Consolidation Operator for Intuitionistic Fuzzy SetPremchand S. Nair

Department of Computer ScienceCreighton UniversityOmaha, NE 68178psnair@creighton.edu

Abstract—Intuitionistic Fuzzy Set (IFS) is the first majorattempt to capture both membership as well as the non-membership in a comprehensive way. The currently availableoperators of IFS, at the very least, cannot be used for of knowl-edge consolidation. In this paper the consolidation operator forIFS has been introduced.Key words: Intuitionistic Fuzzy Set, Data Mining, Data

Fusion, Decision Making Application, Fuzzy Math Modeling,Soft Computing, Database Systems.

I. INTRODUCTION

Fuzzy set was introduced by Zadeh as a generalization ofBoolean logic. In the field of image processing or computergraphics, a pixel is better represented using values in a rangerather than just 0 and 1. For example, if we consider a blackand white picture, each pixel can be better represented usinga value between 0 and 1 to fully caputure the gray level ofthe pixel. In the case of color images, if we are using RBGcolor scheme, each color has three components: Red, Blue andGreen. Further, each color component is represented using avalue in the range 0 and 1. Thus, fuzzy set theory effectivelymodeled the real life situations in the case of image processingand computer graphics. From a set theoritical point of view,fuzzy set generalized the concept of membership from either 0or 1 to any value in the range 0 to 1. In due course, it becamequite obvious to many researchers that there must be someway to represent the negative set membership and the fuzzy settheory is not capable of handling those situations. In our dailylife, there are many situations where set membership is bothpositive and negative. In physics, the fundamental particles haselectric charges that may be positive or negative. In a politicalcontext, every issue has its supporting lobbies, neutral lobbiesand opposing lobbies. In the field of information processing, adata item may have both positive and negative membership ifthe data is collected through multiple sources. The confidenceindex set [5,6,7] is a natural extension of both set theory andfuzzy set theory to accommodate both positive and neagtivemembership values. Histrorically, Intuitionistic Fuzzy Set (IFS)[1] is the first major attempt to capture both membership as wellas the non-membership in a comprehensive way and ciset is ageneralization of IFS. Therefore, for the rest of this paper weshall be dealing with ciset and any statement we make aboutciset is equally applicable to IFS.

The confidence index and confidence index set (ciset) can beused to extend any classical theory. In particular, ciset can beapplied to extend relational database theory to store and processconflicting pieces of information. This extension is quite pow-erful and its usefulness can be explained as follows. Let ussay one of the CIA members has informed the headquartersthat Facility X in country ABC is used to produce biologicalweapons and the agent has 60% confidence. Later on, anothersource has informed the CIA headquarters that the facility is notused for the production of biological weapons and the sourcehas 30% confidence. Present day relational model cannot treatthese two facts in an integrated way. In a relational databasesystem they remain as two independent tuples. Ciset relationaldatabase can integrate the supporting and the opposing factsbased on the subject matter. In a ciset relational databasethere will be only one tuple with a confidence index (0.3,0.6) assigned to it, where 0.3 indicates the degree of opposingevidence and 0.6 indicate the degree of supporting evidence.In [3], the consolidation operator for fuzzy sets have been

shown to be a better operator than the traditional union operatorto merge information. In this paper, we show the limitationsof the existing ciset operators to model real life situations. Weintroduce a consolidation operator for cisets and use it in a cisetrelational database. First we review some of the relevant workon cisets and explore the desirable properties a consolidationoperator must have. Then we introduce such an operator forthe ciset. The usefulness the ciset consolidation operator isdemonstrated by applying it in a ciset relational database.Through out this paper, we denote the complete lattice [0, 1]

under the partial order less than or equal to (≤) as I. Further,we shall use ∨ and ∧ to represent the maximum and minimumoperations on real numbers.

