Extending fuzzy soft sets with fuzzy description logics

Knowledge-Based Systems 24 (2011) 1096–1107

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

Extending fuzzy soft sets with fuzzy description logics Yuncheng Jiang a,b,⇑, Yong Tang a, Qimai Chen a, Hai Liu a, Jianchao Tang a a b

School of Computer Science, South China Normal University, Guangzhou 510631, PR China State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, PR China

a r t i c l e

i n f o

Article history: Received 27 July 2010 Received in revised form 1 May 2011 Accepted 4 May 2011 Available online 8 May 2011 Keywords: Soft sets Fuzzy soft sets Description logics Fuzzy description logics Fuzzy terminology

a b s t r a c t Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In order to handle these types of problem parameters, some fuzzy extensions of soft set theory are presented, yielding fuzzy soft set theory. Fuzzy description logics (DLs) are a family of logics which allow the representation of and the reasoning within structured knowledge affected by vagueness. In this paper we extend fuzzy soft sets with fuzzy DLs, i.e., present an extended fuzzy soft set theory by using the concepts of fuzzy DLs to act as the parameters of fuzzy soft sets. We define some operations for the extended fuzzy soft sets. Moreover, we prove that certain De Morgan’s laws hold in the extended fuzzy soft set theory with respect to these operations. In fact, the extended fuzzy soft set theory based on fuzzy DLs presented in this paper is a fuzzy extension of the extended soft set theory based on DLs.  2011 Elsevier B.V. All rights reserved.

1. Introduction A number of real life problems in engineering, social and medical sciences, economics, etc. involve imprecise data and their solution involves the use of mathematical principles based on uncertainty and imprecision [47]. Classical methods are not always successful, because the uncertainties appearing in these domains may be of various types [2]. There are theories, e.g., theory of probability [53], theory of fuzzy sets [59], theory of intuitionistic fuzzy sets [4,5], theory of vague sets [19], and theory of rough sets [46] which can be considered as mathematical tools for dealing with uncertainties. However, all these theories have their inherent difficulties as pointed out in [39,45]. The reason for these difficulties is, possibly, the inadequacy of the parameterization tool of the theories [39,45]. Consequently, Molodtsov [45] proposed a completely new approach for modeling vagueness and uncertainty. This socalled soft set theory is free from the difficulties affecting existing methods. Up to the present, research on the soft sets has been very active and many important results have been achieved in the theoretical aspects [16]. The concept and basic properties of soft set theory were presented in [39,45]. Concretely, Maji et al. [39] defined some algebraic operations on soft set theory and published a detail theoretical study on soft sets. Based on the analysis of several oper-

⇑ Corresponding author at: School of Computer Science, South China Normal University, Guangzhou 510631, PR China. E-mail addresses: [email protected], [email protected] (Y. Jiang). 0950-7051/$ - see front matter  2011 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2011.05.003

ations on soft sets introduced in [39], Ali et al. [3] presented some new algebraic operations for soft sets and proved that certain De Morgan’s laws hold in soft set theory with respect to these new definitions. Aktas and Cagman [2] introduced the basic properties of soft sets, compared soft sets to the related concepts of fuzzy sets [59] and rough sets [46], pointed out that every fuzzy set and every rough set may be considered as a soft set, and gave a definition of soft groups, and derived their basic properties. Acar et al. [1] introduced the concepts of soft rings. Feng et al. [17] defined soft semirings and several related notions to establish a connection between soft sets and semirings. Jun [26] applied soft sets to the theory of BCK/BCI-algebras, and introduced the concept of soft BCK/BCIalgebras. Jun and Park [30] and Jun et al. [28,29] reported the applications of soft sets in ideal theory of BCK/BCI-algebras and d-algebras. Xiao et al. [54] proposed the notion of exclusive disjunctive soft sets and studied some of its operations. At the same time, there has been some progress concerning practical applications of soft set theory. Maji et al. [42] described the application of soft set theory to a decision making problem. Chen et al. [15] presented a new definition of soft set parameterization reduction, and compared this definition to the related concept of attribute reduction in rough set theory [46]. Kong et al. [32] analyzed the problems of suboptimal choice and introduced the definition of normal parameter reduction in soft sets. Cagman and Enginoglu [13] defined soft matrices and products of soft matrices, constructed a soft max-min decision making method. Cagman and Enginoglu [14] also defined products of soft sets and uni-int decision function. By using these new definitions they then constructed a uni-int decision making method.

Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

However, in real life many problems are imprecise in nature. The classical soft set theory is not capable of successfully dealing with such problems. Maji et al. [40] initiated the study on hybrid structures involving both fuzzy sets and soft sets. In [40] the notion of fuzzy soft sets was introduced as a fuzzy generalization of classical soft sets and some basic properties were discussed. Afterwards, many researchers have worked on this concept. Fuzzy soft set theory has been used to deal with imprecision, and most results of fuzzy soft sets may be found in [18,40,57,58]. In the theoretical aspects, Aygunoglu and Aygun [6] introduced the concept of fuzzy soft group and in the meantime, discussed some properties and structural characteristics of fuzzy soft group. Jun et al. [27] applied fuzzy soft sets to deal with several kinds of theories in BCK/BCI-algebras. The notions of fuzzy soft BCK/BCIalgebras, (closed) fuzzy soft ideals and fuzzy soft p-ideals were introduced [27]. Feng et al. [18] investigated the problem of combining soft sets with fuzzy sets and rough sets. In general, three different types of hybrid models were presented, which were called rough soft sets, soft rough sets and soft rough fuzzy sets, respectively. Yang et al. [58] defined some generalized operators on fuzzy soft sets by using basic fuzzy logic operators and proved certain De Morgan’s laws. Maji et al. [38,41,43] and Xu et al. [56] extended (classical) soft sets to intuitionistic fuzzy soft sets and vague soft sets, respectively. Yang et al. [57] presented the concept of the interval-valued fuzzy soft sets by combining the interval-valued fuzzy set and soft set models. Jiang et al. [22] combined the interval-valued intuitionistic fuzzy sets and soft sets, from which a new soft set model, i.e., interval-valued intuitionistic fuzzy soft set theory, was obtained. Majumdar and Samanta [44] further generalised the concept of fuzzy soft sets as introduced by Maji et al. [40], that is, the concept of generalized fuzzy soft sets was presented. In application aspects, Kong et al. [32] studied the problem of the reduction of fuzzy soft sets by introducing a definition for normal parameter reduction. Zou and Xiao [61] presented some data analysis approaches of soft sets under incomplete information. Xiao et al. [55] proposed a combined forecasting approach based on fuzzy soft sets. Roy and Maji [47] proposed the concept of a fuzzy soft set and provided its properties and an application in decision making under an imprecise environment. Kong et al. [31] argued that the Roy–Maji method [47] was incorrect and presented a revised algorithm. Feng et al. [16] gave deeper insights into decision making based on fuzzy soft sets. They discussed the validity of the Roy–Maji method [47] and showed its limitations. By means of level soft sets, Feng et al. presented an adjustable approach to fuzzy soft sets based decision making. Recently, Jiang et al. [21] presented an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and gave some illustrative examples. However, all of the soft sets and fuzzy soft sets mentioned above have a common characteristic: the parameters in soft sets and fuzzy soft sets are very simple. Concretely, each parameter is only a word or a sentence, and expressive (or complex) parameters are not considered. In order to extend the expressive power of soft sets, Jiang et al. [23] used the concepts of DLs [7] to act as the parameters of soft sets. That is, an extended soft set theory based on DLs was presented in [23]. Obviously, it will be necessary to carry out further research to extend fuzzy soft sets with DLs. The purpose of this paper is to extend fuzzy soft sets [18,40,58] with fuzzy DLs [8,10,11,24,49–51], from which we can obtain a new fuzzy soft set model: the extended fuzzy soft set based on fuzzy DLs. The rest of this paper is organized as follows. The following section briefly reviews some background on fuzzy DLs and fuzzy soft sets. In Section 3, we extend fuzzy soft sets with fuzzy DLs. In Section 4, we present an approach to semantic decision making by using extended fuzzy soft sets presented in Section 3. Section 5 discusses some extensions of extended fuzzy soft sets. Finally, in Sec-

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tion 6, we draw the conclusion and present some topics for future research. 2. Preliminaries In the current section we will briefly recall the notions of fuzzy DLs and fuzzy soft sets. See especially [8,10,11,18,24,40,47, 49–51,58] for further details and background. 2.1. Fuzzy description logics DLs [7,33,60] are a logical reconstruction of the so-called framebased knowledge representation languages, with the aim of providing a simple well-established Tarski-style declarative semantics to capture the meaning of the most popular features of structured representation of knowledge [11]. Fuzzy DLs are fuzzy extensions of DLs. That is, fuzzy DLs are a family of logics which allow the representation of (and the reasoning within) structured knowledge affected by vagueness [52]. Formally, we introduce the fuzzy DL FALCðDÞ (Fuzzy ALCðDÞ) [50]. It should be noted that the approach for extending fuzzy soft sets with fuzzy DLs presented in this paper is not restricted to FALCðDÞ. It applies to arbitrary (decidable) fuzzy DLs, provided that the fuzzy DL allows for negation. That is, the general approach of extending fuzzy soft sets will be independent of any particular fuzzy DL. Of course, if we incorporate numbers as values of roles (i.e., binary predicates) and allow also numbers to instances of concepts, we have to have fuzzy concrete domains [8,11,50] in the fuzzy DL. Recall that ALCðDÞ is the basic DL ALC [48] extended with concrete domains [35,36] allowing to deal with data types such strings and integers. In FALCðDÞ [50], however, concrete domains are fuzzy sets. A fuzzy concrete domain (or simply fuzzy domain) D is a pair hDD, UDi, where DD is an interpretation domain and UD is the set of fuzzy domain predicates d with a predefined arity n D and an interpretation d : DnD ! ½0; 1, which is a n-ary fuzzy relation over DD. To the ease of presentation, we assume the fuzzy predicates have arity one, the domain is a subset of the real numbers R and the range is [0, 1]. Now, Let C, Ra, Rc, Ia and Ic be non-empty finite and pair-wise disjoint sets of concept names (denoted by A), abstract role names (denoted by R), concrete role names (denoted by T), abstract individual names (denoted by a) and concrete individual names (denoted by u). The concepts of the FALCðDÞ (denoted by C) can be built inductively from concept names (A), top concept >, bottom concept \, abstract role names and concrete role names according to the following syntax rule (d is a unary fuzzy domain predicate):

C ! >j ? jAj:CjC 1 u C 2 jC 1 t C 2 j9R  Cj8R  Cj9T  Dj8T  D; D ! dj:d: By using membership functions that range over the interval [0, 1], classical interpretations can be extended to the concept of fuzzy interpretations [49–51]. The semantics of FALCðDÞ is as follows. A fuzzy interpretation I with respect to a concrete domain D = hDD, UDi is a pair (DI, I) consisting of a non empty set DI (called the interpretation domain), disjoint from DD, and of a fuzzy interpretation function I that assigns:      

to each abstract concept C 2 C a function CI: DI ? [0, 1], to each abstract role R 2 Ra a function RI: DI  DI ? [0, 1], to each abstract individual a 2 Ia an element in DI, to each concrete individual u 2 Ic an element in DD, to each concrete role T 2 Rc a function TI: DI  DD ? [0, 1], to each unary concrete predicate d the fuzzy relation dD: DD ? [0, 1] and to :d the negation of dD.

