Similarity and dissimilarity measures between fuzzy sets: a formal

Similarity and dissimilarity measures between fuzzy sets: a formal relational study In´es Couso∗ Dep. of Statistics and O.R., University of Oviedo, Spain

Laura Garrido∗ Dep. of Statistics and O.R., University of Oviedo, Spain

Luciano S´anchez∗ Dep. of Computer Sciences, University of Oviedo, Spain

Abstract The paper deals with the well-known notion of (dis)similarity measures between fuzzy sets. We provide three separate lists of axioms that fit with the respective notions of “general comparison measure”, “similarity measure” and “dissimilarity measure”. Then we review some of the most important axiomatic definitions of (dis)similarity measures in the literature, by referring to the axioms in those lists satisfied by each specific definition. This common framework will make our study about the formal relationships among different axiomatic definitions much easier: some of them, which are apparently different, do in fact share many commonalities. We provide a self-contained picture of these relationships, by providing formal results and counterexamples that reflect which of the (dis)similarity definitions in the literature are connected by implication relations and which of them are not. We finalize the paper with an in-depth study about the notion of “duality” between similarity and dissimilarity measures as well as with some concluding remarks. Keywords.- Fuzzy set; similarity measure; divergence measure; dissimilarity ∗ Corresponding

author Email addresses: [email protected] (In´ es Couso), [email protected] (Laura Garrido), [email protected] (Luciano S´ anchez)

Preprint submitted to Elsevier

December 7, 2012

measure; comparison measure.

1. Introduction The notion of “similarity” and “proximity” are common terms in classical Set Theory, as well as in Statistics. Wang [31], transferred the notion to Fuzzy Sets theory in 1983, providing a specific formula to compute the similarity between two fuzzy sets. Later on, in the nineties, some related definitions such as “similarity”, “similitude”, “proximity” or “resemblance” were proposed, as well as some other dual definitions such as “dissimilarity”, “dissimilitude”, “divergence” or “distance”. All of the above kinds of measures assign a quantity (sometimes within a fixed interval, such as the unit interval) to each pair of fuzzy subsets (or, more generally, to each pair of interval-valued, hesitant or intuitionistic fuzzy sets). Such a quantity represents the degree of equality or difference between the compared (generalized) fuzzy sets. We can divide the literature about this kind of measures into two main blocks: one where we can find some papers introducing axiomatic definitions (see [5, 7, 11, 17, 20, 25, 29, 38, 39, 41, 42], for instance). The other where we can find a great number of works introducing or reviewing some (parametric families of) measures (see [1, 2, 3, 6, 8, 9, 10, 13, 14, 15, 16, 19, 26, 30, 36, 37, 32, 33, 34, 35, 40, 43, 44], among others) most of the times applied to specific subsequent problems. We will focus on the former (those ones devoted to axiomatic definitions). Most of the axiomatic definitions have been introduced independently from each other. They have been proposed in different contexts, with different purposes and using different nomenclatures. To the best of our knowledge, there are no formal studies relating different axiomatic definitions of “similarity” between fuzzy sets. However, some very specific results relating different axioms of “inequality” have been provided in the literature. These formal relations will be reviewed in Section 4. Our main goal in this paper is to present all these axioms in a uniform framework and show the formal relationships among them, in order to provide the reader with an overall picture. We will check some implication

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relations among different lists of axioms and we will also find some counterexamples showing the lack of implications in some other cases. These results will be displayed by means of diagrams and tables. The rest of the paper will be organized as follows. In Section 2, we will list the axioms for different kinds of comparison measures considered in the forthcoming sections. We will divide the list of axioms into three blocks (one on general comparison measures, the second on measures of “equality” or “similarity” and the third block on measures of “inequality”.) In Sections 3 and 4, we will analyze the formal relationships among different axiomatic definitions of “equality” and “inequality” between fuzzy sets, respectively. In Section 5, we will discuss some issues regarding the notions of “equivalence” and “duality” of different comparison measures. The paper will end with some concluding remarks and some open problems. 2. Axioms lists In this section, we provide the necessary background. Let us consider an arbitrary referential set or universe U . Let ℘(U ) denote the power set of U (the family of crisp subsets of U ) and F(U ), the family of fuzzy subsets of U . For the sake of clarity, we will use different notations for a fuzzy set A and for its membership function, µA . We will denote by A ∩ B and A ∪ B, respectively, the intersection and the union of A and B according to the minimum T-norm and the maximum T-conorm, i.e., the fuzzy sets whose membership functions are respectively expressed as µA∩B = min{µA , µB } and µA∪B = max{µA , µB }. A \ B will denote the difference between A and B. As mentioned in the Introduction, we will survey a large collection of axiomatic definitions from the literature regarding the notions of “equality” and “inequality” or “difference” between fuzzy sets, and we will study their formal relations. In order to list all these definitions in a compact way, we will first provide three separate lists of properties: the properties included in the first list are general axioms that may be required in any kind of comparison measure between fuzzy sets. The properties in the second list concern the notion of 3