II. CISET AND CISET RELATIONAL DATABASEA. Confidence IndexDefinition 1: Let α, β ∈ L. Then a pair a = hα, βi is called

a confidence index.Definition 2: Let a = hα, βi be a confidence index. Then

l(a), the lower index of a, is α and u(a), the upper index ofa, is β.Two confidence indexes ai = hαi, βii, i = 1, 2 are equal

if and only if l(a1) = l(a2) and u(a1) = u(a2). We nowproceed to define ≺,¹, and º . Confidence indexes a1 ≺ a2,if l(a1) ≥ l(a2) and u(a1) < u(a2) or l(a1) > l(a2) and

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The 28th North American Fuzzy Information Processing Society Annual Conference (NAFIPS2009)Cincinnati, Ohio, USA - June 14 - 17, 2009

u(a1) ≤ u(a2). Further, a1 ¹ a2 if and only if either a1 ≺ a2or a1 = a2. Note that a1 Â a2 if and only if a2 ≺ a1 anda1 º a2 if and only if a2 ¹ a1.There are four operations on the set of all confidence

indexes[4,5,6]. They are three binary operations union (∪), in-tersection (∩), difference (−); and one unary operation negation(−). Let a1 = hα1, β1i, a2 = hα2, β2i be any two confidenceindexes. Then

a1 ∪ a2 = hα1 ∧ α2, β1 ∨ β2i,

a1 ∩ a2 = hα1 ∨ α2, β1 ∧ β2i,

−a1 = hβ1, α1i,and

a1 − a2 = a1 ∩ (−a2).

We use the notation C to denote the set of all confidenceindexes.

B. Confidence Index SetLet S be a set. A confidence index set or ciset [5,6,7] is a

mapping F : S → C. One can think of F as assigning to eachelement x ∈ S, two degrees of confidence α and β such thatα denotes the degree of confidence one has that x ∈ Sc, thecomplement of S; and β denote the degree of confidence onehas that x ∈ S.Let S = {u, v, x, y, z}. Define F : S → C as follows:

F (u) = h.4, .8i, F (v) = h0, 1i, F (x) = h1, 0i, F (y) =h0, 0i, F (z) = h1, 1i. Note that it is possible for an elementbe in Sc and S with same level of confidence and the sum ofthe levels of confidence of an element w defined as l(F (w))+u(F (w)) need not be 1 or less than or equal to 1 (as in IFS);rather it can be any value between 0 (as in y) and 2 (as inz). At first, it may strike as a contradiction. Since ciset isnot asserting the membership of an element in Sc or S, butrather asserting the confidence levels obtained through varioussources, there is no contradiction. For example, S may be aset of faculty members and F may be the performance of thefaculty, as evaluated by their students. It is quite possible thatsome students may find a certain faculty excellent while otherstudents may find the same faculty the worst teacher they everhad. Similarly, Internet auction sites maintain a rating systemfor its patrons. The same person is rated both positively as wellas negatively by different patrons.We say two cisets F and G on a set S are equal, and write

F = G, if F (x) = G(x) for all x ∈ S.Definition 3: Let F and G be two cisets on a set S such that

F (x) ¹ G(x) for all x ∈ S, then F is said to be subset of Gand G is said to be a superset of F. If F is a subset of G andthere exists at least one x ∈ S such that F (x) ≺ G(x) then Fis said to be proper subset of G and G is said to be a propersuperset of F.

There are five operations on cisets [5,6,7]. They are union,intersection, difference, product and complement. These oper-ations allow us to construct new cisets from given cisets. Forexample, operations union, Cartesian product and complementare defined as follows:Definition 4: Let S be a set and let F,G be two cisets on S.