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Given a t-norm , a t-conorm , a negation H and an implication function ) (see [8,10,52] for some specific choices), the fuzzy interpretation function is extended to complex concepts as follows (where x 2 DI):         

>I(x) = 1; \I(x) = 0; (:C)I(x) = HCI(x); ðC 1 u C 2 ÞI ðxÞ ¼ C I1 ðxÞ  C I2 ðxÞ; ðC 1 t C 2 ÞI ðxÞ ¼ C I1 ðxÞ  C I2 ðxÞ; ð9R:CÞI ðxÞ ¼ supIy2D fRI ðx; yÞ  C I ðyÞg; I ð8R:CÞI ðxÞ ¼ inf y2D fRI ðx; yÞ ) C I ðyÞg I ($T  D) (x) = supv2DD{TI(x, v)  DD(v)}; ("T  D)I(x) = infv2DD{TI(x, v) ) DD(v)}.

Let C, D be FALCðDÞ-concepts. An FALCðDÞ TBox TB is a finite set of fuzzy concept axioms of the form C v D, called fuzzy inclusion axioms, and of the form C  D, called fuzzy equivalence axioms. A fuzzy interpretation I satisfies C v D if "x 2 DI, CI(x) 6 DI(x). A fuzzy interpretation I satisfies C  D if "x 2 DI, CI(x) = DI(x). A fuzzy interpretation I satisfies an FALCðDÞ TBox TB iff it satisfies all fuzzy concept axioms in TB; in this case, we say that I is a model of TB (denoted by I TB). An FALCðDÞ ABox AB is a finite set of fuzzy assertions of the form (a : C) ffl a, (ha, bi : R)ffla, or (ha, ui : T) ffl a, where ffl stands for P, >, 6 and <, and a 2 [0, 1] or the form a = b or a – b. Given a fuzzy interpretation I,        

I I I I I I I I

satisfies satisfies satisfies satisfies satisfies satisfies satisfies satisfies

(a : C) P a if CI(aI) P a; (a : C) 6 a if CI(aI) 6 a; (ha, bi : R) P a if RI(aI, bI) P a; (ha, bi : R) 6 a if RI(aI, bI) 6 a; (ha, ui : T) P a if TI(aI, uD) P a; (ha, ui : T) 6 a if TI(aI, uD) 6 a; a = b if aI = bI; a – b if aI – bI.

The satisfiability of fuzzy assertions with >, < is defined analogously. Observe that, we can also simulate assertions of the form (a : C) = a by considering two assertions of the form (a : C) P a and (a : C) 6 a. A fuzzy interpretation I satisfies an FALCðDÞ ABox AB iff it satisfies all fuzzy assertions in AB; in this case, we say that I is a model of AB (denoted by I AB). Finally, we define some reasoning problems of the FALCðDÞ DL. An FALCðDÞ knowledge base KB consists of a TBox TB and an ABox AB. A fuzzy interpretation I satisfies an FALCðDÞ knowledge base KB if it satisfies all axioms in KB; in this case, I is called a model of KB (denoted by I KB). Given a fuzzy concept axiom or a fuzzy assertion w 2 {C v D, C  D, kffla}, an FALCðDÞ knowledge base KB entails w, written KB w, iff all models of KB also satisfy w.

2.2. Fuzzy soft sets Molodtsov [45] defined the soft set in the following way. Let U be an initial universe of objects and E the set of parameters in relation to objects in U. Parameters are often attributes, characteristics, or properties of objects. Let PðUÞ denote the power set of U and A # E. A pair hF, Ai is called a soft set over U, where F is a mapping given by F : A ! PðUÞ. In other words, the soft set is a parameterized family of subsets of the set U [20,39,45,58]. For any parameter e 2 A, F(e) # U may be considered as the set of e-approximate elements of the soft set hF, Ai. Clearly, there is no limited condition to the description of objects in soft set theory, so researchers can choose the form of

parameters they need, which greatly simplifies the decision-making process and make the process more efficient in the absence of partial information. In other words, soft set theory accommodates approximate description of an object as its starting point. This makes soft set theory a natural mathematical formalism for approximate reasoning. We can use any parameterization we prefer: with the help of words, sentences, real numbers, functions, mappings, and so on. This means that the problem of setting the membership function in fuzzy set theory or the probability density function in probability theory does not arise in soft set theory. Maji et al. [40] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of (crisp) soft sets. Let F ðUÞ be the set of all fuzzy subsets in a universe U. Let E be a set of parameters and A # E. A pair hF, Ai is called a fuzzy soft set over U, where F is a mapping given by F : A ! F ðUÞ [47]. In the above definition, fuzzy subsets in the universe U are used as substitutes for the crisp subsets of U. Hence it is easy to see that every (classical) soft set may be considered as a fuzzy soft set [16]. In general, for any parameter e 2 A, F(e) is a fuzzy subset of U and it is called fuzzy value set of parameter e. If for any parameter e 2 A, F(e) is a crisp subset of U, then hF, Ai is degenerated to be the standard soft set. In order to distinguish with the standard soft set, let us denote lF(e)(x) by the degree that object x holds parameter e where x 2 U and e 2 A. In other words, F(e) can be written as a fuzzy P set such that F(e) = x2UlF(e)(x)/x [58]. The class of all fuzzy value sets of fuzzy soft set hF, Ai is called fuzzy value-class of the fuzzy soft set and is denoted by ChF,Ai, then we have ChF,Ai = {e 2 A : F(e)}. Let hF, Ai and hG, Bi be two fuzzy soft sets over a common universe U, we say that hF, Ai is a fuzzy soft subset of hG, Bi if and only if A # B and for e 2 A, F(e) = G(e) holds, it can be represented as hF, Ai b hG, Bi. hF, Ai is said to be a fuzzy soft super set of hG, Bi, if hG, Bi is a fuzzy soft subset of hF, Ai. We denote it by hF, Ai c hG, Bi. Let hF, Ai and hG, Bi be two fuzzy soft sets, hF, Ai, hG, Bi are said to be fuzzy soft equal if and only if hF, Ai b hG, Bi and hF, Ai c hG, Bi, it can be represented as hF, Ai = hG, Bi. In the standard soft set theory, Maji et al. [39] considered two special soft sets, named as null soft set and absolute soft set, respectively. As far as the fuzzy soft set hF, Ai on U is concerned, if for "e 2 A, "x 2 U, we have lF(e)(x) = 0, then hF, Ai is called a null fuzzy soft set; on the other hand, if for "e 2 A, "x 2 U, we have lF(e)(x) = 1, then hF, Ai is called an absolute fuzzy soft set [58]. Now we introduce some operators on fuzzy soft sets [58]. In the rest of this paper, let , , ), and H be an arbitrary but fixed tnorm, t-conorm, implication function, and negation function, respectively (see [8,10,52] for some specific choices). The negator of a fuzzy soft set hF, Ai is denoted by N ðF; AÞ and it is defined by N ðF; AÞ ¼ hF N ;  Ai, where F N : A ! F ðUÞ is a mapping given by

F N ð eÞ ¼

X x2U

lFðeÞ ðxÞ=x ¼

X

HðlFðeÞ ðxÞÞ=x in which for 8e 2 A:

x2U

Lastly, we define the T operator and the S operator on two fuzzy soft sets. If hF, Ai and hG, Bi are two fuzzy soft sets, then hF; AiT hG; Bi is deP fined by hF; AiT hG; Bi ¼ hH; A  Bi where H(g, k)= x2U(lF(g)(x)  lG(k)(x))/x in which for "(g, k) 2 A  B. If hF, Ai and hG, Bi are two fuzzy soft sets, then hF; AiShG; Bi is P defined by hF; AiShG; Bi ¼ hO; A  Bi where O(g, k) = x2U(lF(g)(x)  lG(k)(x))/x in which for "(g, k) 2 A  B.

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3. Extending fuzzy soft sets with fuzzy description logics

1 In the current section we will extend fuzzy soft sets with fuzzy DLs, that is, we use the concepts of fuzzy DLs to act as the parameters of fuzzy soft sets. First we will give the definition of extended fuzzy soft sets and then we will provide some operations and properties of extended fuzzy soft sets.

0

a

x

b

Fig. 1. R-function.

