Robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.1 (1-16)

Available online at www.sciencedirect.com

ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss

Robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures Yingfang Li a , Keyun Qin b , Xingxing He b,∗ , Dan Meng a a School of Economics Information Engineering, Southwestern University of Finance and Economics, Chengdu 611130, Sichuan, China b School of Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, China

Received 15 March 2014; received in revised form 5 July 2015; accepted 9 July 2015

Abstract This paper discusses the robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures. First of all, the concept of DF-metric is proposed. Secondly, several DF-metrics are introduced as well as their properties and some inequalities about them. Then a formula of divergence measure composed by DF-metric is presented. Finally, based on the proposed divergence measures, the concept of perturbations of fuzzy sets is extended. According to the extended concept, the perturbation parameters raised by various fuzzy connectives are studied and the perturbations of fuzzy reasoning are also investigated. © 2015 Elsevier B.V. All rights reserved. Keywords: Dissimilarity function; Biresiduation; Divergence measure; DF-metric; Perturbation; Robustness; Fuzzy reasoning

1. Introduction Since Zadeh [42–44] introduced the so-called Compositional Rule of Fuzzy Inference (briefly, CRI) method, various methods of fuzzy reasoning have been proposed [10,14,20,22,26,32]. Robustness is one of the important evaluation criteria for fuzzy reasoning methods. In fuzzy literature there exist many approaches to evaluating the robustness of fuzzy reasoning. In [41], Ying estimated maximum and average perturbation parameters for various fuzzy reasoning methods. In [8,9], Cai presented δ-equalities with respect to algebraic operators, fuzzy relations, fuzzy modus ponens (briefly, FMP) and fuzzy modus tollens (briefly, FMT). Then Cai and Zhang [10,45] applied control principles to fuzzy reasoning and proposed quantitative robustness measures. In [11], Cheng and Fu evaluated the upper and lower bounds of the output error affected by perturbation parameters of the input under the CRI method. In [18], Georgescu discussed the robustness of fuzzy reasoning based on analysis of δ-equalities and perturbations of fuzzy sets. Nguyen et al. [35] and Li et al. [27] estimated the robustness of fuzzy reasoning through investigating the robustness of fuzzy connectives and presented some methods for finding the most robust elements in classes of fuzzy connectives. In [28], we used a concept similar to the * Corresponding author.

E-mail address: [email protected] (X. He). http://dx.doi.org/10.1016/j.fss.2015.07.008 0165-0114/© 2015 Elsevier B.V. All rights reserved.

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.2 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

2

modulus of continuity to characterize the robustness of fuzzy connectives and presented robustness results for various fuzzy connectives and the CRI method. For a fuzzy reasoning method, if small disturbances of input always cause small changes of output, then we say that the method has a good behavior of robustness. Under this circumstance, one may ask what do the items “small disturbances” and “small changes” mean? To answer this issue, several concepts have been proposed, which can be used to characterize the approximate equalities or perturbations of fuzzy sets. Pappis [36] introduced the concept of proximity measure to estimate the similarity between fuzzy sets. Hong and Hwang [21] proposed the α-similarity of fuzzy variables. Cai [8,9] examined δ-equalities of fuzzy sets based on the Chebyshev distance. Ying [41] proposed concepts of maximum and average perturbation parameters of fuzzy sets based on the Chebyshev distance. Dai et al. [12] investigated perturbations of fuzzy sets and fuzzy reasoning on the basis of the normalized Minkowski distance. Jin et al. [24] discussed perturbations of fuzzy sets based on logical equivalence measures. Bustince et al. [4,5,7] presented some measures to settle equalities between fuzzy sets. Based on regular pairs derived from left-continuous t-norms, Wang and Duan [40] defined a divergence measure to evaluate the robustness of the Triple I Inference Method. By comparing and analyzing the above-mentioned concepts, we find out that the differences between them are largely due to the underlying similarity measures or divergence measures adopted. In [29], we used fuzzy equivalencies to construct similarity measures. In [6], Bustince et al. constructed divergence measures through restricted dissimilarity functions. In [30], we introduced the concept of dissimilarity function, which is a generation of the restricted dissimilarity function and can be used to construct divergence measures. Let T be a t-norm with ET being the biresiduation of T . Then we call dT = 1 − ET a dissimilarity function obtained from the biresiduation. Furthermore, if dT is a metric on [0, 1], then we call it a DF-metric. In [40], some divergence measures for appraising the robustness of fuzzy reasoning were constructed by DF-metrics. Several DF-metrics were also introduced. In Section 3, we propose some DF-metrics that differ from those proposed in [40]. Then we study properties of DF-metrics and present several inequalities about them. Since DF-metrics are special dissimilarity functions, we use them to construct divergence measures and obtain the following computational formula a D(A, B) =

n 

dT (A(xi ), B(xi ))

i=1

nb + (a − b)

n 

, dT (A(xi ), B(xi ))

i=1

where dT is a DF-metric and a > 0, b > 0. If a = b, then we obtain divergence measures that were used to appraise the robustness of fuzzy reasoning in [40]. In this way Wang and Duan’s work [40] becomes special cases of the results presented in this paper. In Section 4, we extend the concept of perturbations of fuzzy sets on the basis of the above-mentioned divergence measures. According to the extended concept, the perturbation parameters raised by various fuzzy connectives are studied and the perturbations of fuzzy reasoning are also investigated. 2. Preliminaries In this section we briefly recall, without proof, some preliminary definitions and results. Throughout this paper, X is the universal set; F (X) is the class of all fuzzy sets of X; A(x) is the membership function of A ∈ F (X); P (X) is the class of all crisp sets of X; Ac is the complement of A with Ac (x) = 1 − A(x) for all x ∈ X. Definition 1. (See [2].) A continuous, strictly increasing function ϕ : [0, 1] −→ [0, 1] with boundary conditions ϕ(0) = 0, ϕ(1) = 1 is called an automorphism of the unit interval. Definition 2. (See [25].) An associative, commutative and increasing function T : [0, 1]2 −→ [0, 1] is called a t-norm if it has the neutral element equal to 1. An associative, commutative and increasing function S : [0, 1]2 −→ [0, 1] is called a t-conorm if it has the neutral element equal to 0. Fodor and Roubens [17] defined fuzzy equivalence as a binary function on the unit interval in the following way.

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.3 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

3

Table 1 Several t-norms with their residuations and biresiduations. T (x, y) TL = (x + y − 1) ∨ 0 TM = x ∧ y TP = x · y  x ∧ y, x + y > 1, T0 = 0, x + y ≤ 1. xy TH = 2 − x − y + xy

→ −

T (x, y)  1 − x + y, x > y, x ≤ y.  1, y, x > y, x ≤ y.  1, y , x x > y,  1, x ≤ y. (1 − x) ∨ y, x > y, x ≤ y.  1, 2y−xy x+y−xy , x > y,

1,

x ≤ y.

ET (x, y) 1 − |x − y|  x ∧ y, x = y, 1, x = y. x ∧y x ∨ y (x ∧ y) ∨ (1 − x ∧ 1 − y), x = y, 1, x = y. |x − y| 1− x + y − xy

Definition 3. (See [17].) A binary function E : [0, 1]2 −→ [0, 1] is called a fuzzy equivalence if it satisfies the following properties: (E1) (E2) (E3) (E4)

E(x, y) = E(y, x) for all x, y ∈ [0, 1]. E(x, x) = 1 for all x ∈ [0, 1]. E(1, 0) = 0. For all x, y, z ∈ [0, 1], if x ≤ y ≤ z, then min(E(x, y), E(y, z)) ≥ E(x, z).

Now we recall further axioms in terms of fuzzy equivalencies. These properties are required in [29]. (E5) (E6) (E7) (E8)

E(x, y) = 1 if and only if x = y for all x, y ∈ [0, 1]. E(x, y) = 0 if and only if min(x, y) = 0 and max(x, y) = 0 for all x, y ∈ [0, 1]. E(x, 1) = x for all x ∈ [0, 1]. E(x, y) = E(1 − x, 1 − y) for all x, y ∈ [0, 1].

