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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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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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)
226
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)
EΦ
N,I (A)(xi )
(29)
(30)
=
EN,J (A)(xi ) =
EΦ
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:
EΦ
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