The union of F and G, denoted by F ∪G is a mapping fromS → C, defined by (F ∪G)(x) = F (x)∪G(x), for all x ∈ S.Definition 5: Let S be a set and let F,G be two cisets on

S. The Cartesian product of F and G, denoted by F × G isa mapping from S × S → C, defined by (F × G)(x, y) =F (x) ∩G(y), for all (x, y) ∈ S × S.Definition 6: Let S be a set and let F be a ciset on S. The

complement of F , denoted by −F is a mapping from S → C,defined by (−F )(x) = −F (x).A ciset can be considered a generalization of set. Let S be a

set and A be any subset of S. Define a mapping FA : S → C byFA(x) = h0, 1i if x ∈ A and FA(x) = h1, 0i if x /∈ A. Thusa ciset can be considered a generalization of set. Similarly,if µ : S → [0, 1] is a fuzzy subset of S, define a mappingFµ : S → C by Fµ(x) = h1− µ(x), µ(x)i, for all x ∈ S. Leta = (s, t) be an ordered pair of real numbers. Define a–cut set,F ts , by F t

s = {x ∈ S | u(F (x)) ≥ t and l(F (x)) < s}. We usethe symbol F t to denote the set {x ∈ S | u(F (x)) ≥ t}.

C. Ciset Relational Database Model

The ciset relational model allows us to organize data centeredon the notion of a ciset relation as the data structure. A cisetrelational model is a blueprint of the database that can storeconflicting information. In a ciset relational model, the dataalways is presented in the form of a table, which we call aciset relation. For example, data on all professors of a universitycan be organized in the form of a table FACULTY as shownin Table I.

FACULTY

FID F_NAME DEPT EVAL123 John Sim Marketing h0.5, 0.7i318 Mary Lee Mathematics h0.4, 0.9i126 Sam Dew Marketing h0.1, 0.8i567 Bea Cox Accounting h0.7, 0.6i

TABLE ICISET RELATION: FACULTY

The name of the table is FACULTY. The column headingsof table are known as ciset relational attributes. Each row ofthe table is known as a ciset tuple. From the user perspective,database model is still the classical relational model. The onlyconceptual difference as far as the user is concerned can besummarized as follows.1) As user inserts a new piece of data, user is asked toprovide the confidence level. Similar is the situation whenthe user tries to delete a piece of information or modifya piece of information.

The 28th North American Fuzzy Information Processing Society Annual Conference (NAFIPS2009)Cincinnati, Ohio, USA - June 14 - 17, 2009

2) In addition to producing results of a query user can usethe confidence index attribute to obtain the level of “trust”one can place on the result itself.

D. Formalization of Ciset Relation

We now proceed to formalize the notion of a table. Let U bethe set of all ciset relational attributes. For each attributeA ∈ U ,letDOM(A), called domain of A, denote the set of all possiblevalues that can occur in that column. The domains are arbitrary,nonempty sets, nonempty fuzzy sets or nonempty cisets ornonempty subset of confidence indexes, finite or countablyinfinite. Note that if there is only one element in DOM(A),we need not include A as an attribute of the table. It is safe toassume that DOM(A) contains at least two different values.Let R = {A1, . . . , An} be a finite set of ciset relational

attributes. Then R is called a ciset relational scheme. A cisetrelation r on a ciset relational scheme R is a finite set of map-pings {t1, . . . , tm} from R to ∪{DOM(Ai) | i = 1, . . . , n.}with the restriction that t(Ai) ∈ DOM(Ai), i = 1, . . . , n forall t ∈ {t1, . . . , tm}. The mappings are called ciset tuples.Depending on the complexity of DOM(Ai), i = 1, . . . , n,

ciset relations can be classified into many categories, namely,Type 0, Type 1, Type 2, Type 3 and so on.• In Type 0 ciset relation, DOM(Ai), i = 1, . . . , n, is a set.Thus in addition to the commonly used data types such asnumbers, strings, Boolean values and dates, a Type 0 cisetrelation can accommodate confidence index. Therefore,DOM(Ai) can be a subset of C.

• In Type 1 ciset relation, DOM(Ai), i = 1, . . . , n, isa ciset. Note that ciset is a generalization of sets andfuzzy sets. Thus in the case of a Type 1 ciset relation,DOM(Ai), i = 1, . . . , n, is a set, or a fuzzy set or aciset.