In [3], Bustince et al. presented the concept of restricted equivalence function, which arises from the concept of fuzzy equivalence and some properties usually demanded from the measures used for comparing images [5,6]. Definition 4. (See [17].) A binary function I : [0, 1]2 −→ [0, 1] is said to be an implication function if it satisfies the following properties: (I1) x ≤ z implies I (x, y) ≥ I (z, y) for all x, y, z ∈ [0, 1]. (I2) y ≤ z implies I (x, y) ≤ I (x, z) for all x, y, z ∈ [0, 1]. (I3) I (0, 0) = I (1, 1) = 1 and I (1, 0) = 0. → −

Definition 5. (See [37].) The residuation of a t-norm T is the function T : [0, 1]2 −→ [0, 1] defined for all x, y ∈ → − [0, 1] by T (x, y) = sup{α ∈ [0, 1] | T (x, α) ≤ y}. The residuation of a t-norm is an implication function called an R-implication. Definition 6. (See [37].) The biresiduation of a t-norm T is the function ET : [0, 1]2 −→ [0, 1] defined for all x, y ∈ → − → − [0, 1] by ET (x, y) = min(T (x, y), T (y, x)). Proposition 1. (See [29].) The biresiduation ET of a t-norm T is indeed a fuzzy equivalence. Example 1. (See [29].) Several examples of t-norms with their residuations and biresiduations are shown in Table 1 (∧ = min, ∨ = max). There exist several ways of measuring the differences between fuzzy sets through some binary functions. In some references these functions are called distance measures [3,16], in some references these functions are called dissimi-

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.4 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

4

larity measures [1,13], but in other references these functions are called divergence measures [30,33,34]. In this paper, we use the name of divergence measure. Definition 7. (See [31].) A function D : F (X) × F (X) −→ [0, ∞) is called a divergence measure if it satisfies the following properties: (D1) (D2) (D3) (D4)

D(A, B) = D(B, A) for all A, B ∈ F (X). D(A, A) = 0 for all A ∈ F (X). For all A, B, C ∈ F (X), if A ⊆ B ⊆ C, then max(D(A, B), D(B, C)) ≤ D(A, C). D(A, B) ≤ D(P , P c ) for all A, B ∈ F (X) and P ∈ P (X).

Definition 8. (See [38].) Let  : [0, 1]2 −→ R be a function with the following properties: (1) (2) (3) (4)

(x, y) ≥ 0 for all x, y ∈ [0, 1]. (positivity) (x, y) = 0 if and only if x = y for all x, y ∈ [0, 1]. (identity property) (x, y) = (y, x) for all x, y ∈ [0, 1]. (symmetry) (x, y) + (y, z) ≥ (x, z) for all x, y, z ∈ [0, 1]. (triangle inequality)

Then we say that  is a metric on [0, 1] and ([0, 1], ) is a metric space. Sometimes the identity condition (2) is weakened to the condition (2 ) (x, x) = 0 for all x ∈ [0, 1]. If  is a function that fulfills conditions (1), (2 ), (3) and (4), then we call it a pseudo-metric on [0, 1]. 3. Constructions of DF-metrics In [6], Bustince et al. presented the concept of restricted dissimilarity function which arises from concepts of dissimilarity and equivalence function. In [30], we proposed the concept of dissimilarity function, which is a generation of the restricted dissimilarity function. Definition 9. (See [30].) A binary function d : [0, 1]2 −→ [0, 1] is called a dissimilarity function if it satisfies the following properties: (d1) (d2) (d3) (d4)

d(x, y) = d(y, x) for all x, y ∈ [0, 1]. d(x, x) = 0 for all x ∈ [0, 1]. d(1, 0) = 1. For all x, y, z ∈ [0, 1], if x ≤ y ≤ z, then max(d(x, y), d(y, z)) ≤ d(x, z).

Remark 1. If E is a fuzzy equivalence, then we can prove that 1 − E is a dissimilarity function. Let Tα be a t-norm and ETα be the corresponding biresiduation. Suppose that dTα = 1 − ETα , then we call dTα a dissimilarity function obtained from biresiduation. Proposition 2. Given a discrete universe X = {x1 , x2 , . . . , xn }. Let d be a dissimilarity function. Suppose D is a function defined for all A, B ∈ F (X) by a D(A, B) =

n 

d(A(xi ), B(xi ))

i=1

nb + (a − b)

n 

, d(A(xi ), B(xi ))

i=1

where a > 0, b > 0, then D is a divergence measure.

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.5 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

5

Proof. It is easy to prove that D is a function from F (X) × F (X) to [0, 1] satisfying D1, D2 and D4. We prove that D3 holds. at nab Considering the function f (t) = nb+(a−b)t for t ≥ 0, we have f (t) = [nb+(a−b)t] 2 > 0. Thus we conclude that f is increasing with respect to t . Since A ⊆ B ⊆ C implies A(xi ) ≤ B(xi ) ≤ C(xi ) for each xi ∈ X, we have n n   d(A(xi ), B(xi )) ≤ d(A(xi ), C(xi )). According to the monotonicity of f we have i=1

i=1

a

n 

a

d(A(xi ), B(xi ))

i=1

nb + (a − b)

n 

≤ d(A(xi ), B(xi ))

i=1

n 

d(A(xi ), C(xi ))

i=1

nb + (a − b)

n 

. d(A(xi ), C(xi ))

i=1

Therefore, D(A, B) ≤ D(A, C) holds. The case of D(B, C) ≤ D(A, C) can be proved similarly.

2

Definition 10. If a dissimilarity function obtained from biresiduation is a metric on [0, 1], then we call it a DF-metric. Lemma 1. (See [19].) Let T be a t-norm with ET being the biresiduation of T . And let dT be the dissimilarity function obtained from ET . Then dT is a pseudo-metric if and only if T ≥ TL . Moreover, if ET satisfies E5, then dT is a metric. Remark 2. Considering four dissimilarity functions dTL , dTM , dTP , dT0 that are obtained from the biresiduations ETL , ETM , ETP , ET0 shown in Table 1. Note that TL ≤ TP ≤ TM and ETL , ETP , ETM satisfy E5. According to Lemma 1, dTL , dTP and dTM are metrics on [0, 1] and thus they are DF-metrics. In [40], Wang and Duan proved that dT0 is a metric on [0, 1] and thus it is a DF-metric. In the following, other DF-metrics different from dTL , dTM , dTP , dT0 are proposed. Lemma 2. Let dT be a dissimilarity function obtained from biresiduation. If dT (x, y) + dT (y, z) ≥ dT (x, z) whenever x ≤ y ≤ z, then it is a DF-metric. Proof. Since dT is a dissimilarity function obtained from biresiduation, it satisfies the first three conditions for a metric on [0, 1]. We prove that dT satisfies triangle inequality for all x, y, z ∈ [0, 1]. For all x, y, z ∈ [0, 1], there exist six permutations, that is, x ≤ y ≤ z, x ≤ z ≤ y, z ≤ x ≤ y, z ≤ y ≤ x, y ≤ z ≤ x, y ≤ x ≤ z. Since dT (x, y) + dT (y, z) ≥ dT (x, z) whenever x ≤ y ≤ z, by the commutativity of dT we have dT (x, y) + dT (y, z) ≥ dT (x, z) whenever z ≤ y ≤ x. In order to prove that dT satisfies triangle inequality for all x, y, z ∈ [0, 1], we only need to prove that dT satisfies triangle inequality whenever x ≤ z ≤ y, or z ≤ x ≤ y, or y ≤ z ≤ x, or y ≤ x ≤ z. If x ≤ z ≤ y, then dT (x, y) ≥ dT (x, z). If z ≤ x ≤ y, then dT (y, z) ≥ dT (x, z). If y ≤ z ≤ x, then dT (x, y) ≥ dT (x, z). If y ≤ x ≤ z, then dT (y, z) ≥ dT (x, z). Therefore, we conclude that dT satisfies triangle inequality for all x, y, z ∈ [0, 1]. 2 Proposition 3. For the t-norm TH and the corresponding biresiduation ETH , the obtained dissimilarity function dTH is a DF-metric. Proof. Since dTH is a dissimilarity function obtained from biresiduation, according to Lemma 2 we need to prove that dTH (x, y) + dTH (y, z) ≥ dTH (x, z) whenever x ≤ y ≤ z. If x = 0, then we have dTH (0, y) + dTH (y, z) ≥ dTH (0, z). If x = 0, then from x ≤ y ≤ z we have dTH (x, y) = 1 − ETH (x, y) = 1

