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2 Interval-Valued Atanassov's Intuitionistic Fuzzy Sets

# 2 Interval-Valued Atanassov's Intuitionistic Fuzzy Sets

Tải bản đầy đủ - 0trang

Interval Version of Generalized Atanassov’s Intuitionistic Fuzzy Index

221

˜ = NI (0, 1) = 1

˜ and NI (1)

˜ = NI (1, 0) = 0;

˜

NI 1: NI (0)

˜ ≥ ˜ Y˜ then NI (˜

x

)

N

y

).

NI 2: If X

˜ I

U

U

Moreover, NI is a strong IvIFN if it also veriﬁes the involutive property:

˜

˜ = X,

˜ ∀X

˜ ∈ U.

NI 3: NI (NI (X))

˜ n → U.

˜ By [18], the NI -dual interval-valued

Consider NI as IvIFN and FI : U

˜ n → U,

˜ is given by:

Atanassov’s intuitionistic function of f˜, denoted by FI NI : U

˜ n.

˜ = NI (FI (NI (X

˜ = (X

˜ 1 ), . . . , NI (X

˜ n ))), ∀X

˜1, . . . , X

˜n) ∈ U

FI NI (X)

(9)

˜ I is a strong IvIFN, f˜ is a self-dual interval-valued intuitionistic function.

When N

˜ →U

˜ such that

And, by [18], taking a strong IvFN N : U → U, a IvIFN NSI : U

˜ = (N(NS (X2 )), NS (N(X1 ))),

NSI (X)

(10)

is a strong IvIFN generated by the IvFNs N and NS . By [6]), a strong IvIFN is

also a representable IvIFN. Additionally, if N = NS , Eq. (10) can be reduced to

˜ = (X2 , X1 ). Moreover, the complement of A-IvIFS AI is deﬁned by

NS I (X)

AI c ={(x, NAI (x), MAI (x)) : x∈χ, MAI (x) + NAI (x))≤U 1},

(11)

An interval-valued Atanassov’s intuitionistic automorphism (A-IvIA) is a

˜ → U.

˜ For all X,

˜ the following hold:

˜ Y˜ ∈ U,

bijection increasing operator Φ : U

AI 1:

AI 2:

AI 3:

˜ =1

˜ and Φ(0)

˜ = 0;

˜

Φ(1)

−1 ˜

˜

Φ ◦ Φ (X) = X;

˜ ≤ ˜ Φ(Y˜ ).

˜ ≤ ˜ Y˜ iﬀ Φ(X)

X

U

U

˜n → U

˜ is

˜

In the set of all A-IvIAs (Aut(U)),

the conjugate function of fI : U

Φ ˜n

˜

a function fI : U → U, deﬁned as follows

˜1, . . . , X

˜ n ) = Φ−1 (fI (Φ(X

˜ 1 ), . . . , Φ(X

˜ n ))).

fIΦ (X

(12)

Reporting main results in [9, Theorem 17], let φ : U → U be an intervalvalued automorphism, φ ∈ Aut(U). Then, a φ-representability of Φ is given by

˜

˜ = (φ(l ˜ (X)),

˜ 1 − φ(1 − r ˜ (X))),

˜

˜ ∈ U;

Φ(X)

∀X

U

U

(13)

˜ a φU -representability of Φ is given by

˜ ∈ U,

Moreover, if φ ∈ Aut(U ), for all X

˜ = [φU (X ), φU (X 1 )], [1 − φU (1 − X ), 1 − φU (1 − X 2 )] .

Φ(X)

1

2

(14)

Thus, if an IvIA is φ-representable, it is also a φU -representable automorphism

[18].

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L. Costa et al.