• In Type 2 ciset relation, DOM(Ai), i = 1, . . . , n, is a setof subsets of a ciset. In the case of a Type 2 ciset relation,DOM(Ai), i = 1, . . . , n, is a set of subsets of a set, or aset of subsets a fuzzy set or a set of subsets a ciset.

• In Type 3 ciset relation, DOM(Ai), i = 1, . . . , n, is aciset of subsets of a ciset. Thus in the case of a Type 3ciset relation, DOM(Ai), i = 1, . . . , n, is a set or a fuzzyset or a ciset of subsets of sets; or set or a fuzzy set ora ciset of subsets fuzzy sets; or a set or a fuzzy set or aciset of subsets of cisets.

• In Type 2j ciset relation, DOM(Ai), i = 1, . . . , n, is aset of subsets of Type (2j − 1) domain, j > 1.

• In Type 2j+1 ciset relation, DOM(Ai), i = 1, . . . , n, isa ciset of subsets of Type (2j − 1) domain, j > 1.

The classification can be summarized as shown in Table II.From a practical point of view, it is quite easy to implement

a Type 0 ciset relation as opposed to Type 1, Type 2 or Typei (i > 2) ciset relations. So, we concentrate on Type 0 cisetrelational database for the rest of this paper.Ciset relational operations are presented in [5, 7] and the

major result can be summarized as follows.

Type DOM(Ai)0 set1 ciset2 set of subsets of a ciset3 ciset of subsets of a ciset...

...2j set of subsets of Type (2j − 1), j > 1.

2j + 1 ciset of subsets of Type (2j − 1), j > 1

TABLE IICLASSIFICATION OF CISET RELATIONS

Theorem 7: Let REL_R,REL_S any two relations on re-lational schemes R and S respectively. Let ⊗ be any binaryrelational operator and ⊕ be any unary relational operator. Thenthe following diagrams commute.

(REL_R,REL_S) F→ (FREL_R, FREL_S)⏐⏐⏐⏐y⊗⏐⏐⏐⏐y⊗

REL_R⊗REL_SFÀ( )1

FREL_R ⊗ FREL_S

REL_R F→ FREL_R⏐⏐⏐⏐y⊕⏐⏐⏐⏐y⊕

⊕REL_R FÀ( )1

⊕FREL_R

E. Semantics of a ciset relationConsider the following Type 0 ciset relation given in Table

III.

FACULTY

FID F_NAME DEPT EVAL CI123 John Sim Mkt. h0.5, 0.7i h0, 1i318 Mary Lee Math. h0.4, 0.9i h0.2, 0.9i126 Sam Dew Mkt. h0.1, 0.8i h0.3, 0.7i567 Bea Cox Acc. h0.7, 0.6i h0.2, 0.9i

TABLE IIICISET RELATION: FACULTY

The semantics of the first tuple is “John Sim is a facultymember of Marketing department and his faculty identificationnumber is 123. John Sim’s student evaluation is h0.5, 0.7i.The fact that John Sim is a faculty member of Marketingdepartment and his faculty identification number is 123 hasa confidence index value h0, 1i.” Note that there is a subtledifference between the ways we interpreted attributes EVALand CI. The attribute EVAL is in all respects, just like anyother attribute and asserts a fact about John Sim. On the other

The 28th North American Fuzzy Information Processing Society Annual Conference (NAFIPS2009)Cincinnati, Ohio, USA - June 14 - 17, 2009

hand, the attribute CI is not asserting a fact about John Sim.Instead, CI asserts a fact about the validity of the tuple itselfand assigns a confidence index for the tuple. Thus CI plays avery different role compared to all other attributes. The attributeCI is a tuple attribute that qualifies the tuple and all otherattributes are entity attributes.Let R = {A1, . . . , An} be a ciset relational scheme. Then

there can be at most one attribute that measures the validity ofthe tuple itself. This being a special attribute, for the rest of thispaper, we shall refer it as CI and we call it a tuple attribute. Theset E(R) = {Ai | Ai 6= CI} is called the set of entity attributesof R and the set R(R) = {Ai | DOM(Ai)∩C = ∅} is calledthe set of relational attributes of R.Given a ciset relation r on a ciset relational scheme R,

let E(r) and R(r) denote ciset relations on E(R) and R(R)respectively. Note thatR(r) degenerates to a traditional relationsince all attributes of R(R) are relational attributes. Let t be atuple of r. Then we use the notation t(E) and t(R) to denote thecorresponding tuples in E(R) and R(R) respectively. Further,by abusing the notation, we say t = (t(E), t(CI)).In Table 3, the ciset relational scheme is