1

y−x x+y−xy

=

1 1 x−y 1 1 x + y −1

, dTH (y, z) = 1 − ETH (y, z) =

z−y y+z−yz

=

1 1 y−z 1 1 y + z −1

, dTH (x, z) = 1 − ETH (x, z) =

z−x 1 1 1 u−v x−z x+z−xz = 1 + 1 −1 . Let x = u, y = v, z = w, then u ≥ v ≥ w ≥ 1. Thus we have dTH (x, y) = u+v−1 , dTH (y, z) x z v−w u−w u−v v−w u−w v+w−1 , dTH (x, z) = u+w−1 . Therefore, dTH (x, y) + dTH (y, z) − dTH (x, z) = u+v−1 + v+w−1 − u+w−1 (u−w)(2v−1) (2v−1) (u+w−1)(2v−1) (u+v−1)(v+w−1) u−w 1 − (u+v−1)(v+w−1) − u+w−1 = (u − w)( (u+v−1)(v+w−1) − u+w−1 ). Note that 2v−1 2v−1 (u−v)(v−w) (2v−1) 1 ≥ 0, thus ≥ . Therefore, d (x, y) + d (y, z) − d (x, z) ≥ 0 whenever x TH TH TH 2v−1 (u+v−1)(v+w−1) u+w−1

y ≤ z.

2

= = = ≤

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.6 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

6

Lemma 3. (See [23].) Let ϕ be an automorphism of the unit interval. Suppose that T is a t-norm and ET is the biresiduation of T . Let Tϕ be the t-norm which is isomorphic to T , i.e., Tϕ (x, y) = ϕ −1 (T (ϕ(x), ϕ(y))), then the biresiduation of Tϕ can be expressed as ETϕ (x, y) = ϕ −1 (ET (ϕ(x), ϕ(y))). Example 2. Suppose that TϕL and TϕP are two t-norms which are isomorphic to TL and TP , respectively, i.e., TϕL (x, y) = ϕ −1 (max(ϕ(x) + ϕ(y) − 1, 0)), TϕP (x, y) = ϕ −1 (ϕ(x) · ϕ(y)). Then the biresiduations of TϕL and TϕP ϕ(y) can be expressed as ETϕL (x, y) = ϕ −1 (1 − |ϕ(x) − ϕ(y)|) and ETϕP (x, y) = ϕ −1 (min( ϕ(x) ϕ(y) , ϕ(x) )), respectively. Under the conditions of Example 2, we have Proposition 4. If the automorphism of the unit interval is defined as ϕ(x) = (λ+1)x 1+λx , where λ ≥ 0, then for the t-norm TϕL and the corresponding biresiduation ETϕL , the obtained dissimilarity function dTϕL is a DF-metric. Proof. Since dTϕL is a dissimilarity function obtained from biresiduation, according to Lemma 2 we need to prove that dTϕL (x, y) + dTϕL (y, z) ≥ dTϕL (x, z) whenever x ≤ y ≤ z. Since ϕ(x) =

(λ+1)x 1+λx ,

we have ϕ −1 (x) =

x 1+λ(1−x) .

If x ≤ y ≤ z, then ϕ(x) ≤ ϕ(y) ≤ ϕ(z) and thus we have

dTϕL (x, y) = 1 − ETϕL (x, y) = 1 − ϕ −1 (1 − |ϕ(x) − ϕ(y)|) = 1 − ϕ −1 (1 + ϕ(x) − ϕ(y)) = 1 − dTϕL (y, z) = 1 − ETϕL (y, z) = 1 − ϕ −1 (1 − |ϕ(y) − ϕ(z)|) = 1 − ϕ −1 (1 + ϕ(y) − ϕ(z)) = 1 − dTϕL (x, z) = 1 − ETϕL (x, z) = 1 − ϕ −1 (1 − |ϕ(x) − ϕ(z)|) = 1 − ϕ −1 (1 + ϕ(x) − ϕ(z)) = 1 −

(λ+1)x 1−( (λ+1)y 1+λy − 1+λx ) (λ+1)x 1+λ( (λ+1)y 1+λy − 1+λx ) (λ+1)y 1−( (λ+1)z 1+λz − 1+λy ) (λ+1)y 1+λ( (λ+1)z 1+λz − 1+λy )

(λ+1)x 1−( (λ+1)z 1+λz − 1+λx )

, ,

. Sup-

(λ+1)x 1+λ( (λ+1)z 1+λz − 1+λx ) (λ+1)y (λ+1)y (λ+1)x (λ+1)z (λ+1)z (λ+1)x pose that 1+λy − 1+λx = u, 1+λz − 1+λy = v, 1+λz − 1+λx = u + v, then we have dTϕL (x, y) = (λ+1)u 1+λu , (λ+1)v (λ+1)(u+v) u + dTϕL (y, z) = 1+λv , dTϕL (x, z) = 1+λ(u+v) . Therefore, dTϕL (x, y) + dTϕL (y, z) − dTϕL (x, z) = (λ + 1)( 1+λu (u+v) uvλ(λ+1)(2+λu+λv) uvλ(λ+1)(2+λu+λv) v 1+λv − 1+λ(u+v) ) = (1+λu)(1+λv)(1+λ(u+v)) . Since λ ≥ 0, u ≥ 0, v ≥ 0, thus (1+λu)(1+λv)(1+λ(u+v)) ≥ 0. Therefore, dTϕL (x, y) + dTϕL (y, z) − dTϕL (x, z) ≥ 0 whenever x ≤ y ≤ z. 2

Proposition 5. If the automorphism of the unit interval is defined as ϕ(x) = (λ+1)x 1+λx , where λ > −1, then for the t-norm TϕP and the corresponding biresiduation ETϕP , the obtained dissimilarity function dTϕP is a DF-metric. Proof. Since dTϕP is a dissimilarity function obtained from biresiduation, according to Lemma 2 we need to prove that dTϕP (x, y) + dTϕP (y, z) ≥ dTϕP (x, z) whenever x ≤ y ≤ z. Since ϕ(x) =

(λ+1)x 1+λx ,

we have ϕ −1 (x) =

x 1+λ(1−x) .

If x ≤ y ≤ z, then ϕ(x) ≤ ϕ(y) ≤ ϕ(z) and thus we have

ϕ(y) −1 ϕ(x) dTϕP (x, y) = 1 − ETϕP (x, y) = 1 − ϕ −1 (min( ϕ(x) ϕ(y) , ϕ(x) )) = 1 − ϕ ( ϕ(y) ) = 1 − ϕ(z) −1 ϕ(y) ETϕP (y, z) = 1 − ϕ −1 (min( ϕ(y) ϕ(z) , ϕ(y) )) = 1 − ϕ ( ϕ(z) ) = 1 − ϕ(z) −1 ϕ(x) ϕ −1 (min( ϕ(x) ϕ(z) , ϕ(x) )) = 1 − ϕ ( ϕ(z) ) = 1 −

x 1+λz 1+λx · z x 1+λ(1− 1+λx · 1+λz z )

y 1+λz 1+λy · z y 1+λ(1− 1+λy · 1+λz z )

. Suppose that

1+λy x 1+λx · y x 1+λ(1− 1+λx · 1+λy y )