3

Interval Extension of the Generalized Atanassov’s

Intuitionistic Fuzzy Index

˜ → U is called a genDeﬁnition 3. Let N be a strong IvFN. A function Π : U

eralized interval-valued intuitionistic fuzzy index (A − GIvIF Ix(N)) if,

for all X1 , X2 , Y1 , Y2 ∈ U, it holds that:

Π1:

Π2:

Π3:

Π4:

3.1

Π(X1 , X2 ) = 1 iﬀ X1 = X2 = 0;

Π(X1 , X2 ) = 0 iﬀ X1 + X2 = 1;

If (Y1 , Y2 ) U˜ (X1 , X2 ) then Π(X1 , X2 ) ≤U Π(Y1 , Y2 );

Π(X1 , X2 ) = Π(NSI (X1 , X2 )) when NSI is given by Eq.(10).

Relationship with Interval-Valued Fuzzy Connnectivess

In the following, Theorem 1 extends main results in [2].

Theorem 1. Let I(J) : U2 → U be a (co)implicator verifying I1(J2),I4(J4),I5(J5)

˜ →U

and I6(J6) and N : U → U be an involutive IvFN. A function ΠN,I (ΠN,J ) : U

is A-GIvIFIx(N) iﬀ it can be given as

ΠN,I (X) = N(I(NS (X2 ), X1 ))

(ΠN,J (X) = J(N(NS (X2 )), N(X1 ))) . (15)

Proof. Equation(15b) is proved below. Analogously, it can be done to Eq.(15a).

(⇒) Consider that J : U2 → U veriﬁes J2, J4, J5 and J6, it holds that:

Π1 : ΠN,J (X1 , X2 ) = 1 ⇔ J(N(NS (X2 )), N(X1 )) = 1(by Eq.(15b))

⇔ NS (X2 ) = 1 and N(X1 ) = 1 ⇔ X2 = X1 = 0(by J6, N1).

Π2 : ΠN,J (X1 , X2 ) = 0 ⇔ J(N(NS (X2 )), N(X1 )) = 0 (by Eq.(15b))

⇔ N(NS (X2 )) ≥U N(X1 )(by J4)

⇔ NS (X2 ) ≤U X1 (by N3) and NS (X2 ) ≥U X1 (by Eq.(7))

⇔ X1 + X2 = 1.

Π3 : (Y1 , Y2 )

(X1 , X2 ) ⇒ Y1 ≤U X1 and Y2 ≤ X2 (by RI 2)

⇒ N(X1 ) ≥U N(Y1 ) and N(NS (X2 )) ≤U N(NS (Y2 ))(by N2)

⇒ J(N(NS (X2 )), N(X1 )) ≤U J(N(NS (Y2 )), N(Y1 ))(by J1, J2)

⇒ ΠN,J (X1 , X2 ) ≤U ΠN,J (Y1 , Y2 )(by Eq.(15))

Π4 : ΠN,I (N(X1 , X2 )) = J(N(NS (X2 )), NS (N(X1 ))(by Eq.(10))

= J(X1 , NS (X2 ))(by Eq.(15))

= J(N(NS (X2 ))), N(X1 )) = ΠN,J (X1 , X2 )(by J5 and Eq.(15))

(⇐) Considering the function J : U2 → U given as J(X1 , X2 ) = 1, if X1 > X2 ;

and J(X1 , X2 ) = ΠN,J (X2 , NS (N(X1 ))), otherwise. The following holds:

Interval Version of Generalized Atanassov’s Intuitionistic Fuzzy Index

J2 :Y1 ≥ Y2 ⇔ J(X, Y1 ) =

223

1, if X > Y1 ,

ΠN,J (Y1 , NS (N(X))), otherwise; (by Eq.(15); )

1, if X > Y2 ,

ΠN,J (Y2 , NS (N(X))) = J(X, Y2 ), otherwise; (by Π 3 andEq.(15))

J4 : Straightforward.