FACULTY _SCHEME ={FID,F_NAME,DEPT,EV AL,CI}.The tuple attribute is CI, the entity attributes areE(FACULTY _SCHEME)= {FID,F_NAME,DEPT,EV AL}and the relational attributes areR(FACULTY _SCHEME) ={FID,F_NAME,DEPT}.Further,E(FACULTY)FID F_NAME DEPT EVAL123 John Sim Marketing h0.5, 0.7i318 Mary Lee Mathematics h0.4, 0.9i126 Sam Dew Marketing h0.1, 0.8i567 Bea Cox Accounting h0.7, 0.6i

and

R(FACULTY)FID F_NAME DEPT123 John Sim Marketing318 Mary Lee Mathematics126 Sam Dew Marketing567 Bea Cox Accounting

.

Let t = (123, John Sim,Marketing, h0.5, 0.7i, h0, 1i).Thent(E) = (123, John Sim,Marketing, h0.5, 0.7i)andt(R) = (123, John Sim,Marketing).Further,t = (123, John Sim,Marketing, h0.5, 0.7i, h0, 1i) =(t(E), t(CI)) where t(CI) = h0, 1i.Thus the tuple t can also be considered as a ciset

on E(FACULTY ) or equivalently, a mapping from

E(FACULTY ) in to C.

III. CONSOLIDATION OPERATOR FOR CISETS

A. Motivating Example

Currently available ciset operators can not be used to mergedata from multiple sources in a consistent manner. This factcan be explained through the following example. Assume thatfour different objects O1, O2, O3, O4 are being evaluated bytwo experts for a certain specific property, say P . Experts areasked to give a confidence index value, instead of just a number.Let P_EXPERT_1 denote the table corresponding to expert 1and P_EXPERT_2 denote the table corresponding to expert 2respectively.P_EXPERT_1

OBJECT_ID EVALO1 h0.6, 0.8iO2 h0.2, 0.5iO3 h0, 0.8iO4 h0.8, 0i

P_EXPERT_2OBJECT_ID EVAL

O1 h0.4, 0.7iO2 h0.1, 1iO3 h0.3, 0.6iO4 h0.9, 0.2i

If we use the traditional union operator to consolidate bothexpert opinions, we have the following:P_TABLEOBJECT_ID EVAL

O1 h0.4, 0.8iO2 h0.1, 1iO3 h0, 0.8iO4 h0.8, 0.2i

Now suppose that the same four different objects O1, O2,O3, O4 were in fact being evaluated by the same two expertsfor the opposite property of P , P . In that case, the tables wouldhave looked as follows:PBAR_EXPERT_1

OBJECT_ID EVALO1 h0.8, 0.6iO2 h0.5, 0.2iO3 h0.8, 0iO4 h0, 0.8i

PBAR_EXPERT_2OBJECT_ID EVAL

O1 h0.7, 0.4iO2 h1, 0.1iO3 h0.6, 0.3iO4 h0.2, 0.9i

If we use the traditional union operator to consolidate bothexpert opinions, we would have the following:PBAR_TABLE

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OBJECT_ID EVALO1 h0.7, 0.6iO2 h0.5, 0.2iO3 h0.6, 0.3iO4 h0, 0.9i

