtV ðxÞ  1Þ2 þ ðf V ðxÞÞ

i

x2Y 2 2

¼ ðt V ðx1;r1 þ1 Þ  1Þ2 þ ðf V ðx1;r1 þ1 ÞÞ þ ðt V ðx1;r1 þ2 Þ  1Þ2 2

2

þ ðf V ðx1;r1 þ2 ÞÞ þ    þ ðt V ðx1n1 Þ  1Þ2 þ ðf V ðx1n1 ÞÞ ¼ ðt V ðx1;r1 þ1 ÞÞ2 þ ðt V ðx1;r1 þ2 ÞÞ2 þ    þ ðt V ðx1n1 ÞÞ2    2

2

2

þ ðf V ðx1;r1 þ1 ÞÞ þ ðf V ðx1;r1 þ2 ÞÞ þ    þ ðf V ðx1n1 ÞÞ

 2ðtV ðx1;r1 þ1 Þ þ tV ðx1;r1 þ2 Þ þ    þ t V ðx1n1 ÞÞ þ n1  r 1 2

2

P ðtV ðx1;r1 þ1 ÞÞ2 þ ðtV ðx1;r1 þ2 ÞÞ2 þ    þ ðtV ðx1n1 ÞÞ2    þ ðf V ðx1;r1 þ1 ÞÞ þ ðf V ðx1;r1 þ2 ÞÞ þ    þ ðf V ðx1n1 ÞÞ

2

 2ðf V ðx1;r1 þ1 Þ þ f V ðx1;r1 þ2 Þ þ    þ f V ðx1n1 ÞÞ þ n1  r 1 ¼ ðt V ðx1;r1 þ1 Þ  0Þ2 þ ðtV ðx1;r1 þ2 Þ  0Þ2 þ    þ ðtV ðx1n1 Þ  0Þ2 þ    2

2

2

þ ðf V ðx1;r1 þ1 Þ  1Þ þ ðf V ðx1;r1 þ2 Þ  1Þ þ    þ ðf V ðx1n1 Þ  1Þ  X 2 ¼ ðt V ðxÞ  t V R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

x2Y 2

0:5

Therefore,

  X X 2 2 ðtV ðxÞ  tV R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ P ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ 0:5

x2X 1

0:5

x2Y 1

0:5

0:5

 X 2 ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : þ x2Y 2

0:5

0:5

t If

X

tV ðxÞ <

x2Y 1

X

f V ðxÞ;

x2Y 1

X x2Y 2

t V ðxÞ P

X

f V ðxÞ;

x2Y 2

the proof is similar to s, and we have the following inequality,

X x2X 1

2

ðt V ðxÞ  t V R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5



0:5

 X 2 P ðt V ðxÞ  t V R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ 0:5

x2Y 1

0:5

 X 2 þ ðt V ðxÞ  t V R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

x2Y 2

0:5

2

2

1 2 Because X 2 ¼ Y 3 ; X 3 ¼ Y 4 ; . . . ; X m ¼ Y k ðk ¼ m þ 1Þ, and based on above r, s and t, the inequality d ðV; V R0:5 Þ P d ðV; V R0:5 Þ is held. (2) If

X x2X 1

tV ðxÞ <

X x2X 1

f V ðxÞ;

16

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

that is, for any x 2 X 1 , we have

tV R1 ðxÞ ¼ 0; 0:5

f V R1 ðxÞ ¼ 1; 0:5

then

X

2

ðtV ðxÞ  tV R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

x2X 1



Xh

¼

0:5

i 2 ðt V ðxÞÞ2 þ ðf V ðxÞ  1Þ :

x2X 1

Here we have 3 cases to discuss as follows. r If

X

t V ðxÞ P

x2Y 1

X

f V ðxÞ;

x2Y 1

X

tV ðxÞ <

x2Y 2

X

f V ðxÞ;

x2Y 2

then

 Xh i X 2 2 ðt V ðxÞ  t V R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ ¼ ðtV ðxÞÞ2 þ ðf V ðxÞ  1Þ 0:5