J5 : J(N(X2 ), N(X1 )) =

=

1, if N(X2 ) > N(X1 ),

ΠN,J (N(NS (N(X1 )), X2 ), otherwise; (by Eqs.(15) and (10))

1, if X1 ≥ X2 ,

Π N,J (N(NS (X2 ), N(X1 )), otherwise (by Π4 and N3)

= J(X1 , X2 ), (Eq.(15))

J6 : J(X1 , X2 ) = 1 ⇔ΠN,J (N(X2 ), NS N(X1 )) = 1 (by Eq.(15))

⇔N(X2 ) = NS (N(X1 )) = 0 ⇔ X1 = 0 and X2 = 1 (by Π1 )

Therefore, Theorem 1 holds.

The Φ-representability and N-dual IvIFIx constructions are discussed below.

Proposition 2. Let IN (JN ) be the N-dual operator of a (co)implication I(J). The

following holds:

˜ = ΠN,I (X)

˜

ΠN,IN (X)

˜ = ΠN,J (X)

˜ .

ΠN,JN (X)

(16)

˜ = IN (N(NS (X2 )), N(X1 )) = N(I(NS (X2 ), X1 )) = ΠN,I (X),

˜ ∀X

˜∈

Proof. ΠN,IN (X)

˜

U.

Corollary 1. When N = NS , Eq.(15) in Theorem 1 is given as

˜ = NS (I(NS (X2 ), X1 ))

ΠNS ,I (X)

˜ = J(X2 , NS (X1 ) .

ΠNS ,J (X)

(17)

˜ → U be AProposition 3. Let N be an N -representable IvFN and πN,I : U

IFIx(N). If I, J are representable (co)implications given by Eq.(3), a function

˜ → U given by Eq.(17) can be expressed as

ΠN,I : U

˜

˜

ΠN,I (X)=[Π

N,I (X 2 , X 1 ),ΠN,I (X 2 , X 1 )] (ΠN,J (X)=[ΠN,J (X 2 , X 1 ),ΠN,J (X 2 , X 1 )]).

Proof. We proof Eq.(18a), the other one can be analogously done. By taking

˜ X1 = [X , X 1 ], X2 = [X , X 2 ] then X1 + X2 = [X + X , X 1 +

(X1 , X2 ) ∈ U,

1

2

1

2

X 2 ] ≤ 1, meaning that X 1 +X 2 ≤ 1 and X 1 +X 2 ≤ 1. Then, we have the result

˜ = N(I([1−X 2 , 1−X ], [X , X 1 ])) = [N (I(1−X 2 , X 1 )), N (I(1−X , X ))].

ΠN,I (X)

2

1

2

1

˜ = [ΠN,I (X 2 , X 1 ), ΠN,I (X , X )]. So, Proposition 3 holds.

Concluding, ΠN,I (X)

2

1

Example 2. Consider IRC and related NS -dual construction ΠNS ,JRC . By preserving the conditions of Proposition 3, Eq.(18) can be expressed as

Π NS ,IRC(X1 , X2 )=

0, if X1 + X2 = 1,

1−[1−X 2 −X 1 +X 2 X 1 , 1−X 2 −X 1 +X 2 X 1 ], otherwise.

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L. Costa et al.

3.2

Relationship with Interval-Valued Automorphisms

Proposition 4. Let NΦ : U → U be the φ-conjugate of a strong IvFN N : U → U

˜ →U

˜ is a

and φ : U → U be a φ-representable IvA given by Eq.(14). When Φ : U

Φ ˜

Φ-representable IvIFa given by Eq.(13), a function Π : U → U given by

Π Φ (X1 , X2 ) = (φ−1 (Π(φ(X1 )), 1 − φ(1 − X2 )),

(18)

˜ → U is also a A − GIvIF Ix(NI ).

is a A − GIvIF Ix(NI ) whenever Π : U

˜ → U be a φ-representable A-IvA and Π : U

˜ → U be a A −

Proof. Let φ : U

GIvIF Ix(NI ). It holds that:

Π1 :Π Φ (X1 , X2 ) = 1 ⇔ φ−1 (Π(φ(X1 ), 1 − φ(1 − X2 ))) = 1 (by Eq.(18))

⇔ Π(φ(X1 ), 1 − φ(1 − X2 )) = 1 (by AI 1)

⇔ φ(X1 ) = 0 and 1 − φ(1 − X2 ) = 0 (by Π1 )

⇔ X1 = 0 and X2 = 0(by AI 1)

Π2 :It is analogous to Π1 .