3.1. Definitions In the rest of this paper, we assume that FDL is an arbitrary (decidable) fuzzy description logic such as FALCðDÞ [50] and FSROIQðDÞ [8], provided that the FDL allows for negation and fuzzy concrete domains. Definition 1. Let FDL be an arbitrary fuzzy DL, X be a set of FDLconcepts, and KB be an FDL-knowledge base, I = (DI, I) be a model of KB. Let F ðUÞ denote the set of all fuzzy subsets of U, U # DI, and M # X. A pair hI, Mi is called an extended fuzzy soft set (EFSS for short) over DI. Clearly, IjM is a mapping given by IjM: M ? F(DI), i.e., "C 2 M, CI can be written as a fuzzy set over DI such that P CI = x2DICI(x)/x. In the above definition, fuzzy subsets in the universe DI are used as substitutes for the crisp subsets of DI. Hence it is easy to see that every extended soft set [23] may be considered as an extended fuzzy soft set. In general, for any FDL-concept C 2 M, CI is a fuzzy subset of DI and it is called fuzzy approximate value set of the FDL-concept C. If for any FDL-concept C 2 M, CI is a crisp subset of DI, then hI, Mi is degenerated to be the extended soft set [23]. In this paper, note that for any extended fuzzy soft set hI, Mi and C 2 M, we abbreviate I(C) by CI using the notation of fuzzy DLs (since C is an FDL-concept). To illustrate this idea, let us consider the following example. Some of it is quoted from [2,16,23,39,45,47,58]. Example 1. Let us consider an EFSS hI, Mi which describes the ‘‘attractiveness of houses’’ that Mr. X is considering for purchase. Suppose that there are six houses in the domain DI = {h1, h2, h3, h4, h5, h6} under consideration, and that X = {C1, C2, C3, C4, C5} is a set of fuzzy decision parameters (i.e., FDL-concepts). The Ci (i = 1, 2, 3, 4, 5) stand for the fuzzy parameters (i.e., FDL-concepts) ‘‘expensive’’, ‘‘beautiful’’, ‘‘large’’, ‘‘convenient traffic’’, and ‘‘in the green surroundings’’, respectively, where these FDL-concepts are defined in the following fuzzy TBox TB:

TB ¼ fexpensiv e  8hasPricePerSquare:highPrice; beautiful  8hasColor:red t 8hasColor:white; large  9hasRoomNumber:multiRooms t 9hasTotalArea:largeArea; conv enient traffic  9hasBusNumber:multiBusest 9hasSubwayNumber:sev eralSubways; in the green surroundings  9hasHillNumber:multiHills t 9hasRiv erNumber:sev eralRiv ersg: In the above TBox TB, the FDL-concepts highPrice, multiRooms, largeArea, multiBuses, severalSubways, multiHills and severalRivers are atomic fuzzy concepts (atomic fuzzy domain predicates). Concerning non crisp fuzzy domain predicates, we recall that in fuzzy set theory and practice there are many membership functions for fuzzy sets membership specification (see [8,10,52] for more details). However, the triangular, the trapezoidal, the L-function

and the R-function are simple, yet most frequently used to specify membership degrees. The functions are defined over the set of non-negative reals Rþ [ f0g. In the following, we use the R-function to specify membership degrees. The R-function, R(x; a, b), is defined as follows (see Fig. 1):

8 if > <0 Rðx; a; bÞ ¼ ðx aÞ=ðb aÞ if > : 1 if

x 6 a; x 2 ½a; b; x P b:

More concretely, highPrice can be defined using an R-function highPrice(x) = R(x; 5000, 8000), multiRooms may be defined as multiRooms(x) = R(x; 1, 4), largeArea may be defined as largeArea(x) = R(x; 100, 200), multiBuses may be defined as multiBuses(x) = R(x; 1, 4), severalSubways may be defined as severalSubways(x) = R(x; 0, 3), multiHills may be defined as multiHills(x) = R(x; 0, 3), and severalRivers may be defined as severalRivers(x) = R(x; 0, 2). On the other hand, we have the following fuzzy ABox AB: AB ¼ fhasPricePerSquareðh1 ;10000Þ ¼ 1:0;hasPricePerSquare ðh2 ; 7500Þ ¼ 1:0; hasPrice PerSquareðh3 ; 9000Þ ¼ 1:0; hasPricePerSquare ðh4 ;7000Þ ¼ 1:0;hasPricePerSquareðh5 ;6000Þ ¼ 1:0;hasPricePerSquare ðh6 ; 6700Þ ¼ 1:0, hasColor(h1, red) = 0.9, hasColor(h2, red) = 0.7, hasColor(h3, white) = 0.8, hasColor(h4, white) = 0.6, hasColor(h5, red) = 0.5, hasColor(h6, white) = 0.6, hasRoomNumber(h1, 5) = 1.0, hasRoomNumber(h2, 3) = 1.0, hasRoomNumber(h3, 4) = 1.0, hasRoomNumber(h4, 3) = 1.0, hasRoomNumber(h5, 2) = 1.0, hasRoomNumber(h6, 3) = 1.0, hasTotalArea(h1, 260) = 1.0, hasTotalArea(h2, 180) = 1.0, hasTotalArea(h3, 220) = 1.0, hasTotalArea(h4, 210) = 1.0, hasTotalArea(h5, 170) = 1.0, hasTotalArea(h6, 200) = 1.0, hasBusNumber(h1, 5) = 1.0, hasBusNumber(h2, 3) = 1.0, hasBusNumber(h3, 4) = 1.0, hasBusNumber(h4, 3) = 1.0, hasBusNumber(h5, 2) = 1.0, hasBusNumber(h6, 3) = 1.0, hasSubwayNumber(h1, 3) = 1.0, hasSubwayNumber(h2, 1) = 1.0, hasSubwayNumber(h4, 2) = 1.0, hasSubwayNumber(h6, 1) = 1.0, hasHillNumber(h1, 3) = 1.0, hasHillNumber(h3, 2) = 1.0, hasHillNumber(h5, 1) = 1.0, hasHillNumber(h6, 2) = 1.0, hasRiverNumber(h1, 2) = 1.0, hasRiverNumber(h2, 1) = 1.0, hasRiverNumber(h3, 2) = 1.0, hasRiverNumber(h4, 1) = 1.0}. Suppose that we have a fuzzy interpretation I such that I TB and I AB. For example, we can obtain this interpretation I accordI I ing to the ABox AB. Firstly, Let hasPricePerSquare ðh1 ; 10000I Þ ¼ I I I I I 1:0; hasPricePerSquare ðh2 ; 7500 Þ ¼ 1:0; hasPricePerSquare ðh3 ; I I I I I 9000 Þ ¼ 1:0; . . . ; hasRiv erNumber ðh3 ; 2 Þ ¼ 1:0; hasRiv erNumber I ðh4 ; 1Þ ¼ 1:0. According to the definitions of the membership functions highPrice(x), multiRooms(x), largeArea(x), multiBuses(x), severalSubways(x), multiHills(x), and severalRivers(x), we have the following: highPriceI(10000I) = 1.0, highPriceI(7500I) = 0.83, highPriceI (9000I) = 1.0, highPriceI(7000I) = 0.67, highPriceI(6000I) = 0.33, highPriceI(6700I) = 0.57, multiRoomsI(5I) = 1.0, multiRoomsI(3I) = 0.67, multiRoomsI(4I) = 1.0, multiRoomsI(2I) = 0.33, largeAreaI(260I) = 1.0, largeAreaI(180I) = 0.8, largeAreaI(220I) = 1.0, largeAreaI(210I) = 1.0,

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largeAreaI(170I) = 0.7, largeAreaI(200I) = 1.0, multiBusesI(5I) = 1.0, multiBusesI(3I) = 0.67, multiBusesI(4I) = 1.0, multiBusesI(2I) = 0.33, severalSubwaysI(3I) = 1.0, severalSubwaysI(1I) = 0.33, severalSubwaysI(2I) = 0.67, multiHillsI(3I) = 1.0, multiHillsI(2I) = 0.67, multiHillsI(1I) = 0.33, severalRiversI(2I) = 1.0, severalRiversI(1I) = 0.5. Finally, according to the semantics of FDL-concepts (see Section 2.1), we obtain the following results (assume that a  b = min{a, b}, a  b = max{a, b}, Ha = 1 a, and a ) b = max{1 a, b}, that is, we use Zadeh semantics of fuzzy operations): I

I

I

I

I

expensiv e ðh1 Þ ¼ inf fhasPricePerSquare ðh1 ; yÞ ) highPrice ðyÞg y2D

I

¼ hasPricePerSquare ðh1 ; 10000Þ I

) highPrice ð10000Þ ¼ 1:0 ) 1:0 ¼ 1:0: I

I

I

I

Similarly, we have that expensiv e ðh2 Þ ¼ 0:83; expensiv e ðh3 Þ ¼ I I I I I 1:0; expensiv e ðh4 Þ ¼ 0:67; expensiv e ðh5 Þ ¼ 0:33, and expensiv e I ðh6 Þ ¼ 0:57. I

beautiful

I ðh1 Þ

I

I

¼ ðhasColor ðh1 ; redÞÞ  ðhasColor ðh1 ; whiteÞÞ ¼ 0:9  0:0 ¼ 0:9: I

I

I

I

Likewise, we have that beautiful ðh2 Þ ¼ 0:7; beautiful ðh3 Þ ¼ I I I I I I 0:8; beautiful ðh4 Þ ¼ 0:6; beautiful ðh5 Þ ¼ 0:5, and beautiful ðh6 Þ ¼ 0:6. I

I

I

I

large ðh1 Þ ¼ ðsupIy2D fhasRoomNumber ðh1 ; yÞ  multiRooms ðyÞgÞ I

I

 ðsupIy2D fhasTotalArea ðh1 ; yÞ  largeArea ðyÞgÞ I

I

¼ ðhasRoomNumber ðh1 ; 5Þ  multiRooms ð5ÞÞ I

I

 ðhasTotalArea ðh1 ; 260Þ  largeArea ð260ÞÞ ¼ ð1:0  1:0Þ  ð1:0  1:0Þ ¼ 1:0: I

I

In a similar way, we have that large ðh2 Þ ¼ 0:8; I I I I I I I I large ðh3 Þ ¼ 1:0; large ðh4 Þ ¼ 1:0; large ðh5 Þ ¼ 0:7, and large ðh6 Þ ¼ 1:0. I