, dTϕP (y, z) = 1 −

, dTϕP (x, z) = 1 − ETϕP (x, z) = 1 −

x 1+λx

y 1+λz x · 1+λy y = u, 1+λy · z = v, 1+λx ·

1+λz u z = uv, then 0 ≤ u ≤ 1, 0 ≤ v ≤ 1, 0 ≤ uv ≤ 1. Thus we have dTϕP (x, y) = 1 − 1+λ(1−u) , dTϕP (y, z) = 1 − v uv uv u 1+λ(1−v) , dTϕP (x, z) = 1 − 1+λ(1−uv) . Therefore, dTϕP (x, y) + dTϕP (y, z) − dTϕP (x, z) = 1 + 1+λ(1−uv) − 1+λ(1−u) − 2 2 (1+λ)(λ uv−(1+λ ))(u−1)(1−v) v 2 2 1+λ(1−v) = (1+λ(1−uv))(1+λ(1−u))(1+λ(1−v)) . Since 1 + λ > 0, λ uv − (1 + λ ) < 0, u − 1 ≤ 0, 1 − v ≥ 0, 1 + λ(1 − 2 2 (1+λ)(λ uv−(1+λ ))(u−1)(1−v) ≥ 0. Therefore, dTϕP (x, y) + uv) > 0, 1 + λ(1 − u) > 0, 1 + λ(1 − v) > 0, we have (1+λ(1−uv))(1+λ(1−u))(1+λ(1−v)) dTϕP (y, z) − dTϕP (x, z) ≥ 0 whenever x ≤ y ≤ z. 2

Remark 3. In Propositions 3, 4 and 5, we have proved that dTH , dTϕL and dTϕP are DF-metrics through Lemma 2. Actually, these three propositions can be proved in another way. Note that ETH , ETϕL and ETϕP satisfy E5. According to Lemma 1, in order to prove that dTH , dTϕL and dTϕP are DF-metrics, we can also prove that TH , TϕL and TϕP are bigger or equal than TL .

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.7 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

7

Lemma 4. (See [30].) Given a dissimilarity function d, for all x1 , x2 , y1 , y2 ∈ [0, 1] we have (1) d(max(x1 , y1 ), max(x2 , y2 )) ≤ max(d(x1 , x2 ), d(y1 , y2 )). (2) d(min(x1 , y1 ), min(x2 , y2 )) ≤ max(d(x1 , x2 ), d(y1 , y2 )). The following proposition shows properties of DF-metrics. Proposition 6. Let T be a t-norm with ET being the biresiduation of T . And let dT be the dissimilarity function obtained from ET . If dT is a DF-metric, then we have dT (max(x1 , y1 ), max(x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ) for all x1 , x2 , y1 , y2 ∈ [0, 1]. dT (min(x1 , y1 ), min(x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ) for all x1 , x2 , y1 , y2 ∈ [0, 1]. dT (T (x1 , y1 ), T (x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ) for all x1 , x2 , y1 , y2 ∈ [0, 1]. If S is the t-conorm which is dual to T and ET satisfies E8, then we have dT (S(x1 , y1 ), S(x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ) for all x1 , x2 , y1 , y2 ∈ [0, 1]. → − → − → − (5) If T is the residuation of T , then we have dT (T (x1 , y1 ), T (x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ) for all x1 , x2 , y1 , y2 ∈ [0, 1]. (1) (2) (3) (4)

Proof. (1) Since dT is a DF-metric, we have dT (max(x1 , y1 ), max(x2 , y2 )) ≤ dT (max(x1 , y2 ), max(x2 , y2 )) + dT (max(x1 , y1 ), max(x1 , y2 )). By Lemma 4, we have dT (max(x1 , y2 ), max(x2 , y2 )) ≤ max(dT (x1 , x2 ), dT (y2 , y2 )) = dT (x1 , x2 ), dT (max(x1 , y1 ), max(x1 , y2 )) ≤ max(dT (x1 , x1 ), dT (y1 , y2 )) = dT (y1 , y2 ). Thus dT (max(x1 , y1 ), max(x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ). (2) The proof of (2) is similar to that of (1). (3) Since dT is a DF-metric, we conclude that dT (T (x1 , y1 ), T (x2 , y2 )) ≤ dT (T (x1 , y2 ), T (x2 , y2 )) + dT (T (x1 , y1 ), T (x1 , y2 )). It was proved in [39] that ET (T (x1 , y1 ), T (x2 , y2 )) ≥ T (ET (x1 , x2 ), ET (y1 , y2 )) for all x1 , x2 , y1 , y2 ∈ [0, 1]. Thus dT (T (x1 , y1 ), T (x2 , y2 )) = 1 − ET (T (x1 , y1 ), T (x2 , y2 )) ≤ 1 − T (ET (x1 , x2 ), ET (y1 , y2 )). Since S is the t-conorm which is dual to T , we have 1 − T (ET (x1 , x2 ), ET (y1 , y2 )) = S(1 − ET (x1 , x2 ), 1 − ET (y1 , y2 )) = S(dT (x1 , x2 ), dT (y1 , y2 )). Therefore, we have dT (T (x1 , y1 ), T (x2 , y2 )) ≤ S(dT (x1 , x2 ), dT (y1 , y2 )). Hence, we obtain that dT (T (x1 , y2 ), T (x2 , y2 )) ≤ S(dT (x1 , x2 ), dT (y2 , y2 )) = dT (x1 , x2 ) and that dT (T (x1 , y1 ), T (x1 , y2 )) ≤ S(dT (x1 , x1 ), dT (y1 , y2 )) = dT (y1 , y2 ). Thus dT (T (x1 , y1 ), T (x2 , y2 )) ≤ dT (x1 , x2 ) + dT (y1 , y2 ). (4) Since S is the t-conorm which is dual to T and ET satisfies E8, we have dT (S(x1 , y1 ), S(x2 , y2 )) = 1 − ET (S(x1 , y1 ), S(x2 , y2 )) = 1 − ET (1 − T (1 − x1 , 1 − y1 ), 1 − T (1 − x2 , 1 − y2 )) = 1 − ET (T (1 − x1 , 1 − y1 ), T (1 − x2 , 1 − y2 )). From inequality ET (T (x1 , y1 ), T (x2 , y2 )) ≥ T (ET (x1 , x2 ), ET (y1 , y2 )) for all x1 , x2 , y1 , y2 ∈ [0, 1], we have 1 − ET (T (1 − x1 , 1 − y1 ), T (1 − x2 , 1 − y2 )) ≤ 1 − T (ET (1 − x1 , 1 − x2 ), ET (1 − y1 , 1 − y2 )) = S(1 − ET (1 − x1 , 1 − x2 ), 1 − ET (1 − y1 , 1 − y2 )) = S(dT (x1 , x2 ), dT (y1 , y2 )). Hence, we obtain that dT (S(x1 , y1 ), S(x2 , y2 )) ≤ S(dT (x1 , x2 ), dT (y1 , y2 )). Since dT is a DF-metric, we have dT (S(x1 , y1 ), S(x2 , y2 )) ≤ dT (S(x1 , y2 ), S(x2 , y2 )) + dT (S(x1 , y1 ), S(x1 , y2 )) ≤ S(dT (x1 , x2 ), dT (y2 , y2 )) + S(dT (x1 , x1 ), dT (y1 , y2 )) = dT (x1 , x2 ) + dT (y1 , y2 ). → − → − (5) The inequality ET (T (x1 , y1 ), T (x2 , y2 )) ≥ T (ET (x1 , x2 ), ET (y1 , y2 )) for all x1 , x2 , y1 , y2 ∈ [0, 1] was proved in [39]. Based on this inequality, we can prove that (5) holds in the same manner with (3). 2 By using induction method in (1) and (2) in Proposition 6 we have Corollary 1. Let T be a t-norm with ET being the biresiduation of T . And let dT be the dissimilarity function obtained from ET . If dT is a DF-metric, then for all xi , yi ∈ [0, 1](i = 1, 2, . . . , n) we have (1) dT (

n 

i=1

xi ,

n  i=1

yi ) ≤

n  i=1

dT (xi , yi ).

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.8 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

8

(2) dT (

n 

i=1

xi ,

n  i=1

yi ) ≤

n 

dT (xi , yi ).

i=1

4. Robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures In [8] and [41], Cai and Ying used the concepts of δ-equality and maximum ε-perturbation of fuzzy sets to estimate the robustness of fuzzy connectives and fuzzy reasoning, respectively. Definition 11. (See [8].) Let X be a universe of discourse. Let A and B be two fuzzy subsets of X, A(x) and B(x) their membership functions, respectively. Then A and B are said to be δ-equal, denoted by A = (δ)B, if sup |A(x) − x∈X

B(x)| ≤ 1 − δ, δ ∈ [0, 1].