Π3 :(X1 , X2 )

(Y1 , Y2 ) ⇒ X1 ≤U Y1 and X2 ≤U Y2 by

−relation

⇒ φ(X1 ) ≤U φ(Y1 ) and 1 − φ(1 − X2 ) ≤U 1 − φ(1 − Y2 ) by AI 1

⇒ Π(φ(X1 ), 1 − φ(1 − X2 )) ≤U Π(φ(Y1 ), 1 − φ(1 − Y2 )) by Π3

⇒ φ−1 (Π(φ(X1 ), 1 − φ(1 − X2 ))) ≤U φ−1 (Π(φ(Y1 ), 1 − φ(1 − Y2 ))) by A1

φ

φ

⇒ ΠG

(X1 , X2 ) ≤U ΠG

(Y1 , Y2 ) (by Eq.(13)).

Let NI be a strong IvIFN given by Eq.(10) and NΦ

I its Φ−conjugate function.

−1

Π4 :Π Φ NΦ

Π(Φ ◦ Φ−1 (NI (Φ(X1 , X2 )))) (by Eq. (18))

I (X1 , X2 ) = Φ

= Φ−1 (Π(NI (φ(X1 , X2 )))) = Φ−1 (Π(Φ(X1 , X2 )) = Π(X1 , X2 ) (by Π4 )

The new results follow from Proposition 4 and Theorem 1.

Corollary 2. In conditions of Proposition 4 and also considering φrepresentable IvA given by Eq.(14), we can express Eq.(18) as follows:

Π Φ (X1 , X2 ) = Π φ (X1 , X2 ), Π φ (X1 , X2 ) .

(19)

˜ and I(J) :

Corollary 3. Let Φ be a φ-representable automorphism in Aut(U)

2

˜

˜

U → U be the corresponding φ-conjugate operator related to a (co)implication

I(J) : U2 → U, verifying the conditions of Theorem 1. And, let NΦ be a strong

˜ → U given by

φ-conjugate IvFN negation. A function ΠN,Iφ (ΠN,Jφ ) : U

Φ

(X1 , X2 ) = NΦ (IΦ (NS (X2 ), X1 ))

ΠN,I

Φ

(X1 , X2 )

ΠN,J

φ

Φ

Φ

= J (N (NS (X2 ), N (X1 )) .

(20)

(21)

˜ → U is also a A−GIvIF Ix(N).

is an A−IvGIF Ix(N ) whenever ΠN,I (ΠN,J ) : U

Interval Version of Generalized Atanassov’s Intuitionistic Fuzzy Index

225

Example 3. Consider IRC and related Φ-conjugate construction ΠNΦS ,JRC given by

Eq.(18). For a φ-representable IvIA, taking φ(X) = X n and n as an integer

non-negative integer, we have the following:

Π Φ NS ,IRC (X1 , X2 )=

4

n

n

(1 − X 1 )(1 − X 2 )n ;

n

(1 − X n1 )(1 − X 2 )n . (22)

Interval-Valued Intuitionistic Fuzzy Entropy

This section generalizes results from [7, Deﬁnition 2] also discussing properties

related to the Atanassov’s interval-valued intuitionistic fuzzy entropy (A-IvIFE)

which are obtained by action of an interval-valued aggregation of A-GIvIFIx.

Deﬁnition 4. An interval-valued function E : AI → U is called an A-IvIFE if

E veriﬁes the following properties:

E2:

E2:

E3:

E4:

E(AI ) = 0 ⇔ AI ∈ A;

E(AI ) = 1 ⇔ MAI (x) = NAI (x) = 0, ∀x ∈ χ;

E(AI ) = E(AI c );

If AI U˜ BI then E(AI ) ≥U E(BI ), ∀AI , BI ∈ AI .