If hα, βi is the evaluation of the object O in P_TABLE, inorder for the system to be consistent, we would have expectedhβ, αi as the evaluation of the same object O in PBAR_TABLEand vice-versa. However, that is not the case. Thus, from thesame data, we arrive at two different conclusions dependingupon whether we consider P or P the property we are interestedin. Such a situation is not acceptable under any circumstances.In other words, the consolidation operator must be "symmetric".That is, if a1, a2 are two confidence index values and ⊗ is theconsolidation operator, then we must have a1⊗a2 =−((−a1)⊗(−a2)). We shall call this the symmetric property.It is easy to see that combining two confidence indexes ai =

hαi, βii, i = 1, 2 must produce a new confidence index a1⊗a2,with following properties:1) [Closure]If a1, a2 ∈ C then a1 ⊗ a2 ∈ C.2) [Commutativity]If a1, a2 ∈ C then a1 ⊗ a2 = a2 ⊗ a1.3) [Associativity]If a1, a2, a3 ∈ C then (a1 ⊗ a2) ⊗ a3 =

a1 ⊗ (a2 ⊗ a3).4) [Incremental Effect]Let a1, a2 ∈ C. If 0 <

l(a1), l(a2) < 1 then l(a1), l(a2) < l(a1⊗a2). Similarly,if 0 < u(a1), u(a2) < 1 then u(a1), u(a2) < u(a1⊗a2).

5) [Zero Gain]h0, 0i⊗ a1 = a1.6) [Solid Evidence]Let a1, a2 ∈ C. If l(a1 ⊗ a2) = 1 then

l(a1) or l(a2) = 1. Similarly, If u(a1 ⊗ a2) = 1 thenu(a1) or u(a2) = 1.

7) [Monotone]Let a1, a2, a3 ∈ C. If l(a2) ≤ l(a3) thenl(a1⊗a2) ≤ l(a1⊗a3). Similarly, if u(a2) ≤ u(a3) thenu(a1 ⊗ a2) ≤ u(a1 ⊗ a3).

8) [Symmetry] Let a1, a2 ∈ C. Then a1⊗a2 = −((−a1)⊗(−a2)).

The first property states that the consolidation operation musthave closure property. Thus no justification is required. Astwo pieces of information is received their order should notplay on the final outcome. In other words, commutativity isquite essential. The third property will ensure the consistencyof our model. Without associativity, people may arrive atdifferent results while the underlying pieces of informationis the same. If both the values are between 0 and 1, theconsolidation operator must produce a result that is strictlygreater than both of them. If one of the values is zero, thereshould not be any gain and that is the reason for property 5.Finally, property 6 states that sold evidence or value 1 can beproduced only if one of the values is 1. Once the value 1 isattained, no more incremental effect is possible. Property 7 isquite straightforward. There must be symmetry in dealing withsupporting and opposing facts. Observe that if x is a statementwith confidence index ai = hαi, βii then the complementstatement x0 of x has confidence index hβi, αii. Therefore,whether the system keeps the fact x or x0 must not affect theresult as shown in the motivating example. In other words,

symmetry is essential.One possible consolidation operator is presented next.Definition 8: If ai = hαi, βii ∈ C, i = 1, 2 then a1 ⊗ a2

defined by hα1 + α2 − α1α2, β1 + β2 − β1β2i.Theorem 9: The binary operator ⊗ satisfies all the desired

properties of a consolidation operator. In other words, ⊗ is apossible consolidation operator.

IV. CONCLUSIONIn this paper, we advocated the need for a consolidation

operator for IFS. The consolidation operator introduced in thispaper has certain unique properties such as incremental effect,zero gain, solid evidence, monotone and symmetry besidesmost common properties such as closure, commutativity andassociativity. The consolidation operator is shown to be thebetter choice than traditional union value operator in dealingwith information acquired from multiple sources.

V. REFERENCES1) Atanassov, K., Intuitionistic fuzzy sets. Fuzzy Sets andSystems 20 (1986) 87–96

2) Klir, G.J. and Yuan, B., Fuzzy Sets and Fuzzy Logic:Theory and Applications, Prentice Hall, Upper SaddleRiver, N.J. 1995.

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