Definition 12. (See [41].) Let X be a universe of discourse. Let A and B be two fuzzy subsets of X, A(x) and B(x) their membership functions, respectively. If sup |A(x) − B(x)| ≤ ε, then B is called a maximum ε-perturbation of A, x∈X

denoted by A ≡ (ε)B, ε ∈ [0, 1]. Remark 4.

(1) According to the concept of δ-equality, A ≡ (ε)B means that A is (1 − ε)-equal to B, thus the notion of maximum ε-perturbation of fuzzy set is dual to that of δ-equality of fuzzy sets. (2) In Cai’s and Ying’s methods, the Chebyshev distance was used to calculate the divergence degree between fuzzy sets. In [12], the approximate equality of fuzzy sets was extended based on the normalized Minkowski distance. In [40], a so-called finer measure was used to calculate the approximate equality of fuzzy sets. Note that the Chebyshev distance, the normalized Minkowski distance and the so-called finer measure are special cases of divergence measures. In Proposition 2, we have presented a way of constructing divergence measures by means of dissimilarity functions. Let dT be a DF-metric. If dT is used in Proposition 2, then a divergence measure is obtained n  a dT (A(xi ), B(xi )) i=1 , a > 0, b > 0. (1) D(A, B) = n  nb + (a − b) dT (A(xi ), B(xi )) i=1

In order to estimate the robustness of fuzzy connectives and fuzzy reasoning, three problems should be discussed: (1) Let ◦ denote a fuzzy connective. Suppose that A ≡ (ε1 )A1 , B ≡ (ε2 )B1 , then what is the relationship between A ◦ B and A1 ◦ B1 ? (2) Suppose that A1 ≡ (ε1 )A2 , B1 ≡ (ε2 )B2 and A1 ≡ (ε3 )A2 , then what is the relationship between B1 and B2 , where Bk (k = 1, 2) is the CRI solution of the FMP model? (3) Suppose that A1 ≡ (ε1 )A2 , B1 ≡ (ε2 )B2 and B1 ≡ (ε3 )B2 , then what is the relationship between A1 and A2 , where Ak (k = 1, 2) is the CRI solution of the FMT model? Next we extend the concept of perturbations of fuzzy sets on the basis of divergence measures defined as Equation (1) and then discuss the above-mentioned problems. Definition 13. Given a discrete universe X = {x1 , x2 , . . . , xn }. Let dT be a DF-metric and D be a divergence measure defined for all A, B ∈ F (X) as Equation (1). If a D(A, B) =

n 

dT (A(xi ), B(xi ))

i=1

nb + (a − b)

n 

≤ ε, dT (A(xi ), B(xi ))

i=1

where a > 0, b > 0, ε ∈ [0, 1], then B is called an ε-perturbation of A, denoted by A ≈ (ε)B. Example 3. Let X = {x1 , x2 , x3 , x4 } and let A, B and B ∈ F (X) with A = {1, 1, 1, 1}, B = {0.2, 0.2, 0.2, 0.2} and B = {1, 1, 1, 0.2}. It follows from Definition 12 that A ≡ (0.8)B and A ≡ (0.8)B . Although B and B are quite

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.9 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

9

different, they still satisfy the same ε-perturbation with respect to A for ε = 0.8. Let us take dT = dTP and a = b n  |A(xi )−B(xi )| on Equation (1), then we have D(A, B) = n1 max(A(xi ),B(xi )) . According to Definition 13 we have A ≈ (0.8)B and i=1

A ≈ (0.2)B . The results are consistent with the fact that B is close to A while B is far from A. → −

Proposition 7. Let T be a t-norm with T and ET being the residuation and biresiduation of T , respectively. And let dT be the dissimilarity function obtained from ET . Suppose that dT is a DF-metric and D is a divergence measure defined as Equation (1) with a ≥ b > 0. If A ≈ (ε1 )A1 , B ≈ (ε2 )B1 , then we have A ◦ B ≈ ( )A1 ◦ B1 , where → − ◦ ∈ {∪, ∩, T , T } and  = min(

a 2 ε1 + a 2 ε2 + 2abε1 ε2 − 2a 2 ε1 ε2 , 1). 2abε1 ε2 − a 2 ε1 ε2 − b2 ε1 ε2 + a 2

Proof. Since D is defined as Equation (1), we have n  dT (A(xi ), A1 (xi )) a i=1 ≤ ε1 , D(A, A1 ) = n  nb + (a − b) dT (A(xi ), A1 (xi )) i=1

a D(B, B1 ) =

n 

dT (B(xi ), B1 (xi ))

i=1

nb + (a − b)

≤ ε2 .

n 

dT (B(xi ), B1 (xi ))

i=1

If ε1 , ε2 ∈ (0, 1], then we have 1 a − b nb = + ( n D(A, A1 ) a a 

1

)≥

1 > 0, ε1

)≥

1 > 0. ε2

dT (A(xi ), A1 (xi ))

i=1

a − b nb 1 = + ( n D(B, B1 ) a a 

1 dT (B(xi ), B1 (xi ))

i=1

Thus we have n  dT (A(xi ), A1 (xi )) ≤ i=1 n 

dT (B(xi ), B1 (xi )) ≤

i=1

nbε1 , a + bε1 − aε1 nbε2 . a + bε2 − aε2

If ε1 = ε2 = 0, then we have n 

dT (A(xi ), A1 (xi )) = 0 =

nbε1 , a + bε1 − aε1

dT (B(xi ), B1 (xi )) = 0 =

nbε2 . a + bε2 − aε2

i=1 n  i=1

Therefore, we obtain that for all ε1 , ε2 ∈ [0, 1] n  i=1 n  i=1

dT (A(xi ), A1 (xi )) ≤

nbε1 , a + bε1 − aε1

dT (B(xi ), B1 (xi )) ≤

nbε2 . a + bε2 − aε2

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.10 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

10

Note that 1 a − b nb = + ( n D(A ◦ B, A1 ◦ B1 ) a a 

1

).

dT ((A ◦ B)(xi ), (A1 ◦ B1 )(xi ))

i=1

→ −

Since ◦ ∈ {∪, ∩, T , T }, according to Proposition 6 we have dT ((A ◦ B)(xi ), (A1 ◦ B1 )(xi )) = dT (A(xi ) ◦ B(xi ), A1 (xi ) ◦ B1 (xi )) ≤ dT (A(xi ), A1 (xi )) + dT (B(xi ), B1 (xi )), where xi ∈ X. Then we obtain that n 

dT ((A ◦ B)(xi ), (A1 ◦ B1 )(xi )) ≤

i=1

dT (A(xi ), A1 (xi )) +

i=1

≤ If max(ε1 , ε2 ) = 0, then

nbε1 a+bε1 −aε1

+

n 

dT (B(xi ), B1 (xi ))

i=1

nbε2 nbε1 + . a + bε1 − aε1 a + bε2 − aε2

nbε2 a+bε2 −aε2

1 n 

n 

> 0 and thus 1

≥

nbε1 a+bε1 −aε1

dT ((A ◦ B)(xi ), (A1 ◦ B1 )(xi ))

+

nbε2 a+bε2 −aε2

i=1

(a + bε1 − aε1 )(a + bε2 − aε2 ) > 0. nbε1 (a + bε2 − aε2 ) + nbε2 (a + bε1 − aε1 )

= Since a ≥ b > 0, thus we have

1 a − b nb (a + bε1 − aε1 )(a + bε2 − aε2 ) ≥ + ( ) D(A ◦ B, A1 ◦ B1 ) a a nbε1 (a + bε2 − aε2 ) + nbε2 (a + bε1 − aε1 ) a−b (a + bε1 − aε1 )(a + bε2 − aε2 ) = + > 0. a aε1 (a + bε2 − aε2 ) + aε2 (a + bε1 − aε1 ) Therefore, we obtain that D(A ◦ B, A1 ◦ B1 ) ≤ =