Now, main properties of A-IvIFE obtained by A-GIvIFI are studied [15].

Theorem 2. Consider χ = {x1 , . . . , xn }. Let M : Un → U be an automorphism,

˜ ). A function E : AI → U given by

N be a strong IvFN and Π ∈ Aut(U

E(AI ) = Mni=1 Π(AI (xi )), ∀xi ∈ χ,

(23)

is an A-IvIFE in the sense of Deﬁnition 4.

Proof. Let AI c be the complement of AI given by Eq.(11). For all xi ∈ χ and

AI , BI ∈ AI , we have that:

E1 : E(AI ) = 0 ⇔ Mni=1 Π(AI (xi )) = 0. By M1, E(AI ) = 0 ⇔ MAI (xi ) +

NAI (xi ) = 1, ∀xi ∈ χ. Then, by Π2, E(AI ) = 0 ⇔ AI ∈ A.

E2 : E(AI ) = 1 ⇔ Mni=1 Π(AI (xi )) = 1. By M1, E(AI ) = 1 ⇔ MAI (xi ) +

NAI (xi ) = 0, meaning that MAI (xi ) = NAI (xi ) = 0.

E3 : E(AI )c = Mni=1 Π(AI c (xi )) = Π(NI (X1 , X2 )). By Π3, the following holds

E(AI )c = Π(X1 , X2 ). Concluding, E(AI )c = E(AI ).

E4 : If AI

˜ BI then AI (xi )

˜ BI (xi ). Based on Π3, it holds that

U

U

Π(BI (xi )) ≤U Π(AI (xi )). By M3, we obtain that Mni=1 Π(BI (xi )) ≤U

Mni=1 Π(BI (xi )). As conclusion, E(AI ) ≥U E(BI ).

Therefore, Theorem 2 is veriﬁed.

Proposition 5. Consider χ = {x1 , . . . , xn }. Let M : Un → U be an IvA, N be

˜ → U is A-GIvIFIx(N) given by Eq.(15). Then,

a strong IvFN and ΠN,I (ΠN,J ) : U

for all xi ∈ χ, an A-IvIFE E : AI → U can be given by

EΠN,I (AI )=Mni=1 ΠN,I (AI (xi ))

EΠN,I (AI )=Mni=1 ΠN,J (AI (xi )) .

(24)

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L. Costa et al.

Proof. Straightforward Theorems 1 and 2.

Corollary 4. Consider N = NS , A-GIvIFIx (NS ) ΠN,I given by Eq.(15). Then,

by taking AI (xi ) = (MAI (xi ), NAI (xi )) = (X1i , X2i ) for all xi ∈ χ, an A-IvIFE

E : AI → U which is given in Eq.(24) can be expressed as

EΠN,I (AI )=Mni=1 (NS (I(NS (X2i ), X1i )) EΠN,J (AI )=Mni=1 J(X2i , NS (X1i ) . (25)

Proof. Straightforward Proposition 5 and Theorem 1.

Example 4. By taking the arithmetic mean as an aggregation operator, IRC

in Eq.(4) and related IvIFIx given in Eq.(18). Let AI be an IvIFS deﬁned

˜ for all xi ∈ χ, an IvIFE as EΠ

(X1i , X2i ) =

by pairs (X1i , X2i ) ∈ U,

NS ,IRB

n

1

N

(I

(N

(X

),

X

)

can

be

given

as

follows:

S

RB

S

2i

1i

i=1

n

EΠNS ,IRB (X1i , X2i ) =

4.1

1

n

n

[1−X 2i −X 1i +X 2i X 1i , 1−X 2i −X 1i +X 2i X 1i ]. (26)

i=1

Relationship with Intuitionistic Index and Conjugate Operators

Conjugation operator and duality properties related to generalized Atanassov’s

Intuitionistic Fuzzy Index are reported from [10].