I

I

conv enient traffic ðh1 Þ ¼ ðsupIy2D fhasBusNumber ðh1 ; yÞ I

 multiBuses ðyÞgÞ I

 ðsupIy2D fhasSub wayNumber ðh1 ; yÞ I

 sev eralSubways ðyÞgÞ I

¼ ðhasBusNumber ðh1 ; 5Þ I

 multiBuses ð5ÞÞ I

 ðhasSubwayNumber ðh1 ; 3Þ I

 sev eralSubways ð3ÞÞ ¼ ð1:0  1:0Þ  ð1:0  1:0Þ ¼ 1:0: I

I

I

in the green surroundings ðh1 Þ I

I

¼ ðsupIy2D fhasHillNumber ðh1 ; yÞ  multiHills ðyÞgÞ I

I

 ðsupIy2D fhasRiv erNumber ðh1 ; yÞ  sev eralRiv ers ðyÞgÞ I

U

C1

C2

C3

C4

C5

h1 h2 h3 h4 h5 h6

1.0 0.83 1.0 0.67 0.33 0.57

0.9 0.7 0.8 0.6 0.5 0.6

1.0 0.8 1.0 1.0 0.7 1.0

1.0 0.67 1.0 0.67 0.33 0.67

1.0 0.5 1.0 0.5 0.33 0.67

    I I I I surroundings h4 ¼ 0:5; in the green surroundings h5 ¼ 0:33,   I I and in the green surroundings h6 ¼ 0:67. Clearly, we have I TB and I AB. Therefore, we can view the EFSS hI, Xi as consisting of the following collection of fuzzy approximations: expensive houses = 1.0/h1 + 0.83/h2 + 1.0/h3 + 0.67/h4 + 0.33/ h5 + 0.57/h6, beautiful houses = 0.9/h1 + 0.7/h2 + 0.8/h3 + 0.6/h4 + 0.5/h5 + 0.6/ h6, large houses = 1.0/h1 + 0.8/h2 + 1.0/h3 + 1.0/h4 + 0.7/h5 + 1.0/h6, convenient traffic houses = 1.0/h1 + 0.67/h2 + 1.0/h3 + 0.67/h4 + 0.33/h5 + 0.67/h6, in the green surroundings houses = 1.0/h1 + 0.5/h2 + 1.0/h3 + 0.5/h4 + 0.33/h5 + 0.67/h6. Table 1 gives the tabular representation of the EFSS hI, Xi. It is well-known that fuzzy DL systems [8,10,11,24,49,51,52] provide their users with inference services (like computing the fuzzy subsumption hierarchy and the fuzzy entailment problem) that deduce implicit knowledge from explicitly represented knowledge. Now we show the approach to construct a fuzzy soft set using fuzzy DL reasoning. Given a fuzzy DL knowledge KB ¼ hTB; ABi, where TB ¼ fD1  E1 ; D2  E2 ; . . . ; Dn  En g; AB ¼ fF 1 ða1 Þ ¼ a1 ; F 2 ða2 Þ ¼ a2 ; . . . ; F m ðam Þ ¼ am ; R1 ðb1 ; c1 Þ ¼ b1 ; R2 ðb2 ; c2 Þ ¼ b2 ; . . . ; Rk ðbk ; ck Þ ¼ bk g; fP1 ; P2 ; . . . ; P l g is the set of primitive fuzzy concepts of KB, and {S1, S2, . . . , Sh} is the set of primitive fuzzy roles of KB. Hence, to obtain a model of KB, we have to give a primitive fuzzy interpretation J for {P1, P2, . . . , Pl} and {S1, S2, . . . , Sh} firstly. Next, we can obtain a fuzzy interpretation I for KB according to the semantics of fuzzy DLs [8,10,11,24,49,51,52]. If I TB and I AB, then we have I is a model of KB. Assume that the set of decision parameters M = {C1, C2, . . . , Cj}. Hence, for each parameter C i 2 M; i 2 f1; 2; . . . ; jg; C Ii can be written as a fuzzy set over DI such that C Ii ¼ P I x2D IC i ðxÞ=x. Thus, we obtain an extended fuzzy soft set as follows:

hI; Mi ¼ fðC 1 ; C I1 Þ; ðC 2 ; C I2 Þ; . . . ; ðC j ; C Ij Þg:

Analogously, we have that conv enient traffic ðh2 Þ ¼ I I I I 0:67; conv enient traffic ðh3 Þ ¼ 1:0; conv enient traffic ðh4 Þ ¼ I I I 0:67; conv enient traffic ðh5 Þ ¼ 0:33, and conv enient traffic I ðh6 Þ ¼ 0:67. I

Table 1 Tabular representation of extended fuzzy soft set hI, Xi.

There is a point we have to point out here. To decide whether I TB and I AB, we must rely on fuzzy DL reasoning systems such as DeLorean [9] and fuzzyDL [12]. Obviously, similarly with fuzzy soft sets [58], we have the following definition. Definition 2. The class of all fuzzy value sets of EFSS hI, Mi is called fuzzy value-class of the EFSS and is denoted by VChI,Mi, then we have VChI,Mi = {C 2 M : CI}.

I

¼ ðhasHillNumber ðh1 ; 3Þ  multiHills ð3ÞÞ I

I

 ðhasRiv erNumber ðh1 ; 2Þ  sev eralRiv ers ð2ÞÞ ¼ ð1:0  1:0Þ  ð1:0  1:0Þ ¼ 1:0:

  I I Similarly, we have that in the green surroundings h2 ¼   I I 0:5; in the green surroundings h3 ¼ 1:0; in the green

Example 2. For the extended fuzzy soft set that is represented in Example 1, VChI,Mi can be written as follows:

C I1 ¼ 1:0=h1 þ 0:83=h2 þ 1:0=h3 þ 0:67=h4 þ 0:33=h5 þ 0:57=h6 ; C I2 ¼ 0:9=h1 þ 0:7=h2 þ 0:8=h3 þ 0:6=h4 þ 0:5=h5 þ 0:6=h6 ;

Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

C I3 ¼ 1:0=h1 þ 0:8=h2 þ 1:0=h3 þ 1:0=h4 þ 0:7=h5 þ 1:0=h6 ; C I4 ¼ 1:0=h1 þ 0:67=h2 þ 1:0=h3 þ 0:67=h4 þ 0:33=h5 þ 0:67=h6 ; C I5 ¼ 1:0=h1 þ 0:5=h2 þ 1:0=h3 þ 0:5=h4 þ 0:33=h5 þ 0:67=h6 : 3.2. Operations and properties

1101

(2) M1 \L M2 = {E 2 M2j $Ci 2 M1, for any model I of KB; 8x 2 DI , we have that EI ðxÞ ¼ C Ii ðxÞg. Let us consider the above sets of decision parameters (i.e., FDLconcepts) M1 and M2. If we use the first definition of \L, we have the following

M1 \L M 2 ¼ fcostly; pretty; conv enient communications; Similarly with extended soft sets [23], we need to redefine the operations in fuzzy soft set theory [58] and extend the operations in soft set theory [3]. The reason is that fuzzy DLs have reasoning power such as computing the fuzzy subsumption hierarchy and the fuzzy entailment problem [8,10,11,24,49–51]. For example, given an arbitrary fuzzy DL knowledge base KB, we have I  :(C u D)  (:C t :D) and I  :$R  C  "R  :C under Zadeh semantics and Gödel semantics for any FDL-concepts C, D and FDL-role R in KB, and any model I of KB. Definition 3. Given two sets of decision parameters (i.e., FDLconcepts) M1 = {C1, C2, . . . , Cm} and M2 = {D1, D2, . . . , Dn}, where Ci (1 6 i 6 m) and Dj (1 6 j 6 n) are FDL-concepts in KB ¼ hTB; ABi. If "Ci 2 M1, there exists Dj 2 M2 satisfies: for any model I = (DI, I) of KB; 8x 2 DI ; C Ii ðxÞ ¼ DIj ðxÞ (denoted by Ci  Dj), then we say that M1 is a logical subset of M2 (denoted by M1 # L M2). If M1 # L M2 and M2 # L M1, then we say that M1 and M2 are logical equivalent (denoted by M1 =L M2). The logical intersection of M1 and M2 (denoted by M1 \L M2) is defined as follows: M1 \L M2 = {E 2 M1j $Dj 2 M2, for any model I = (DI, I) of KB; 8x 2 DI , we have that EI ðxÞ ¼ DIj ðxÞg. Obviously, we also may defined the logical intersection as follows: M1 \L M2 = {E 2 M2j $Ci 2 M1, for any model I of KB, "x 2 DI, we have that EI ðxÞ ¼ C Ii ðxÞg. It is easy to know that these two definitions are equivalent from a semantic point of view. However, we can choose only one of the two definitions in EFSSs. In this paper, we use the first definition. The logical union of M1 and M2 (denoted by M1 [L M2) is defined as follows: M1 [L M2 = M1 [ M2 (M1 \L M2). The logical difference of M1 and M2 (denoted by M1 L M2) is defined as follows: M1 L M2 = M1 (M1 \L M2). Obviously, Definition 3 is a fuzzy extension of Definition 2 of [23]. There are some remarks here. It should be noted that in Definition 3 we implicitly assume that a concept name always has the same semantics. That is, it is impossible that a decision parameter matches another decision parameter but their semantics is different. Form Definition 3 we know that if M1 # M2 then M1 # L M2. However, there are some cases where M1 6 # M2, we can also have that M1 # L M2. For example, let M1 = {costly, pretty, convenient communications, in the green environment} and M2 = {expensive, beautiful, large, convenient traffic, in the green surroundings}. Clearly, M1 6 # M2. Suppose that a fuzzy TBox (or fuzzy knowledge base) TB is as follows: TB ¼ fexpensiv e  costly; beautiful  pretty; conv enienttraffic  conv enientcommunications; inthegreensurroundings  inthegreen env ironmentg. By Definition 3, we have that M1 # L M2. Thus, # L and # are different notions, i.e., # L – # . Regarding the logical intersection \L, there are two equivalent definitions from a semantic point of view as follows (see also Definition 3): (1) M1 \L M2 = {E 2 M1j $Dj 2 M2, for any model I of KB; 8x 2 DI , we have that EI ðxÞ ¼ DIj ðxÞg, or

in the green env ironmentg: If we use the second definition of \L, we have the following