1 (a+bε1 −aε1 )(a+bε2 −aε2 ) a−b a + aε1 (a+bε2 −aε2 )+aε2 (a+bε1 −aε1 ) a 2 ε1 + a 2 ε2 + 2abε1 ε2 − 2a 2 ε1 ε2 . 2abε1 ε2 − a 2 ε1 ε2 − b2 ε1 ε2 + a 2

Furthermore, we note that D(A ◦ B, A1 ◦ B1 ) ≤ 1. Therefore, we conclude that D(A ◦ B, A1 ◦ B1 ) ≤ min(

a 2 ε1 + a 2 ε2 + 2abε1 ε2 − 2a 2 ε1 ε2 , 1). 2abε1 ε2 − a 2 ε1 ε2 − b2 ε1 ε2 + a 2

ε1 +a ε2 +2abε1 ε2 −2a ε1 ε2 a ε1 +a ε2 +2abε1 ε2 −2a ε1 ε2 In this case,  = min( a2abε 2 2 2 , 1). If max(ε1 , ε2 ) = 0, then we have 2abε ε −a 2 ε ε −b2 ε ε +a 2 = 0 1 ε2 −a ε1 ε2 −b ε1 ε2 +a 1 2 1 2 1 2 and dT (A(xi ), A1 (xi )) = dT (B(xi ), B1 (xi )) = 0 for all xi ∈ X. Since dT is a DF-metric, we have A = A1 and B = B1 . 2 ε +a 2 ε +2abε ε −2a 2 ε ε a 2 ε1 +a 2 ε2 +2abε1 ε2 −2a 2 ε1 ε2 1 2 1 2 1 2 In this case,  = 0 = a2abε 2 2 2 . Therefore, we conclude that  = min( 2abε ε −a 2 ε ε −b2 ε ε +a 2 , 1) 1 ε2 −a ε1 ε2 −b ε1 ε2 +a 1 2 1 2 1 2 for all ε1 , ε2 ∈ [0, 1]. 2 2

2

2

2

2

2

Proposition 8. Let T be a t-norm with ET being the biresiduation of T and S being the t-conorm which is dual to T . And let dT be the dissimilarity function obtained from ET . Suppose that dT is a DF-metric, D is a divergence measure defined as Equation (1) with a ≥ b > 0 and ET satisfies E8. If A ≈ (ε1 )A1 , B ≈ (ε2 )B1 , then we have S(A, B) ≈ ( )S(A1 , B1 ), where  = min(

a 2 ε1 + a 2 ε2 + 2abε1 ε2 − 2a 2 ε1 ε2 , 1). 2abε1 ε2 − a 2 ε1 ε2 − b2 ε1 ε2 + a 2

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.11 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

Proof. It can be proved in the same manner with Proposition 7.

11

2

Corollary 2. Under the conditions of Propositions 7 and 8, if ε1 = ε2 = ε and A ≈ (ε)A1 , B ≈ (ε)B1 , then we have → − A ◦ B ≈ ( )A1 ◦ B1 , where ◦ ∈ {∪, ∩, T , T , S} and  = min(

2aε , 1). a − bε + aε

Corollary 3. Under the conditions of Propositions 7 and 8, if a = b and A ≈ (ε1 )A1 , B ≈ (ε2 )B1 , then we have → − A ◦ B ≈ (min(ε1 + ε2 , 1))A1 ◦ B1 , where ◦ ∈ {∪, ∩, T , T , S}. Remark 5. If a = b in Equation (1), then D(A, B) =

1 n

B ≈ (ε)B1 , then we have A ◦ B ≈ (min(2ε, 1))A1 ◦ B1 .

n 

dT (A(xi ), B(xi )). Suppose ε1 = ε2 = ε and A ≈ (ε)A1 ,

i=1

As we know, there are two basic inference models in fuzzy reasoning: FMP and FMT models. Let X = {x1 , x2 , . . . , xn } and Y = {y1 , y2 , . . . , ym } be two universes of discourse, then the FMP and FMT models can be written as follows: Antecedent: Fact:

If x is A, then y is B x is A

Conclusion:

y is B

Antecedent: Fact:

If x is A, then y is B y is B

Conclusion:

x is A

where A, A ∈ F (X) and B, B ∈ F (Y ), and x and y are variables valued in X and Y , respectively. In [42–44], Zadeh proposed the CRI method to implement the FMP model in the following way: B (yj ) =

n 

T (A (xi ), I (A(xi ), B(yj ))), xi ∈ X, yj ∈ Y,

i=1

where T and I represent a t-norm and an implication function, respectively. The FMT model is implemented as follows: A (xi ) =

m 

T (B (yj ), I (A(xi ), B(yj ))), xi ∈ X, yj ∈ Y.

j =1 → −

Proposition 9. Let T be a t-norm with T and ET being the residuation and biresiduation of T , respectively. And let dT be the dissimilarity function obtained from ET . Suppose that dT is a DF-metric and D is a diver→ − gence measure defined as Equation (1) with a ≥ b > 0. And suppose that T and T are used in FMP model, i.e., n  → − T (Ak (xi ), T (Ak (xi ), Bk (yj ))) (k = 1, 2) for all xi ∈ X, yj ∈ Y . If A1 ≈ (ε1 )A2 , B1 ≈ (ε2 )B2 and Bk (yj ) = i=1

A1 ≈ (ε3 )A2 , then we have B1 ≈ ( )B2 , where  = min( a−b a

1 +

1 an (

1

ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

)

, 1).

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.12 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

12

Proof. Since D is defined as Equation (1), we have a

n 

dT (A1 (xi ), A2 (xi ))

i=1

D(A1 , A2 ) =

n 

nb + (a − b)

≤ ε1 , dT (A1 (xi ), A2 (xi ))

i=1

a

m  j =1

D(B1 , B2 ) =

dT (B1 (yj ), B2 (yj ))

mb + (a − b) a



D(A1 , A2 ) =

n 

≤ ε2 ,

m  j =1

dT (B1 (yj ), B2 (yj ))

dT (A1 (xi ), A2 (xi ))

i=1

nb + (a − b)

n 

dT (A1

(x

i ), A2

(x

≤ ε3 . i ))

i=1

If ε1 , ε2 , ε3 ∈ (0, 1], then we have 1 a − b nb = + ( n D(A1 , A2 ) a a 

1

)≥

dT (A1 (xi ), A2 (xi ))

1 > 0, ε1

i=1

a − b mb 1 = + ( m D(B1 , B2 ) a a  j =1

a − b nb 1 = + ( n D(A1 , A2 ) a a 

1

1 dT (A1 (xi ), A2 (xi ))

Thus we have dT (A1 (xi ), A2 (xi )) ≤

nbε1 , a + bε1 − aε1

dT (B1 (yj ), B2 (yj )) ≤

mbε2 , a + bε2 − aε2

dT (A1 (xi ), A2 (xi )) ≤

nbε3 . a + bε3 − aε3

i=1 m  j =1 n  i=1

If ε1 = ε2 = ε3 = 0, then we have n 

dT (A1 (xi ), A2 (xi )) = 0 =

nbε1 , a + bε1 − aε1

dT (B1 (yj ), B2 (yj )) = 0 =

mbε2 , a + bε2 − aε2

dT (A1 (xi ), A2 (xi )) = 0 =

nbε3 . a + bε3 − aε3

i=1 m  j =1 n  i=1

1 > 0, ε2

)≥

1 > 0. ε3

dT (B1 (yj ), B2 (yj ))

i=1

n 

)≥

Therefore, we obtain that for all ε1 , ε2 , ε3 ∈ [0, 1]

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.13 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–••• n 

dT (A1 (xi ), A2 (xi )) ≤

nbε1 , a + bε1 − aε1

dT (B1 (yj ), B2 (yj )) ≤

mbε2 , a + bε2 − aε2

dT (A1 (xi ), A2 (xi )) ≤

nbε3 . a + bε3 − aε3

i=1 m  j =1 n  i=1

Note that 1 a − b mb = + ( m D(B1 , B2 ) a a  j =1

1

),

dT (B1 (yj ), B2 (yj ))

and m 

dT (B1 (yj ), B2 (yj ))

j =1

=

m 

dT (

j =1

n 

→ −

T (A1 (xi ), T (A1 (xi ), B1 (yj ))),

i=1

n 

→ −

T (A2 (xi ), T (A2 (xi ), B2 (yj )))).