x2X 1

¼

Xh

0:5

2

ðt V ðxÞÞ þ ðf V ðxÞ  1Þ

2

i

x2Y 1

þ

Xh

x2X 1

i 2 ðt V ðxÞÞ þ ðf V ðxÞ  1Þ : 2

x2Y 2

Obviously,

Xh

2

ðtV ðxÞÞ2 þ ðf V ðxÞ  1Þ

i

x2Y 2

¼

X x2Y 2

 2 ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

0:5

But,

Xh

2

ðtV ðxÞÞ2 þ ðf V ðxÞ  1Þ

i

2

2

¼ ðt V ðx11 ÞÞ2 þ ðf V ðx11 Þ  1Þ þ ðt V ðx12 ÞÞ2 þ ðf V ðx12 Þ  1Þ þ    þ ðtV ðx1r1 ÞÞ2

x2Y 1

þ ðf V ðx1r1 Þ  1Þ

2 2

2

¼ ðt V ðx11 ÞÞ2 þ ðt V ðx12 ÞÞ2 þ    þ ðt V ðx1r1 ÞÞ2    þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1r1 ÞÞ

2

 2ðf V ðx11 Þ þ f V ðx12 Þ þ    þ f V ðx1r1 ÞÞ þ r1 2

2

P ðt V ðx11 ÞÞ2 þ ðt V ðx12 ÞÞ2 þ    þ ðt V ðx1r1 ÞÞ2 þ    þ ðf V ðx11 ÞÞ þ ðf V ðx12 ÞÞ þ    þ ðf V ðx1r1 ÞÞ  2ðtV ðx11 Þ þ t V ðx12 Þ þ    þ t V ðx1r1 ÞÞ þ r 1 2

¼ ðt V ðx11 Þ  1Þ2 þ ðtV ðx11 Þ  1Þ2 þ    þ ðt V ðx1r1 Þ  1Þ2    þ ðf V ðx11 Þ  0Þ þ ðf V ðx12 Þ  0Þ 2

þ    þ ðf V ðx1r1 Þ  0Þ  X 2 ¼ ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

x2Y 1

0:5

So,

X

2

ðtV ðxÞ  tV R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

x2X 1

 P

0:5

X x2Y 1

þ

2

ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ 0:5

X

x2Y 2



0:5

 2 ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

0:5

s If

X

t V ðxÞ <

x2Y 1

X

f V ðxÞ;

x2Y 1

X x2Y 2

t V ðxÞ P

X

f V ðxÞ;

x2Y 2

the proof is similar to above r, we have the following inequality,

X x2X 1

2

ðtV ðxÞ  tV R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ 0:5

0:5

 P

X x2Y 1

þ

2

ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ

X

x2Y 2

0:5



0:5

 2 ðtV ðxÞ  tV R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

0:5

2

2

17

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

Fig. 7. The Euclidean similarity degree in different knowledge granularity spaces.

t If

X

tV ðxÞ <

x2Y 1

X

f V ðxÞ;

x2Y 1

X

t V ðxÞ <

x2Y 2

X

f V ðxÞ;

x2Y 2

then we easily have the equation,

X x2X 1

¼

2

ðt V ðxÞ  t V R1 ðxÞÞ2 þ ðf V ðxÞ  f V R1 ðxÞÞ

X

x2Y 1

0:5

 ¼

0:5

2

ðt V ðxÞ  t V R2 ðxÞÞ þ ðf V ðxÞ  f V R2 ðxÞÞ 0:5

0:5

2



Xh x2X 1

þ

2

ðt V ðxÞÞ2 þ ðf V ðxÞ  1Þ

X

x2Y 2

i

 2 ðt V ðxÞ  t V R2 ðxÞÞ2 þ ðf V ðxÞ  f V R2 ðxÞÞ : 0:5

0:5

2

2

1 2 Because X 2 ¼ Y 3 , X 3 ¼ Y 4 , . . ., and X m ¼ Y k ðk ¼ m þ 1Þ, we can obtain the inequality d ðV; V R0:5 Þ P d ðV; V R0:5 Þ based on r, R1 R2 R1 R2 s and t. In a word, dðV; V 0:5 Þ P dðV; V 0:5 Þ is held, that is to say SðV; V 0:5 Þ 6 SðV; V 0:5 Þ is proved successfully. Theorem 4 shows that the finer the knowledge granules are, the greater the Euclidean similarity degree between a vague set and its 0.5-crisp set is. This is consistent with the human cognitive mechanism. This conclusion can be shown by Fig. 7. In 1 2 Fig. 7, the bold dashed line stands for SðV; V R0:5 Þ and the bold solid line stands for SðV; V R0:5 Þ.