M1 \L M 2 ¼ fexpensiv e; beautiful; conv enient traffic; in the green surroundingsg: Obviously, {costly, pretty, convenient communications, in the green environment} and {expensive, beautiful, convenient traffic, in the green surroundings} are equivalent from a semantic point of view. Of course, they are not equivalent from a syntax point of view. There is one point we have to point out here. Although there are two semantically equivalent definitions for the logical intersection \L, we can choose only one of the two definitions in EFSSs. On the other hand, by Definition 3 we have that M1 \ M2 # M1 \L M2 and M1 \L M2 # M1. Since we use the first definition of \L (see Definition 3), thus, we cannot obtain that M1 \L M2 # M2. That is to say, for any sets of decision parameters M1 and M2, we cannot get that M1 \L M2 = M1 \ M2. Therefore, \L and \ are also different notions, i.e., \L – \. For any sets of decision parameters M1 and M2, it is clear that we cannot obtain that M1 \L M2 = M2 \L M1. However, we certainly have that M1 \L M2 =L M2 \L M1. Regarding the logical union and logical difference, see the following examples. Consider the above sets of decision parameters M1 and M2, and the above defined fuzzy TBox TB, by Definition 3, we have the following

M1 [L M 2 ¼ M1 [ M2 ðM 1 \L M2 Þ ¼ fcostly; pretty; conv enientcommunications; inthegreenenv ironmentg [ fexpensiv e; beautiful; large; conv enienttraffic; inthegreensurroundingsg fcostly; pretty; conv enientcommunications; inthegreenenv ironmentg ¼ fexpensiv e; beautiful; large; conv enienttraffic; inthegreensurroundingsg: M2 [L M 1 ¼ M2 [ M1 ðM 2 \L M 1 Þ ¼ fexpensiv e; beautiful; large; conv enienttraffic; inthegreensurroundingsg [ fcostly; pretty; conv enientcommunications; inthegreenenv ironmentg fexpensiv e; beautiful; conv enienttraffic; inthegreensurroundingsg ¼ fcostly; pretty; large; conv enientcommunications; inthegreenenv ironmentg: M1 L M 2 ¼ M1 ðM 1 \L M2 Þ ¼ fcostly; pretty; conv enientcommunications; inthegreenenv ironmentg fcostly; pretty; conv enientcommunications; inthegreenenv ironmentg ¼ /:

1102

Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

M 2 L M 1 ¼ M2 ðM 2 \L M 1 Þ ¼ fexpensiv e; beautiful; large; conv enient traffic; inthegreensurroundingsg fexpensiv e; beautiful; conv enient traffic; inthegreensurroundingsg ¼ flargeg: Definition 4. For two extended fuzzy soft sets hI, Mi and hJ, Ni over a common domain D (i.e., D = DI = DJ) and a common fuzzy knowledge base KB, we say that hI, Mi is a fuzzy soft subset of hJ, Ni if (i) M # L N, and (ii) "C 2 M, CI = CJ, i.e., CI(x) = CJ(x) for any x 2 D. We write hI, Mi b hJ, Ni. hI, Mi is said to a fuzzy soft superset of hJ, Ni, if hJ, Ni is a fuzzy soft subset of hI, Mi. We denote it by hI, Mi c hJ, Ni. Definition 5. Two extended fuzzy soft sets hI, Mi and hJ, Ni over a common domain D and a common fuzzy knowledge base KB are said to be fuzzy soft equal if hI, Mi is a fuzzy soft subset of hJ, Ni and hJ, Ni is a fuzzy soft subset of hI, Mi. Definition 6. Let M = {C1, C2, . . . , Cm} be a set of parameters (i.e., FDL-concepts). The NOT set of M denoted by eM is defined by eM = {:C1, :C2, . . . , :Cm}, where :is the negation constructor of fuzzy DLs [8,10,11,24,49,51,52]. The following results are obvious.

Definition 8. The T operator on two extended fuzzy soft sets. If hI, Mi and hJ, Ni are two EFSSs over a common domain D and a common fuzzy knowledge base KB, then hI, MiT hJ, Ni is defined by P hI; MiT hJ; Ni ¼ hH; M  Ni where (C, D)H = x2D(CI(x)  DJ(x))/x, "(C, D) 2 M  N. Example 4. For the EFSS that is represented in Example 1, DI = {h1, h2, h3, h4, h5, h6}, suppose that M = {C1, C2, C3}, N = {C4, C5}, a  b = min{a, b}, then we have the following:

ðC 1 ; C 4 ÞH ¼ 1:0=h1 þ 0:67=h2 þ 1:0=h3 þ 0:67=h4 þ 0:33=h5 þ 0:57=h6 ; ðC 1 ; C 5 ÞH ¼ 1:0=h1 þ 0:5=h2 þ 1:0=h3 þ 0:5=h4 þ 0:33=h5 þ 0:57=h6 ; ðC 2 ; C 4 ÞH ¼ 0:9=h1 þ 0:67=h2 þ 0:8=h3 þ 0:6=h4 þ 0:33=h5 þ 0:6=h6 ; ðC 2 ; C 5 ÞH ¼ 0:9=h1 þ 0:5=h2 þ 0:8=h3 þ 0:5=h4 þ 0:33=h5 þ 0:6=h6 ; ðC 3 ; C 4 ÞH ¼ 1:0=h1 þ 0:67=h2 þ 1:0=h3 þ 0:67=h4 þ 0:33=h5 þ 0:67=h6 ; ðC 3 ; C 5 ÞH ¼ 1:0=h1 þ 0:5=h2 þ 1:0=h3 þ 0:5=h4 þ 0:33=h5 þ 0:67=h6 : Definition 9. The S operator on two extended fuzzy soft sets. If hI, Mi and hJ, Ni are two EFSSs over a common domain D and a common fuzzy knowledge base KB, then hI; MiShJ; Ni is defined by P hI; MiShJ; Ni ¼ hO; M  Ni where (C, D)O = x2D(CI(x)  DJ(x))/x, "(C, D) 2 M  N. Example 5 (Example 4 cont’d). Suppose that a  b = max{a, b}, then we have the following:

ðC 1 ; C 4 ÞO ¼ 1:0=h1 þ 0:83=h2 þ 1:0=h3 þ 0:67=h4 þ 0:33=h5 þ 0:67=h6 ; ðC 1 ; C 5 ÞO ¼ 1:0=h1 þ 0:83=h2 þ 1:0=h3 þ 0:67=h4 þ 0:33=h5 þ 0:67=h6 ;

Theorem 1. For any two sets of parameters (i.e., FDL-concepts) M and N, the following properties are satisfied:

ðC 2 ; C 4 ÞO ¼ 1:0=h1 þ 0:7=h2 þ 1:0=h3 þ 0:67=h4 þ 0:5=h5 þ 0:67=h6 ; ðC 2 ; C 5 ÞO ¼ 1:0=h1 þ 0:7=h2 þ 1:0=h3 þ 0:6=h4 þ 0:5=h5 þ 0:67=h6 ;

(1) (2) (3) (4)

for any FDL-concept C, if :(:(C))  C, then e(eM) =L M; e(M [L N) =L (eM [L eN); e(M \L N) =L (eM \L eN); e(M L N) =L (eM L eN).

Proof. The proof is similar to that of Theorem 1 in [23]. h For example, e(eM) =L M is satisfied under Zadeh semantics and Lukasiewicz semantics, but is not satisfied under Gödel semantics and Product semantics. Regarding Zadeh semantics, Lukasiewicz semantics, Gödel semantics and Product semantics of fuzzy operations, see [8,10,52] for more details.

ðC 3 ; C 4 ÞO ¼ 1:0=h1 þ 0:8=h2 þ 1:0=h3 þ 1:0=h4 þ 0:7=h5 þ 1:0=h6 ; ðC 3 ; C 5 ÞO ¼ 1:0=h1 þ 0:8=h2 þ 1:0=h3 þ 1:0=h4 þ 0:7=h5 þ 1:0=h6 : Definition 10. The extended union of two extended fuzzy soft sets hI, Mi and hJ, Ni over D and KB is the extended fuzzy soft set hK, Gi, where G = M [L N and "C 2 G,

8P I C ðxÞ=x; if > > > > x2D >

> > P > I J > : ðC ðxÞ  C ðxÞÞ=x; if

C 2 M L N; C 2 N L M; C 2 N\L M;

x2D

Definition 7. The complement of an extended fuzzy soft set hI, Mi is denoted by hI, Mic and is defined by hI, Mic = hIc, eMi, where Ic :eM ! F ðDI Þ is a mapping given by

C Ic ¼

X x2DI

C Ic ðxÞ=x ¼

X

Hðð:CÞI ðxÞÞ=x; 8C 2eM:

where if C 2 N \L M;

then we have

C I ðxÞ ¼ DI ðxÞ; D 2 M; KB C  D: This relationship is denoted by hI; MidE hJ; Ni ¼ hK; Gi.

x2DI

Clearly, hhI, Micic = hI, Mi under Zadeh semantics or Lukasiewicz semantics. Example 3. For the extended fuzzy soft set that is represented in Example 1, suppose that H(a) = 1 a, then we have the following: Ic

ð:C 1 Þ ¼ 0=h1 þ 0:17=h2 þ 0=h3 þ 0:33=h4 þ 0:67=h5 þ 0:43=h6 ;

Definition 11. Let hI, Mi and hJ, Ni be two extended fuzzy soft sets over a common domain D and a common fuzzy knowledge base KB such that M \L N – /. The restricted union of hI, Mi and hJ, Ni is denoted by hI; MidR hJ; Ni ¼ hK; Gi, where G = M \L N and for all C 2 G,

CK ¼

X I ðC ðxÞ  C J ðxÞÞ=x; where C J ðxÞ ¼ DJ ðxÞ; D 2 N; KB C  D: x2D

ð:C 2 ÞIc ¼ 0:1=h1 þ 0:3=h2 þ 0:2=h3 þ 0:4=h4 þ 0:5=h5 þ 0:4=h6 ; ð:C 3 ÞIc ¼ 0=h1 þ 0:2=h2 þ 0=h3 þ 0=h4 þ 0:3=h5 þ 0=h6 ; ð:C 4 ÞIc ¼ 0=h1 þ 0:33=h2 þ 0=h3 þ 0:33=h4 þ 0:67=h5 þ 0:33=h6 ; ð:C 5 ÞIc ¼ 0=h1 þ 0:5=h2 þ 0=h3 þ 0:5=h4 þ 0:67=h5 þ 0:33=h6 :