“equality” between fuzzy sets, and they appear in the axiomatic definitions of similarity, resemblance, and proximity. The third list is related to the notion of “difference” or “inequality”. The properties in this list appear in the definitions of divergence, distance and dissimilarity. The specific nomenclature we have chosen to name all the properties listed below will allow us to simplify our analysis about the formal relationships between different axioms. As a convention, the asterisk will be understood as “stronger than”. Some implication relations are straightforward (such as, for instance, G1*⇒G1). Other implication relations such as S2*⇒S2 or S5*⇒S5 will be proved in Section 3. Conversely, Axiom S1*-var is just a variant of S1* (not stronger, but somehow related). 2.1. General properties • G1.- 0 ≤ m(A, B) ≤ 1 • G1*.- 0 ≤ m(A, B) ≤ 1

∀A, B ∈ F(U ). ∀A, B ∈ F(U ) and there exists a pair of fuzzy

sets C, D ∈ F(U ) such that m(C, D) = 1. • G2.- m(A, B) = m(B, A) ∀A, B ∈ F(U ). • G3.- Suppose U finite and let ρ : U → U be a permutation. For each A ∈ F(U ) denote Aρ the fuzzy set whose membership function is defined from µA as follows: µAρ (x) = µA (ρ(x)). Then m(A, B) = m(Aρ , B ρ ). • G3*.- U is finite and there exists h : [0, 1]×[0, 1] → R such that m(A, B) = P x∈U h[µA (x), µB (x)]. • G4.- There exists a mapping f : F(U ) → R such that m(A, B) = f (A ∩ B, A \ B, B \ A). • G4*.- There exists a function Fm : R3 → R and a fuzzy measure1 M : F(U ) → R such that, for all A, B ∈ F(U ), m(A, B) can be written 1 Several formal definitions of this notion can be found in the literature. Here, we will refer to a monotone increasing set-function satisfying the restriction M (∅) = 0.

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as follows: m(A, B) = Fm (M (A ∩ B), M (A \ B), M (B \ A)). • G5: If A∩B = ∅, A0 ∩B 0 = ∅, m(A, ∅) ≤ m(A0 , ∅) and m(B, ∅) ≤ m(B 0 , ∅), then m(A, B) ≤ m(A, B 0 ). 2.2. Axioms for measures of “equality” • S1.- ∀A, B, C ∈ F(U ), if A ⊆ B ⊆ C and maxx∈U µA (x) = maxx∈U µB (x) then s(A, C) ≤ s(A, B). • S1*.- ∀A, B, C ∈ F(U ), A ⊆ B ⊆ C ⇒ s(A, C) ≤ s(A, B) • S1*-var.-∀A, B, C ∈ F(U ), A ( B ( C ⇒ s(A, C) < s(A, B) • S2.- ∀A, B, C ∈ F(U ), A ⊆ B ⊆ C ⇒ s(A, C) ≤ s(B, C). • S2*.- If A, B, C ∈ F(U ) satisfy: – µA (x0 ) < µB (x0 ) ≤ µC (x0 ), for some x0 ∈ U , and – µB (x) = µA (x), ∀ x ∈ U, x 6= x0 , then s(A, C) < s(B, C). • S3.- ∀D ∈ ℘(U ), s(D, Dc ) = 0.

• S3*.- s(D, Dc ) = 0 ⇐⇒ D ∈ ℘(U ).