˜ a φ-representable

Proposition 6. Consider χ = {x1 , . . . , xn } and Φ ∈ Aut(U)

A-IvIFA given by Eq.(13). When Π is A − GIvIF Ix(N), an A-IvIFE is a function EΦ : AI → U deﬁned by

n

EΦ (AI ) = Mφ i=1 Π φ (AI (xi )), ∀xi ∈ χ.

(27)

Proof. Based on Eqs.(12) and (13), the following holds:

EΦ (AI (xi )) = EΦ (AI ) = φ−1 (E(φ(lU˜ (AI (xi ))), 1 − φ(1 − rU˜ (AI (xi )))

= φ−1 Mni=1 (φ ◦ φ−1 )Π(φ(lU˜ (AI (xi ))), 1 − φ(1 − rU˜ (AI (xi ))

n

= φ−1 Mni=1 (φ(Π φ (AI (xi ))) = Mφ i=1 Π φ (AI (xi ))

Figure 1 summarizes the main results related to the classes of A-GIvIFIx(N)

and A-IvIFE denoted by C(Π) and C(E), respectively. This A-IvIFE is obtained

not only from generalized IvIFIx [4] but also from dual and conjugate operators.

˜ and I(J) :

Proposition 7. Let Φ be a φ-representable automorphism in Aut(U)

2

˜

˜

U → U be the corresponding φ-conjugate operator related to a (co)implication

I(J) : U2 → U, verifying the conditions of Theorem 1. Additionally, let NΦ be a

strong φ-conjugate IvFN negation and M : Un → U be an aggregation function.

Φ

Then, for A ∈ A, the functions EN,I , EΦ

N,I (EN,I , EN,I ) : A → U given by

EN,I (A)(xi ) = Mni=1 N(I(1 − NAI (xi ), MAI (xi ))),

(28)

N,I (A)(xi )

(29)

(30)

=

EN,J (A)(xi ) =

N,J (A)(xi ) =

n

Mφ i=1 Nφ (Iφ (1 − NAI (xi ), MAI (xi )));

Mni=1 J(NAI (xi ), 1 − MAI (xi ))

n

Mφ i=1 Jφ (NAI (xi ), 1 − MAI (xi )).

express an interval-valued Atanassov’s intuitionistic fuzzy entropy.

(31)

Interval Version of Generalized Atanassov’s Intuitionistic Fuzzy Index

227

˜)

Fig. 1. Conjugate construction of A − GIF Ix(N ) and A − IF E on Aut(U

Proof. Straightforward from Proposition 6.

Example 5. By Eqs.(28) and (26), an IvIFE expression is obtained as follows:

NS ,IRB (A)(xi ) =

4.2

1

n

n

n

n

(1 − X 1i )(1 − X 2i )n ;

n

(1 − X n1i )(1 − X 2i )n (. 32)

i=1

Preserving Fuzzyness and Intuitionism Based on IvIFE

Based on [12], assuming that χ = {u}, A1 = {(u, [0.1, 0.2], [0.3, 0.4])} and A2 =

{(u, [0.2, 0.3], [0.4, 0.5])} in order to calcule the entropies by equations below

EY (A) =

1

n

n

i=1

μA (xi ) + μA (xi ) − νA (xi ) − νA (xi )

1

π − 1]√

2 cos

8

2−1

1

EG (A) =

n

n

cos

i=1

|μA (xi ) − νA (xi )| + |μA (xi ) − νA (xi )|

π.

8

(33)

(34)

Thus, E(A1 ) and (E(A2 )) contains the diﬀerence between the membership and

non-membership degrees related to the hesitancy degree. However, despite the

diﬀerences, the same value for related IvIFEs are matched, making impossible

to distinguish the fuzziness and intuitionism of these two cases. Intuitively, it

is easy to observe that A1 is more fuzzy than A2 , meaning that πA1 ≥ πA2 .