Definition 12. The extended intersection of two extended fuzzy soft sets hI, Mi and hJ, Ni over a common domain D and a common fuzzy knowledge base KB is the extended fuzzy soft set hK, Gi, where G = M [L N, "C 2 G,

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Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

8P I C ðxÞ=x; if > > > x2D > >

> > P > > : ðC I ðxÞ  C J ðxÞÞ=x; if

C 2 M L N C 2 N L M ; C 2 N\L M

x2D

where if C 2 N \L M; then we have C I ðxÞ ¼ DI ðxÞ; D 2 M; KB C  D: We write hI; Mi eE hJ; Ni ¼ hK; Gi. Definition 13. Let hI, Mi and hJ, Ni be two extended fuzzy soft sets over a common domain D and a common fuzzy knowledge base KB such that M \L N – /. The restricted intersection of hI, Mi and hJ, Ni is denoted by hI; Mi eR hJ; Ni, and is defined as hI; MieR hJ; Ni ¼ hK; Gi, where G = M \L N and for all C 2 G,

Proof. Let ,  and H be an arbitrary but fixed t-norm, t-conorm (i.e., s-norm) and negation function, respectively. (1) Since hI, Mic = hIc, eMi and hJ, Nic = hJc, eNi, then we have

hhI; Mic ShJ; Nic i ¼ hhIc ; eMiShJ c ; eNii: Assume that hhIc ; eMiShJ c ; eNii ¼ hO; eMeNi. P Then, "(C, D) 2 eM  eN, we have (C, D)O = x2D(CIc(x)  DJc(x))/x. Suppose that hhI; MiT hJ; Niic ¼ hH; M  Nic ¼ hHc ; eðM  NÞi. Since (C, D) 2 eM  eN, then we obtain (C, D) 2 e(M  N) and :(C, D)=(:C, :D) 2 M  N. Hence, we have the following P (:C, :D)H = x2D((:C)I(x)  (:D)J(x))/x. Then,

ðC; DÞHc ¼

X X ðC; DÞHc ðxÞ=x ¼ Hðð:ðC; DÞÞH ðxÞÞ=x x2D

X I CK ¼ ðC ðxÞ  C J ðxÞÞ=x; where C J ðxÞ ¼ DJ ðxÞ; D 2 N; KB C  D:

¼

X

x2D

Hðð:C; :DÞH ðxÞÞ=x

x2D

x2D

¼

X

Hðð:CÞI ðxÞ  ð:DÞJ ðxÞÞ=x

x2D

Definition 14. Let hI, Mi and hJ, Ni be two extended fuzzy soft sets over a common domain D and a common fuzzy knowledge base KB such that M \L N – /. The restricted difference of hI, Mi and hJ, Ni is denoted by hI; Mi^R hJ; Ni ¼ hK; Gi, where G = M \L N and for all C 2 G,

CK ¼

X

maxf0; C I ðxÞ C J ðxÞg=x;

x2D

where C J ðxÞ ¼ DJ ðxÞ; D 2 N; KB C  D: Definition 15. Let D be an initial domain set, X be the universe set of parameters (i.e., FDL-concepts), and M L X. hI, Mi is called a relative null extended fuzzy soft set (with respect to the parameter set M), denoted by UM, if CI(x) = 0 for "C 2 M, "x 2 D. hJ, Mi is called a relative whole extended fuzzy soft set (with respect to the Q I parameter set M), denoted by M , if C (x) = 1Qfor all C 2 M, "x 2 D. The relative whole extended fuzzy soft set X with respect to the universe set of parameters X is called the absolute extended fuzzy soft set over D.

¼

X ðHðð:CÞI ðxÞÞ  Hðð:DÞJ ðxÞÞÞ=x

¼

X Ic ðC ðxÞ  DJc ðxÞÞ=x:

x2D

x2D

Therefore, hhI; MiT hJ; Niic ¼ hhI; Mic ShJ; Nic i. (2) Since hI, Mic = hIc, e Mi and hJ, Nic = hJc, eNi, then we have

hhI; Mic T hJ; Nic i ¼ hhIc ; eMiT hJ c ; eNii: Suppose that hhIc ; eMiT hJ c ; eNii ¼ hH; eMeNi. P Then, "(C, D) 2 eM  eN, we have (C, D)H = x2D(CIc(x)  DJc(x))/x. Assume that hhI; MiShJ; Niic ¼ hhO; M  Niic ¼ hOc ; eðM  NÞi. Since (C, D) 2 eM  eN, then we obtain (C, D) 2 e(M  N) and :(C, D)=(:C, :D) 2 M  N. Hence, we have the following P (:C, :D)O = x2D((:C)I(x)  (:D)J(x))/x. Then,

ðC; DÞOc ¼

X X ðC; DÞOc ðxÞ=x ¼ Hðð:ðC; DÞÞO ðxÞÞ=x x2D

¼

X

x2D O

Hðð:C; :DÞ ðxÞÞ=x ¼

x2D

Definition 16. The relative complement of an extended fuzzy soft set hI, Mi is denoted by hI, Mir and is defined by hI, Mir = hIr, Mi, where Ir: M ? F(DI) is a mapping given by

C Ir ¼

X

¼

X

Hðð:CÞI ðxÞ  ð:DÞJ ðxÞÞ=x

x2D

X ðHðð:CÞI ðxÞÞ  Hðð:DÞJðxÞÞÞ=x x2D

X Ic ¼ ðC ðxÞ  DJc ðxÞÞ=x: x2D

HðC I ðxÞÞ=x;

8C 2 M:

Therefore, hhI; MiShJ; Niic ¼ hhI; Mic T hJ; Nic i.

h

x2DI

Q Clearly, hI; Mir ¼ X ^R hI; Mi and hhI, Mirir = hI, Mi under Zadeh semantics or Lukasiewicz semantics. From the definitions defined above (from Definition 3 to Definition 16), it is obvious to know that these definitions are the extensions of the corresponding definitions in [3,23,39,58]. By the suggestions given by Ali et al. [3], Jiang et al. [23], and Yang et al. [58], in the following we will show that some De Morgan’s type of results hold in extended fuzzy soft set theory for the above defined operations. Theorem 2. Let hI, Mi and hJ, Ni be two extended fuzzy soft sets over a common domain D and a common fuzzy knowledge base KB. Then we have the following properties: (1) hhI; MiT hJ; Niic ¼ hhI; Mic ShJ; Nic i; (2) hhI; MiShJ; Niic ¼ hhI; Mic T hJ; Nic i.

From Theorem 2 we know that the complement, T operator, and S operator in extended fuzzy soft set theory presented in this paper satisfy the De Morgan’s laws. In fact, these properties are the same as that of the fuzzy soft set theory presented in [58] (see Theorem 1 in literature [58]). In other words, the extended fuzzy soft set theory preserves the corresponding properties of the fuzzy soft set theory presented in [58]. Theorem 3. Let hI, Mi and hJ, Ni be two extended fuzzy soft sets over a common domain D and a common fuzzy knowledge base KB such that M \L N – /. Then (1) hhðI; Mi dR hJ; Niir ¼ hI; Mir eR hJ; Nir ; (2) hhI; MÞ eR hJ; Niir ¼ hI; Mir dR hJ; Nir . Proof. Let ,  and H be an arbitrary but fixed t-norm, t-conorm (i.e., s-norm) and negation function, respectively.

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Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

P (1) Let hI; Mi dR hJ; Ni ¼ hK; Gi, where CK = x2D(CI(x)  CJ(x))/x for all C 2 G = M \L N – /. Since hhðI; Mi dR hJ; Niir ¼ hK; Gir , by the definition of relative complement, for all C 2 G, we P P have that CKr = x2DIH(CK(x))/x = x2DIH(CI(x)  CJ(x))/x. r r r Let hI; Mi eR hJ; Ni ¼ hI ; Mi eR hJ r ; Ni ¼ hL; Gi, where G = P M \L N and for all C 2 G, CL = x2D(CIr(x)  CJr(x))/x. By the definition of relative complement, for all C 2 G, we have that P P CL = x2D((H(CI(x)))  (H(CJ(x))))/x = x2DIH(CI(x)  CJ(x))/ Kr x=C . Therefore, hhðI; Mi dR hJ; Nðiir ¼ hI; Mir eR hJ; Nir . (2) Let hI; MÞ eR hJ; Ni ¼ hK; Gi, where G = M \L N and for all P C 2 G, CK = x2D(CI(x)  CJ(x))/x. Since hhI; MÞ eR hJ; Niir ¼ r hK; Gi , by the definition of relative complement, for all P P C 2 G, we have that CKr = x2DIH(CK(x))/x = x2DIH(CI(x)  CJ(x))/x.