i=1

According to Proposition 6 and Corollary 1 we have m 

dT (

j =1

≤ ≤ ≤ =

n 

→ −

T (A1 (xi ), T (A1 (xi ), B1 (yj ))),

i=1 n m 

n 

→ −

T (A2 (xi ), T (A2 (xi ), B2 (yj ))))

i=1 → −

→ −

dT (T (A1 (xi ), T (A1 (xi ), B1 (yj ))), T (A2 (xi ), T (A2 (xi ), B2 (yj ))))

j =1 i=1 n m  

→ −

→ −

(dT (A1 (xi ), A2 (xi )) + dT (T (A1 (xi ), B1 (yj )), T (A2 (xi ), B2 (yj ))))

j =1 i=1 n m  

(dT (A1 (xi ), A2 (xi )) + dT (A1 (xi ), A2 (xi )) + dT (B1 (yj ), B2 (yj )))

j =1 i=1 m  n 

(

dT (A1 (xi ), A2 (xi )) +

n 

j =1 i=1 n 

i=1 n 

i=1

i=1

dT (A1 (xi ), A2 (xi )) + m

=m

dT (A1 (xi ), A2 (xi )) +

n 

dT (B1 (yj ), B2 (yj ))) i=1 m 

dT (A1 (xi ), A2 (xi )) + n

dT (B1 (yj ), B2 (yj ))

j =1

mnbε1 mnbε2 mnbε3 + + a + bε3 − aε3 a + bε1 − aε1 a + bε2 − aε2 ε3 ε1 ε2 = mnb( + + ). a + bε3 − aε3 a + bε1 − aε1 a + bε2 − aε2 ≤

If max(ε1 , ε2 , ε3 ) = 0, then mnb( a+bεε33−aε3 + 1 m  j =1

dT (B1 (yj ), B2 (yj ))

Since a ≥ b > 0, thus we have

≥

ε1 a+bε1 −aε1

+

ε2 a+bε2 −aε2 ) > 0

and thus

1 mnb( a+bεε33−aε3

+

ε1 a+bε1 −aε1

+

ε2 a+bε2 −aε2 )

> 0.

13

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.14 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

14

1 a − b mb = + ( m D(B1 , B2 ) a a  j =1

1

)

dT (B1 (yj ), B2 (yj ))

1 a−b + ( ≥ a an a+bεε33−aε3 +

1 ε1 a+bε1 −aε1

ε2 a+bε2 −aε2

+

) > 0.

Therefore, we obtain that D(B1 , B2 ) ≤

1 a−b a

+

1 an (

1

)

ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

.

Furthermore, we note that D(B1 , B2 ) ≤ 1. Therefore, we have D(B1 , B2 ) ≤ min( a−b a

In this case,  = min( a−b a

+

1

ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

1 1 + an (

1 a−b 1 a + an (

1 1 an (

1 ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

1 ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

)

ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

)

)

)

, 1).

, 1). Meanwhile, if max(ε1 , ε2 , ε3 ) = 0, then we have

= 0 and dT (A1 (xi ), A2 (xi )) = dT (B1 (yj ), B2 (yj )) = dT (A1 (xi ), A2 (xi )) = 0

for all xi ∈ X, yj ∈ Y . Since dT is a DF-metric, we have A1 = A2 , B1 = B2 and A1 = A2 . In this case,  = 0 = 1 1 . Therefore, we conclude that  = min( a−b 1 , 1) a−b 1 1 1 a

+ an (

for all ε1 , ε2 , ε3 ∈ [0, 1].

a

2

+ an (

ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

)

Corollary 4. Under the conditions of Proposition 9, if ε1 = ε2 = ε3 = ε and A1 ≈ (ε)A2 , B1 ≈ (ε)B2 , A1 ≈ (ε)A2 , then we have B1 ≈ ( )B2 , where  = min(

3naε , 1). (3na − 3nb + b − a)ε + a

Corollary 5. Under the conditions of Proposition 9, if a = b and A1 ≈ (ε1 )A2 , B1 ≈ (ε2 )B2 , A1 ≈ (ε3 )A2 , then we have B1 ≈ (min(nε1 + nε2 + nε3 , 1))B2 . Remark 6. If a = b in Equation (1), then D(A, B) =

1 n

n 

dT (A(xi ), B(xi )). Suppose ε1 = ε2 = ε3 = ε and A1 ≈

i=1

(ε)A2 , B1 ≈ (ε)B2 , A1 ≈ (ε)A2 , then we have B1 ≈ (min(3nε, 1))B2 . → −

Proposition 10. Let T be a t-norm with T and ET being the residuation and biresiduation of T , respectively. And let dT be the dissimilarity function obtained from ET . Suppose that dT is a DF-metric and D is a diver→ − gence measure defined as Equation (1) with a ≥ b > 0. And suppose that T and T are used in FMT model, i.e., m  → − T (Bk (yj ), T (Ak (xi ), Bk (yj ))) (k = 1, 2) for all xi ∈ X, yj ∈ Y . If A1 ≈ (ε1 )A2 , B1 ≈ (ε2 )B2 and Ak (xi ) = j =1

B1 ≈ (ε3 )B2 , then we have A1 ≈ ( )A2 , where  = min( a−b a

1 +

1 am (

1

ε3 ε1 ε2 a+bε3 −aε3 + a+bε1 −aε1 + a+bε2 −aε2

)

, 1).

Proof. It can be proved in the same manner with Proposition 9.

2

Corollary 6. Under the conditions of Proposition 10, if ε1 = ε2 = ε3 = ε and A1 ≈ (ε)A2 , B1 ≈ (ε)B2 , B1 ≈ (ε)B2 , then we have A1 ≈ ( )A2 , where

JID:FSS AID:6853 /FLA

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.15 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

 = min(

15

3maε , 1). (3ma − 3mb + b − a)ε + a

Corollary 7. Under the conditions of Proposition 10, if a = b and A1 ≈ (ε1 )A2 , B1 ≈ (ε2 )B2 , B1 ≈ (ε3 )B2 , then we have A1 ≈ (min(mε1 + mε2 + mε3 , 1))A2 . Remark 7. If a = b in Equation (1), then D(A, B) =

1 n

n 

dT (A(xi ), B(xi )). Suppose ε1 = ε2 = ε3 = ε and A1 ≈

i=1

(ε)A2 , B1 ≈ (ε)B2 , B1 ≈ (ε)B2 , then we have A1 ≈ (min(3mε, 1))A2 . 5. Conclusions Fuzzy reasoning deals with gradual properties of variables represented by fuzzy sets. When they are provided by experts, their membership functions may be not precisely defined. The same expert can sometimes use a membership function A or another A that is a slight modification of the former for a given variable [15]. It is therefore reasonable to require that fuzzy reasoning results should not change much for a slight change in inputs. Thus the robustness of fuzzy reasoning has become a particular focus in fuzzy research. By comparing and analyzing lots of existing works about robustness of fuzzy reasoning, we find out that the differences between them are largely due to the underlying similarity measures or divergence measures that are adopted to calculate the perturbation parameters between fuzzy sets. The Chebyshev distance is a divergence measure that has been most widely used. Many fuzzy reasoning scheme have been proved to be robust based on the Chebyshev distance. However, we do not know whether they are still robust based on a more complex divergence measure. Therefore, unlike lots of existing works about robustness of fuzzy reasoning, our work uses a more general and complex divergence measure to calculate the perturbation parameters between fuzzy sets. In this paper we presented the concept of DF-metric and introduced some DF-metrics that differ from those proposed in [40]. We analyzed properties of DF-metrics through several inequalities about them. Since DF-metrics are special dissimilarity functions, we used them to construct divergence measures and obtained a new computational formula of divergence measures. It is shown that the divergence measures that were used to appraise the robustness of fuzzy reasoning in [40] are special cases of our proposed computational formula of divergence measures. In this way Wang and Duan’s work [40] becomes special cases of the results presented in this paper. Based on the proposed divergence measures, we obtained an extension of the concept of perturbations of fuzzy sets. According to the extended concept, the perturbation parameters raised by various fuzzy connectives were studied and the perturbations of fuzzy reasoning were also discussed. Those perturbation parameters could help formulate certain criteria useful in selecting some specific fuzzy connectives in fuzzy reasoning. The obtained results also provide us with criteria to judge whether a mathematical fuzzy reasoning scheme is robust or perturbation resistant against the deviation of human expertise from its corresponding mathematically quantitative representations. Acknowledgements The authors are very grateful to the Editor and referees, for their constructive comments and useful suggestions that helped us to improve the quality of the paper. This research is supported by the National Natural Science Foundation of China (Grant Nos. 61305074, 61473239, 71301129, 61175044, 61175055 and 61100046), the 2013 Cultivation Program for the Excellent Doctoral Dissertation of Southwest Jiaotong University. References [1] [2] [3] [4]