However, this cannot be seen by using the above Eqs.(33) and (34). So, a more

sensitive deﬁnition of IvIFE is introduced in order to deal with this problem.

In our proposed methodology, we calculate the related IvIFEs by using

Eqs.(26) and (32) together with corresponding IvIFIx given by Eqs.(18) and (22).

See these results presented in 1st and 2nd columns of Table 1 when the inputs are

given as A1 and A2 . Since χ is singleton IvIFS, the resulting hesitant degree and

corresponding entropy measure coincide. Additionally, it is possible to naturally

preserve properties of related interval entropy, meaning that IvIFE is an order

preserving index, by including IFE. Moreover, taking A3 = [0.2, 0.2], [0.3, 0.3]

and A4 = [0.3, 0.3], [0.4, 0.4] as inputs, the entropy values obtained with the

degenerate intervals related to membership and non-membership degrees are

included in the interval entropy obtained with non-degenerated interval-valued

inputs. See these results in the 3rd and 4th columns of Table 1.

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L. Costa et al.

Table 1. IvIFIxs and IvIFEs related to IvIFSs from A1 to A4 IvIFSs.

IvIF Ix

A1

A2

A3

A4

Π(Ai ) = EΠ(Ai )

[0, 48; 0, 63]

[0, 35; 0, 48]

[0, 56; 0, 56]

[0, 42; 0, 42]

Π φ (Ai ) = EφΠ (Ai ) [0, 5879; 0, 6965] [0, 4769; 0, 5879] [0, 4704; 0, 4704] [0, 3276; 0, 3276]

5

Conclusion

The generalized concept of the Atanassov’s interval-valued intuitionistic fuzzy

index was studied by dual and conjugate construction methods. We also extend

the study of Atanassov’s intuitionistic fuzzy entropy based on such two constructors. Further work considers the extension of such study related to properties

veriﬁed by the A − GIvIF Ix(N ) and A − IvIF E and also the use of admissible

linear orders to compare the results of the interval entropy, since, in some cases,

the values of interval entropy cannot be compared using the Moore’s method.

Acknowledgment. Work supported by the Brazilian funding agencies CAPES,

MCTI/CNPQ, Universal (448766/2014-0), PQ (310106/2016-8), CNPq/PRONEX/

FAPERGS and PqG 02/2017.

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Fuzzy Ontologies: State of the Art Revisited

Valerie Cross(&) and Shangye Chen

Computer Science and Software Engineering,

Miami University, Oxford, OH 45056, USA

crossv@miamioh.edu

Abstract. Although ontologies have become the standard for representing

knowledge on the Semantic Web, they have a primary limitation, the inability to

represent vague and imprecise knowledge. Much research has been undertaken

to extend ontologies with the means to overcome this and has resulted in

numerous extensions from crisp ontologies to fuzzy ontologies. The original

web ontology language, and tools were not designed to handle fuzzy information; therefore, additional research has focused on modiﬁcations to extend them.

A review of the fuzzy extensions to allow fuzziness in ontologies, web languages, and tools as well as several very current examples of fuzzy ontologies in

real-world applications is presented.

Keywords: Ontologies Á Fuzzy logic Á Web ontology language

OWL Á Fuzzy formal concept analysis Á Ontology tools Á Semantic web

1 Introduction

An ontology is a shared explicit speciﬁcation of a conceptualization formalizing

concepts pertaining to a domain, properties of these concepts, and relationships existing

between the concepts [1]. An ontology is the main knowledge representation method

for describing information on the Semantic Web and promotes the inclusion of

semantic content in web pages. Ontologies are both understandable to humans and

expressed in a machine-readable format using a web ontology language. Being

understandable to humans is important to representing knowledge, but one capability

initially missing in ontologies is the ability to represent and manage imprecision and

vagueness that often exists in domain knowledge as understood by humans.