DK

C

8P > Hðð:DÞI ðxÞÞ=x; if > > > x2 D > >

> >P > > > Hðð:DÞI ðxÞ  ð:DÞJ ðxÞÞ=x; if :

:D 2 M L N :D 2 N L M :D 2 N\L M

x2D

8P > Hðð:DÞI ðxÞÞ=x; if > > x2D > > >

> > >P > I J > : ðHð:DÞ ðxÞÞ  ðHð:DÞ ðxÞÞ=x; if

D 2 :M L :N; D 2 :N L :M; D 2 :N\L :M:

x2D

Again, hI; Mic eE hJ; Nic ¼ hIc ; eMi eE hJ c ; eNi ¼ hL; Hi, where H = eM [L eN, "D 2 H = G,

8 P Ic > D ðxÞ=x; if D 2 :M L :N > > > x2D > > < P Jc D ðxÞ=x; if D 2 :N L :M DL ¼ x2D > > > > c P c > > : ðDI ðxÞ  DJ ðxÞÞ=x; if D 2 :N\L :M

Let hI; Mir dR hJ; Nir ¼ hIr ; Mi dR hJ r ; Ni ¼ hL; Gi, where G = P M \L N and for all C 2 G, CL = x2D(CIr(x)  CJr(x))/x. By the definition of relative complement, for all C 2 G, we have that P P CL = x2D((H(CI(x)))  (H(CJ(x))))/x = x2DIH(CI(x)  CJ(x))/x = CKr. r Therefore, hhI; MÞ eR hJ; Nii ¼ hI; Mir dR hJ; Nir . h

x2D

8P > Hðð:DÞI ðxÞÞ=x; if D 2 :M L :N > > x2D > > >

> > >P > I J > : ðHð:DÞ ðxÞÞ  ðHð:DÞ ðxÞÞ=x; if D 2 :N\L :M

From Theorem 3 we know that the relative complement, the restricted union, and the restricted intersection in extended fuzzy soft set theory presented in this paper satisfy the De Morgan’s laws. Theorem 4. Let hI, Mi and hJ, Ni be two extended fuzzy soft sets over a common domain D and a common fuzzy knowledge base KB. Then under Zadeh semantics or Lukasiewicz semantics we have the following properties:

x2D

Therefore, hhðI; MidE hJ; Niic ¼ hI; Mic eE hJ; Nic . (2) Let hI; Mi eE hJ; Ni ¼ hK; Gi, where G = M [L N, "C 2 G,

8P I > C ðxÞ=x; if > > > x2D > >

> >P > > I J > : ðC ðxÞ  C ðxÞÞ=x; if

(1) hhðI; Mi dE hJ; Niic ¼ hI; Mic eE hJ; Nic ; (2) hhI; Mi eE hJ; Niic ¼ hI; Mic dE hJ; Nic .

Proof

C 2 M L N;

DK

C

C 2 N\L M:

Then we have that hhI; Mi eE hJ; Niic ¼ hK; Gic ¼ hK c ; eGi ¼ P P hK ; eM [L eNi, where DKc = x2DDKc(x)/x = x2DH((:D)K(x))/x, "D 2 (eM [L eN). Thus, we have the following:

C 2 N L M;

ðÞ

C 2 N\L M:

c

c

Kc

DK

Then we have hhðI; Mi dE hJ; Nii ¼ hK; Gi ¼ hK ; eGi, where D = P P Kc K x2DD (x)/x = x2DH((:D) (x))/x, "D 2 eG. Since G = M [L N, by the property (2) of Theorem 1 we have eG = eM [L eN, thus hKc, eGi = hKc, eM [L eNi. Since eM [L eN = {eM L eN} [ {eN-LeM} [ {eN \L eM} and {eM L e N} \ {eN-LeM} \ {eN \L eM} = /, therefore, we obtain

8P Hðð:DÞK ðxÞÞ=x; if > > > x2D > > >

> >P > > > : Hðð:DÞK ðxÞÞ=x; if

ð Þ

c

x2D

c

C 2 N L M;

x2D

(1) Let hðI; Mi dE hJ; Ni ¼ hK; Gi, where G = M [L N, "C 2 G,

8P I C ðxÞ=x; if > > > > x2D >

> > P > I J > : ðC ðxÞ  C ðxÞÞ=x; if

C 2 M L N;

D 2 :M L :N; D 2 :N L :M;

C

8P > Hðð:DÞK ðxÞÞ=x; if > > > x2D > >

> > > P > > Hðð:DÞK ðxÞÞ=x; if :

D 2 :M L :N; D 2 :N L :M; D 2 :N\L :M:

x2D

According to (⁄⁄) we obtain the following:

DK

C

8P > Hðð:DÞI ðxÞÞ=x; if > > > x2D > >

> > > P > > Hðð:DI ÞðxÞ  ð:DJ ÞðxÞÞ=x; if :

:D 2 M L N :D 2 N L M :D 2 N\L M

x2D

D 2 :N\L :M:

x2D

Since D 2 eG, then we have :D 2 G. By the properties (1), (3), and (4) of Theorem 1, we have D 2 (eM L eN) iff :D 2 e(eM L eN) iff :D 2 e(e(M L N)) iff :D 2 (M L N), D 2 (eN L eM) iff :D 2 (N L M), and D 2 (eN \L eM) iff :D 2 e(eN \L eM) iff :D 2 e(e(N \L M)) iff :D 2 (N \L M). Hence, according to (⁄) we have the following:

8P > Hðð:DÞI ðxÞÞ=x; if > > > x2 D > < P ¼ Hðð:DJ ÞðxÞÞ=x; if > > x2D > > > : ðHð:DI ÞðxÞÞ  ðHð:DJ ÞðxÞÞ=x; if c

c

c

c

D 2 :M L :N; D 2 :N L :M; D 2 :N\L :M: c

Again, hI; Mi dE hJ; Ni ¼ hI ; eMi dE hJ ; eNi ¼ hL; Hi, where H = eM [L eN, "D 2 H = eG,

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Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

8 P Ic D ðxÞ=x; if D 2 :M L :N > > > x2D > > < P Jc D ðxÞ=x; if D 2 :N L :M DL ¼ x2D > > > c P c > > : ðDI ðxÞ  C J ðxÞÞ=x; if D 2 :N\L :M x2D 8P I > if D 2 :M L :N > > x2D Hðð:DÞ ðxÞÞ=x; > < P KC ¼ Hðð:DJ ÞðxÞÞ=x; if D 2 :N L :M ¼ D : > > x2 D > > : ðHð:DI ÞðxÞÞ  ðHð:DJ ÞðxÞÞ=x; if D 2 :N\L :M Therefore, hhI; Mi eE hJ; Niic ¼ hI; Mic dE hJ; Nic . h From Theorem 4 we know that the extended intersection, the extended union, and the complement in extended fuzzy soft set theory presented in this paper satisfy the De Morgan’s laws. In fact, the properties in Theorem 3 (resp., Theorem 4) are the same as that of Theorem 4.1 (resp., Theorem 4.2) in [3] and Theorem 2 (resp., Theorem 3) in [23]. That is to say, the extended fuzzy soft set theory preserves the corresponding properties of the soft set theory presented in [3] and the extended soft set theory presented in [23].

4. Semantic decision making using EFSSs In this section, we present an approach to semantic decision making by using extended fuzzy soft sets presented in Section 3. Let us continue to consider the Purchase House example (Example 1). Now we also consider that Mr. X is going to buy a house. Suppose that there are two different extended fuzzy soft sets hI, Mi and hJ, Ni in different Real Estate Companies, where hI, Mi is an extended fuzzy soft set defined in Table 1, hJ, Ni is an extended fuzzy soft set defined in Table 2, where the Di (i = 1, 2, 3, 4, 5) stand for the fuzzy parameters (i.e., FDL-concepts) ‘‘costly’’, ‘‘pretty’’, ‘‘wooden’’, ‘‘convenient communications’’, and ‘‘in the green environment’’, respectively. Suppose that a fuzzy TBox (or fuzzy knowledge base) TB is as follows: TB ¼ fexpensiv e  costly; beautiful  pretty; conv enienttraffic  conv enientcommunications; inthegreensurroundings  inthegreenen v ironmentg. In order to help Mr. X to make decision, firstly we have to merge the extended fuzzy soft sets hI, Mi and hJ, Ni, that is, we must obtain an extended fuzzy soft set hK, Gi from hI, Mi and hJ, Ni by using the extended union operation of extended fuzzy soft set theory (see Definition 10). By Definition 10, the hK, Gi is shown in Tables 3 or 4. Obviously, hK, Gi is also a traditional fuzzy soft set from a syntax point of view. Therefore, we can help Mr. X to make decision by using the approaches to traditional fuzzy soft sets based decision making [16,31,47]. If we do not use the above-proposed method, that is, we adopt the union operation of traditional fuzzy soft set theory (see [18,40,57,58] for more details), then a new fuzzy soft set hL, Hi is obtained from hI, Mi and hJ, Ni, where the hL, Hi is shown in Table 5. Obviously, the above-given fuzzy soft set hL, Hi is not correct from a semantics point of view since there exist some semantic relations between parameters of hI, Mi and parameters of hJ, Ni. For example, expensive  costly, beautiful  pretty, and convenient traffic  convenient communications. The correct fuzzy soft set should be shown in Tables 3 or 4. In other words, Mr. X should make decision using Tables 3 or 4 (not Table 5). Analogously, we can also use the other operations such as the T operator, the S operator, the restricted union, the extended intersection, and the restricted intersection for the other convenient problems.

Table 2 Tabular representation of extended fuzzy soft set hJ, Ni. U

D1

D2

D3

D4

D5

h1 h2 h3 h4 h5 h6

1.0 0.83 1.0 0.67 0.33 0.57

0.9 0.7 0.8 0.6 0.5 0.6

0.8 0.77 0.9 0.75 0.63 0.69

1.0 0.67 1.0 0.67 0.33 0.67

1.0 0.5 1.0 0.5 0.33 0.67

Table 3 Tabular representation of extended fuzzy soft set hK; Gi ¼ hJ; Ni dE hI; Mi. U

C1

C2

C3

C4

C5

D3

h1 h2 h3 h4 h5 h6

1.0 0.83 1.0 0.67 0.33 0.57

0.9 0.7 0.8 0.6 0.5 0.6

1.0 0.8 1.0 1.0 0.7 1.0

1.0 0.67 1.0 0.67 0.33 0.67

1.0 0.5 1.0 0.5 0.33 0.67

0.8 0.77 0.9 0.75 0.63 0.69

Table 4 Tabular representation of extended fuzzy soft set hK; Gi ¼ hI; Mi dE hJ; Ni. U