B. Bouchon-Meunier, M. Rifqi, S. Bothorel, Towards general measures of comparison of objects, Fuzzy Sets Syst. 84 (2) (1996) 143–153. H. Bustince, P. Burillo, F. Soria, Automorphisms, negations and implication operators, Fuzzy Sets Syst. 134 (2) (2003) 209–229. H. Bustince, E. Barrenechea, M. Pagola, Restricted equivalence functions, Fuzzy Sets Syst. 157 (17) (2006) 2333–2346. H. Bustince, M. Pagola, E. Barrenechea, Construction of fuzzy indices from fuzzy DI-subsethood measures: application to the global comparison of images, Inf. Sci. 177 (3) (2007) 906–929. [5] H. Bustince, E. Barrenechea, M. Pagola, Image thresholding using restricted equivalence functions and maximizing the measures of similarity, Fuzzy Sets Syst. 158 (5) (2007) 496–516.

JID:FSS AID:6853 /FLA

16

[m3SC+; v1.211; Prn:24/07/2015; 13:32] P.16 (1-16)

Y. Li et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

[6] H. Bustince, E. Barrenechea, M. Pagola, Relationship between restricted dissimilarity functions, restricted equivalence functions and normal EN -functions: image thresholding invariant, Pattern Recognit. Lett. 29 (4) (2008) 525–536. [7] H. Bustince, J. Fernandez, J. Sanz, M. Baczy´nski, R. Mesiar, Construction of strong equality index from implication operators, Fuzzy Sets Syst. 211 (2013) 15–33. [8] K.Y. Cai, δ-Equalities of fuzzy sets, Fuzzy Sets Syst. 76 (1) (1995) 97–112. [9] K.Y. Cai, Robustness of fuzzy reasoning and δ-equalities of fuzzy sets, IEEE Trans. Fuzzy Syst. 9 (5) (2001) 738–750. [10] K.Y. Cai, L. Zhang, Fuzzy reasoning as a control problem, IEEE Trans. Fuzzy Syst. 16 (3) (2008) 600–614. [11] G. Cheng, Y. Fu, Error estimation of perturbations under CRI, IEEE Trans. Fuzzy Syst. 14 (6) (2006) 709–715. [12] S.S. Dai, D.W. Pei, S.M. Wang, Perturbation of fuzzy sets and fuzzy reasoning based on normalized Minkowski distance, Fuzzy Sets Syst. 189 (1) (2011) 63–73. [13] J. Diatta, Description-meet compatible multiway dissimilarities, Discrete Appl. Math. 154 (3) (2006) 493–507. [14] D. Dubois, H. Prade, Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions, Fuzzy Sets Syst. 40 (1) (1991) 143–202. [15] D. Dubois, H. Prade, Gradualness, uncertainty and bipolarity: making sense of fuzzy sets, Fuzzy Sets Syst. 192 (2012) 3–24. [16] J.L. Fan, W.X. Xie, Distance measure and induced fuzzy entropy, Fuzzy Sets Syst. 104 (2) (1999) 305–314. [17] J.C. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht, 1994. [18] I. Georgescu, (δ, H )-Equality of fuzzy sets, J. Mult.-Valued Log. Soft Comput. 14 (1) (2008) 1–32. [19] S. Gottwald, On t-norms which are related to distances of fuzzy sets, Busefal 50 (1992), online update at http://www.listic.univ-savoie.org/ busefal/Papers/50.zip/50_04.pdf. [20] J.W. Guan, D.A. Bell, Approximate reasoning and evidence theory, Inf. Sci. 96 (3) (1997) 207–235. [21] D.H. Hong, S.Y. Hwang, A note on the value similarity of fuzzy systems variables, Fuzzy Sets Syst. 66 (3) (1994) 383–386. [22] W.H. Hsiao, S.M. Chen, C.H. Lee, A new interpolative reasoning method in sparse rule-based systems, Fuzzy Sets Syst. 93 (1) (1998) 17–22. [23] J. Jacas, J. Recasens, Maps and isometries between indistinguishability operators, Soft Comput. 6 (1) (2002) 14–20. [24] J.H. Jin, Y.M. Li, C.Q. Li, Robustness of fuzzy reasoning via logically equivalence measure, Inf. Sci. 177 (22) (2007) 5103–5117. [25] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. [26] E.S. Lee, Q.G. Zhu, Fuzzy and Evidence Reasoning, Springer, Heidelberg, 1995. [27] Y.M. Li, D.C. Li, W. Pedrycz, J.J. Wu, An approach to measure the robustness of fuzzy reasoning, Int. J. Intell. Syst. 20 (4) (2005) 393–413. [28] Y.F. Li, K.Y. Qin, X.X. He, Robustness of fuzzy connectives and fuzzy reasoning, Fuzzy Sets Syst. 225 (2013) 93–105. [29] Y.F. Li, K.Y. Qin, X.X. He, Some new approaches to constructing similarity measures, Fuzzy Sets Syst. 234 (2014) 46–60. [30] Y.F. Li, K.Y. Qin, X.X. He, Dissimilarity functions and divergence measures between fuzzy sets, Inf. Sci. 288 (2014) 15–26. [31] X.C. Liu, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets Syst. 52 (3) (1992) 305–318. [32] Y. Liu, E.E. Kerre, An overview of fuzzy quantifiers (II): reasoning and applications, Fuzzy Sets Syst. 95 (1) (1998) 135–146. [33] S. Montes, P. Miranda, J. Jiménez, P. Gil, A measure of the difference between two fuzzy partitions, Tatra Mt. Math. Publ. 21 (133) (2001) 133–152. [34] S. Montes, I. Couso, P. Gil, C. Bertoluzza, Divergence measure between fuzzy sets, Int. J. Approx. Reason. 30 (2) (2002) 91–105. [35] H.T. Nguyen, V. Kreinovich, D. Tolbert, On robustness of fuzzy logics, in: Proceedings of the 1993 IEEE International Conference on Fuzzy Systems, vol. 1, FUZZ-IEEE’93, San Francisco, California, 1993, pp. 543–547. [36] C.P. Pappis, Value approximation of fuzzy systems variables, Fuzzy Sets Syst. 39 (1) (1991) 111–115. [37] J. Recasens, Indistinguishability Operators—Modelling Fuzzy Equalities and Fuzzy Equivalence Relations, Springer, Heidelberg, 2010. [38] W.A. Sutherland, Introduction to Metric and Topological Spaces, Oxford University Press, 1975. [39] G.J. Wang, Nonclassical Mathematical Logic and Approximate Reasoning, Science in China Press, Beijing, 2000 (in Chinese). [40] G.J. Wang, J.Y. Duan, On robustness of the full implication triple I method with respect to finer measurements, Int. J. Approx. Reason. 55 (3) (2014) 787–796. [41] M.S. Ying, Perturbation of fuzzy reasoning, IEEE Trans. Fuzzy Syst. 7 (5) (1999) 625–629. [42] L.A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning, I, Inf. Sci. 8 (3) (1975) 199–249. [43] L.A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning, II, Inf. Sci. 8 (4) (1975) 301–357. [44] L.A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning, III, Inf. Sci. 9 (1) (1975) 43–80. [45] L. Zhang, K.Y. Cai, Optimal fuzzy reasoning and its robustness analysis, Int. J. Intell. Syst. 19 (11) (2004) 1033–1049.