This need to handle uncertainty in ontologies motivated researchers in knowledge

representation to propose fuzzy ontologies. With fuzzy ontologies, the ability is provided to model real world environments that naturally include uncertainty with fuzzy

set theory and mathematics and through “computing with words” [2], that is, the use of

linguistic terms represented by fuzzy sets. This ability is extremely important to

knowledge being extracted from human experts since they often are more comfortable

with the use of inexact, fuzzy linguistic terms rather than precise numbers. Besides

human imprecision in domain knowledge modeling, some concepts themselves cannot

be precisely model and require a formalism to allow for a vague speciﬁcation and yet

still require the ability to be used in a reasoning process. Other factors also contribute to

the need for modeling uncertainty in ontologies. For example, a wide variety of

© Springer International Publishing AG, part of Springer Nature 2018

G. A. Barreto and R. Coelho (Eds.): NAFIPS 2018, CCIS 831, pp. 230–242, 2018.

https://doi.org/10.1007/978-3-319-95312-0_20

Fuzzy Ontologies: State of the Art Revisited

231

knowledge sources may require the integration of diverse inexact speciﬁcations in

modeling the domain knowledge. Although many different logical formalisms have

been proposed to extend ontologies for handling uncertainty, this paper addresses only

extensions using type-1 fuzzy sets.

The more recent research in ontological knowledge representation for the Semantic

Web has had a proliferation of approaches to deﬁning, constructing and using fuzzy

ontologies. Numerous places exists where uncertainty can occur in the speciﬁcation of

a fuzzy ontology since there are many ways that domain knowledge can contain

uncertainty and vagueness from human description. This paper outlines some of the

progression of the development of fuzzy ontologies from simplest to the more complex

and provides examples illustrating the fuzzy extensions. It is hoped that this paper can

provide readers an introduction and an a more intuitively understanding of fuzzy

ontology development so that formalisms used in other fuzzy ontology research papers

which are referenced further in the paper are better understood. Section 2 examines

where and why an ontology becomes a fuzzy ontology and how “fuzzy” is the

ontology, that is, can only certain features of the ontology be allowed to have uncertainty. Once an ontology designer has identiﬁed places and/or levels where uncertainty

needs to be speciﬁed what languages and tools are available to specify this uncertainty

in the fuzzy ontology. Section 3 discusses current fuzzy languages and tools to describe

fuzzy ontologies and examples of the different approaches to actually building a fuzzy

ontology. Several very current uses of fuzzy ontologies in a wide variety of domains

ranging from medical and transportation and for different tasks such as information

retrieval and opinion-mining of social media platforms are described in Sect. 4. Section 5 summarizes and discusses some limitations of fuzzy ontologies that hinder them

from becoming more widespread and presents areas for future work to increase their

use.

2 What Makes an Ontology Fuzzy?

The answer to the question varies depending what the research on fuzzy ontologies is

focused on. At what place and for what purpose is the uncertainty, imprecision or

vagueness introduced? The simplest answer is given as “a fuzzy ontology is simply an

ontology which uses fuzzy logic to provide a natural representation of imprecise and

vague knowledge and eases reasoning over it” [3]. This vague deﬁnition indicates no

universal standard deﬁnition of fuzzy ontologies exists since the needs of different

applications require different fuzzy extensions to an ontology.

There are many places where fuzzy extensions may be made since an ontology has

many components. An ontology consists of concepts C, instances of those concepts I,

hierarchical or taxonomic relationships between concepts H, attributes speciﬁed in

deﬁning concepts A, properties that are nonhierarchical relationships between concepts

P, and axioms that must hold for the concepts X. Some of these components may not be

present; for example, simple ontologies, do not have axioms speciﬁed. All these

components are similar to those found in an object-oriented database. Research on

fuzzy object oriented databases previously addressed many of these issues when fuzzy

extensions were proposed to object oriented databases [43]. The natural fuzzy

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