D1

D2

D3

D4

D5

C3

h1 h2 h3 h4 h5 h6

1.0 0.83 1.0 0.67 0.33 0.57

0.9 0.7 0.8 0.6 0.5 0.6

0.8 0.77 0.9 0.75 0.63 0.69

1.0 0.67 1.0 0.67 0.33 0.67

1.0 0.5 1.0 0.5 0.33 0.67

1.0 0.8 1.0 1.0 0.7 1.0

Table 5 Tabular representation of fuzzy soft set hL, Hi. U

C1

C2

C3

C4

C5

D1

D2

D3

D4

D5

h1 h2 h3 h4 h5 h6

1.0 0.83 1.0 0.67 0.33 0.57

0.9 0.7 0.8 0.6 0.5 0.6

1.0 0.8 1.0 1.0 0.7 1.0

1.0 0.67 1.0 0.67 0.33 0.67

1.0 0.5 1.0 0.5 0.33 0.67

1.0 0.83 1.0 0.67 0.33 0.57

0.9 0.7 0.8 0.6 0.5 0.6

0.8 0.77 0.9 0.75 0.63 0.69

1.0 0.67 1.0 0.67 0.33 0.67

1.0 0.5 1.0 0.5 0.33 0.67

5. Discussion From Section 3 we know that the extended fuzzy soft set theory presented in this paper is a fuzzy extension of the extended soft set theory [23] based on classical fuzzy soft sets (FSSs) [40,47,58] and classical fuzzy DLs (FDLs) [8,10,11,24,49–52]. In fact, there are several fuzzy soft sets such as vague soft sets (VSSs) [56], interval-valued fuzzy soft sets (IVFSSs) [57] and intuitionistic fuzzy soft sets (IFSSs) [38,41,43] other than classical fuzzy soft sets [40,47,58]. Obviously, we can obtain different extended fuzzy soft sets based on different fuzzy soft sets and different fuzzy DLs [34]. For example, we can extend intuitionistic fuzzy soft sets with intuitionistic fuzzy DLs (IFDLs) [25] (i.e., obtain extended intuitionistic fuzzy soft sets EIFSSs) and extend vague soft sets with vague DLs (VDLs) [37] (i.e., obtain extended vague soft sets EVSSs). That is, we can obtain different extended fuzzy soft sets depicted in Fig. 2. In the following, we roughly discuss the basic definitions of the extended intuitionistic fuzzy soft sets (EIFSSs). That is, we will simply discuss how to extend intuitionistic fuzzy soft sets (IFSSs) [38,41,43] with intuitionistic fuzzy DLs (IFDLs) [25] using the approach presented in Section 3. Let IFDL be an arbitrary intuitionistic fuzzy DL such as IFALC [25], X be a set of IFDL-concepts, KB be an IFDL-knowledge base,

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Y. Jiang et al. / Knowledge-Based Systems 24 (2011) 1096–1107

IFSSs

EIFSSs

VSSs

6. Conclusion

EVSSs EIVFSSs

IVFSSs FSSs

IVChI,Mi, then we have IVC hI;Mi ¼ fC 2 M : C I g ¼ fC 2 M :  P I I I x2D IC ðxÞ=xg ¼ C 2 M : x2D IhlC ðxÞ; cC ðxÞi=x .

by P

Fuzzy soft sets

EFSSs

FDLs

IVFDLs

VDLs

IFDLs Fuzzy DLs

Fig. 2. Different extended fuzzy soft sets.

I = (DI, I) be a model of KB. Let IF ðUÞ denote the set of all intuitionistic fuzzy subsets of U, U # DI, and M # X. A pair hI, Mi is called an extended intuitionistic fuzzy soft set over DI. Clearly, IjM is a mapping given by IjM : M ! IF ðDI Þ, i.e., "C 2 M, CI can be written as an intuitionistic fuzzy set over DI such P P that C I ¼ x2D IC I ðxÞ=x ¼ x2D IhlIC ðxÞ; cIC ðxÞi=x, where lIC ðxÞ 2 ½0; 1 is called the ‘‘degree of membership of the element x 2 DI in CI’’, cIC ðxÞ 2 ½0; 1 is called the ‘‘degree of non-membership of the element x 2 DI in CI’’, and lIC ðxÞ and cIC ðxÞ satisfy the following condition: 8x 2 DI ; 0 6 lIC ðxÞ þ cIC ðxÞ 6 1. For example, regarding the decision parameter (i.e., IFDL-concept) expensive, we have the following expression:

In this paper, we extend fuzzy soft sets with fuzzy DLs, in other words, we present an extended fuzzy soft set theory by using the concepts of fuzzy DLs to act as the parameters of fuzzy soft sets. Furthermore, we define equality of two extended fuzzy soft sets, fuzzy subset and fuzzy superset of an extended fuzzy soft set, complement and restricted difference of an extended fuzzy soft set, and define some operations such as the extended union, the restricted intersection, the restricted union, the extended intersection, the T operator and the S operator of two extended fuzzy soft sets. Moreover, we prove that certain De Morgan’s laws hold in extended fuzzy soft set theory with respect to these operations. In fact, the extended fuzzy soft set theory presented in this paper is a fuzzy extension of the extended soft set theory presented in [23]. As far as future directions are concerned, these will include extending vague soft sets, interval-valued fuzzy soft sets and intuitionistic fuzzy soft sets with vague DLs, interval-valued fuzzy DLs and intuitionistic fuzzy DLs, respectively. It is also desirable to further explore the applications of using the extended fuzzy soft set approach to solve real world problems such as data mining, forecasting, and data analysis.

I

expensiv e ¼ h0:9; 0:1i=h1 þ h0:7; 0:2i=h2 þ h0:6; 0:3i=h3

Acknowledgements

þ h0:7; 0:3i=h4 þ h0:5; 0:4i=h5 þ h0:5; 0:5i=h6 : Similarly with the approach that construct an extended fuzzy soft set (see Section 3.1 for more details), we can construct an extended intuitionistic fuzzy soft set using intuitionistic fuzzy DL reasoning based on an intuitionistic fuzzy DL knowledge base. We also need to redefine the operations in the extended intuitionistic fuzzy soft set theory. In what follows, we only define the notions of the intuitionistic fuzzy soft subset and the intuitionistic fuzzy soft superset. Regarding other operations (such as the intersection, the union, and the complement) and their properties in the extended intuitionistic fuzzy soft sets, we will study them in detail in future. Given two sets of decision parameters (i.e., IFDL-concepts) M1 = {C1, C2, . . . , Cm} and M2 = {D1, D2, . . . , Dn}, where Ci (1 6 i 6 m) and Dj (1 6 j 6 n) are IFDL-concepts in KB ¼ hTB; ABi. If "Ci 2 M1, there exists Dj 2 M2 satisfies: for any model I = (DI, I) of KB; 8x 2 DI ; C Ii ðxÞ ¼ DIj ðxÞ, i.e., lICi ðxÞ ¼ lIDj ðxÞ; cICi ðxÞ ¼ cIDj ðxÞ (denoted by Ci  Dj), then we say that M1 is a logical subset of M2 (denoted by M1 # L M2). For two extended intuitionistic fuzzy soft sets hI, Mi and hJ, Ni over a common domain D (i.e., D = DI = DJ) and a common intuitionistic fuzzy knowledge base KB, we say that hI, Mi is an intuitionistic fuzzy soft subset of hJ, Ni if (i) M # L N, and (ii) "C 2 M, CI = CJ, i.e., x 2 D.

lIC ðxÞ ¼ lJC ðxÞ and cIC ðxÞ ¼ cJC ðxÞ for any

hI, Mi is said to an intuitionistic fuzzy soft superset of hJ, Ni, if hJ, Ni is an intuitionistic fuzzy soft subset of hI, Mi. Two extended intuitionistic fuzzy soft sets hI, Mi and hJ, Ni over a common domain D and a common intuitionistic fuzzy knowledge base KB are said to be intuitionistic fuzzy soft equal if hI, Mi is an intuitionistic fuzzy soft subset of hJ, Ni and hJ, Ni is an intuitionistic fuzzy soft subset of hI, Mi. The class of all intuitionistic fuzzy value sets of extended intuitionistic fuzzy soft set hI, Mi is called intuitionistic fuzzy value-class of the extended intuitionistic fuzzy soft set and is denoted

The authors would like to thank the anonymous referees for their valuable comments as well as helpful suggestions from Professor Jie Lu, Editor-in-Chief which helped in improving this paper significantly. The works described in this paper are supported by The National Natural Science Foundation of China under Grant Nos. 60663001, 60673135, 60970044 and 60736020; The Foundation of the State Key Laboratory of Computer Science of Chinese Academy of Sciences under Grant No. SYSKF0904; The Natural Science Foundation of Guangdong Province of China under Grant No. 10151063101000031; The Natural Science Foundation of Guangxi Province of China under Grant Nos. 0991100; and The Science Research and Technology Development Project of Guangdong Province of China under Grant Nos. 2009B090300326 and 2009B010800036. References [1] U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Computers & Mathematics with Applications 59 (11) (2010) 3458–3463. [2] H. Aktas, N. Cagman, Soft sets and soft groups, Information Sciences 177 (13) (2007) 2726–2735. [3] M.I. Ali, F. Feng, X. Liu, W.K. Min, M. Shabir, On some new operations in soft set theory, Computers & Mathematics with Applications 57 (9) (2009) 1547–1553. [4] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1) (1986) 87–96. [5] K. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg/New York, 1999. [6] A. Aygunoglu, H. Aygun, Introduction to fuzzy soft groups, Computers & Mathematics with Applications 58 (6) (2009) 1279–1286. [7] F. Baader, D. Calvanese, D. McGuinness, D. Nardi, P. Patel-Schneider, The Description Logic Handbook: Theory, Implementation and Applications, second ed., Cambridge University Press, 2007. [8] F. Bobillo, M. Delgado, J. Gomez-Romero, Crisp representations and reasoning for fuzzy ontologies, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 17 (4) (2009) 500–530. [9] F. Bobillo, M. Delgado, J. Gomez-Romero, DeLorean: A reasoner for fuzzy OWL 1.1, in: Proceedings of the 4th International Workshop on Uncertainty Reasoning for the Semantic Web (URSW 2008), CEUR Workshop Proceedings, vol. 423, 2008. [10] F. Bobillo, M. Delgado, J. Gomez-Romero, U. Straccia, Fuzzy description logics under Gödel semantics, International Journal of Approximate Reasoning 50 (3) (2009) 494–514. [11] F. Bobillo, U. Straccia, Fuzzy description logics with general t-norms and datatypes, Fuzzy Sets and Systems 160 (23) (2009) 3382–3402.

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