5. Conclusions Over the past 30 years of rough set research in the field of rough set, both in the system of rough set theory and the development of application system of rough set, the extended models have made great achievements [9]. Rough set and their extended models have been successfully applied in uncertain artificial intelligence field, such as machine learning, expert system, intelligent decision, data analysis, fuzzy reasoning, complex problem solving, medical diagnosis, fault analysis and so on [43,45]. Combining the fuzzy set theory, rough fuzzy set and fuzzy rough set have been proposed and applied in different fields [42]. Rough set describes an uncertain concept with two boundaries, i.e., the lower-approximation set and upper-approximation set, but they can not describe the uncertain concept with a defined set which is a union set of many granules in approximation space ðU; RÞ. In this paper, we describe the target concept which is a vague set with many defined granules or subsets of U in approximation space ðU; RÞ, and this defined set is called 0.5-crisp set. And we draw a conclusion that the Euclidean similarity degree between a vague set and its 0.5-crisp set is better than that of the vague set and the other defined sets in the approximation space ðU; RÞ. Afterwards, the step-average vague set is presented, and it is proved that the step-average vague set is an optimal step-vague set because the Euclidean similarity degree between a vague set and its step-average vague set in the approximation space ðU; RÞ reaches its maximum value. Finally, the change rules of the Euclidean similarity degree with the different knowledge granularities are discussed, and the rules accord with human cognitive mechanism in multi-granularity knowledge space. In particular, if the similarity degree in Theorem 1, Theorem 2, Theorem 3 and Theorem 4 is not Euclidean similarity degree, these conclusions need further discussion. In practice, an uncertain problem caused by unclear boundary can lead to conflict. However, these uncertainties often come from two aspects. One is the uncertainty of knowledge base (or approximation space ðU; RÞ), the other is the uncertain

18

Q. Zhang et al. / Information Sciences 300 (2015) 1–19

concept. The key for solving the uncertain problem (or disputed problem) is obtaining a defined approximation set of the target concept in the existing knowledge base (or approximation space ðU; RÞ) rather than only offering the boundaries of the uncertain problems. For example, in the field of web-document classification, new document is continuously emerging every day, and how to quickly classify these web-documents is very complex problem. Facing to this kind of web big data, people need to make classification decisions in time even if this classification may be inaccurate. Therefore, even if we can give out this classifications boundary, we still cant efficiently solve web-document classification problem due to the uncertainty of boundary web-document. However, if we utilize the approximation set of an uncertain concept (web-documents in boundary), this problem can be easily solved because the approximation set is a defined set in current knowledge granularity space. We hope our research work can promote the development of uncertain artificial intelligence, and further extend the application domain of rough set theory, vague set theory, rough fuzzy set theory as well as fuzzy rough set theory. In the future research, we will focus on the knowledge acquisition by the approximation sets of an uncertain concept which is useful to acquire the fuzzy decision rules and deal with fuzzy image segmentation. Acknowledgements This work is supported by National Natural Science Foundation of PR China (Nos. 61472056, 61272060, 61309014, 61379114), Natural Science Foundation of Chongqing of PR China (Nos. cstc2012jjA40032, cstc2013jcyjA40063) and Science & Technology Research Program of Chongqing Education Committee of PR China (No. KJ130534). References [1] R. Aliev, W. Pedrycz, L. Zeinalova, O. Huseynov, Decision making with second-order imprecise probabilities, Int. J. Intell. Syst. 29 (2014) 137–160. [2] K.T. 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