• S4.- s(C, C) = maxA,B∈F (U ) s(A, B) ∀C ∈ F(U ). 5

• S4*.- C = D ⇐⇒ s(C, D) = maxA,B∈F (U ) s(A, B). • S5.- s(A, B) = 0 ⇒ A ∩ B = ∅. • S5*.- Consider A, B ∈ F(U ) and an arbitrary x0 ∈ U . Define C and D such that: – µC (x) = µA (x) and µD (x) = µB (x), ∀ x 6= x0 – µC (x0 ) = µA (x0 ) + α and µD (x0 ) = µB (x0 ) + α, where 0 ≤ α ≤ 1 − max{µA (x0 ), µB (x0 )} Then: – If maxx∈U µC∩D (x) = maxx∈U µA∩B (x) then s(C, D) = s(A, B). – If maxx∈U µC∩D (x) > maxx∈U µA∩B (x) then s(C, D) > s(A, B). • S6.- Consider A, B ∈ F(U ) and an arbitrary x0 ∈ U . Define C and D such that: – µC (x) = µA (x) and µD (x) = µB (x), ∀ x 6= x0 – µC (x0 ) = µA (x0 ) + α and µD (x0 ) = µB (x) + α. Then, s(A, C) = s(B, D). • S7.- If A, B, C, D ∈ F(U ) satisfy the following restrictions: A∩B ⊆ C ∩D, A \ B ⊇ C \ D, and B \ A ⊇ D \ C then s(A, B) ≤ s(C, D). 2.3. Axioms for measures of “inequality” or “difference” • D1.- If A, B, C ∈ F(U ) satisfy the restrictions A ⊆ B ⊆ C, then: (a) d(A, C) ≥ d(A, B) and (b) d(A, C) ≥ d(B, C). • D2.- d(D, Dc ) = maxA,B∈F (U ) d(A, B) ∀D ∈ ℘(U ). • D3.- A = B =⇒ d(A, B) = 0. • D3*.- A = B ⇐⇒ d(A, B) = 0. 6

• D4.- ∀A, B, C ∈ F(U ), d(A ∩ C, B ∩ C) ≤ d(A, B). • D5.- ∀A, B, C ∈ F(U ), d(A ∪ C, B ∪ C) ≤ d(A, B). • D6.- If A, B, C, D ∈ F(U ) satisfy the following restrictions: A∩B ⊆ C ∩D, A \ B ⊇ C \ D, and B \ A ⊇ D \ C then d(A, B) ≥ d(C, D). • D7.- d(A, B) = d(A \ B, B \ A). 3. Measures of “equality” 3.1. Axiomatic definitions of similarity measures Now we will recall the most relevant axiomatic definitions of similarity measures in the literature. Let the reader notice that we have modified the presentation of each definition taken from the literature, in order to unify the nomenclature. For instance, in most definitions of similarity in the literature, S1* and S2 are jointly presented as a single axiom. As another example, many axiomatic definitions for similarities require conditions G1 and s(A, A) = 1 for any A ∈ F(U ). For the sake of unity, we will express it equivalently here by saying that the similarity measure must satisfy G1* and S4. Something similar happens with Axioms D4 and D5, which are expressed as a single one in [25], as a requirement for divergence measures. We must also mention that some of the original definitions have been introduced in more general contexts (intervalvalued fuzzy sets, intuitionistic fuzzy sets). In order to perform a comprehensive comparison, we will restrict their domain to the family of pairs of fuzzy sets. Definition 1. ([39]) s is called a similarity measure, if it satisfies G2, S1*, S2, S3 and S4. Definition 2. ([11]) s(A, B) is said to be the degree of similarity between A ∈ F(U ) and B ∈ F(U ) if s satisfies G1*, G2, S1*, S2 and S4. Definition 3. ([20, 22] ) s(A, B) is said to be the degree of similarity between A ∈ F(U ) and B ∈ F(U ) if s satisfies G1*, G2, S1*, S2 and S4*.

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Remark 3.1. Definition 3 is just a stronger variant of Definition 2, where S4 is replaced by S4*. (The equality between two fuzzy sets is required in [20, 22], but not in [11], in order to achieve maximum similarity.) Definition 4. ([41]) s is called a similarity measure when it satisfies the properties G1*, G2, S1*, S2, S3 and S4*. Definition 5. ([38]) s is called a f-near-degree, if it satisfies G1*, G2, S1*, S2, S4* and S5. Remark 3.2. A function on F(U ) × F(U ) satisfying Definition 5 is called a similarity measure in [23]. Definition 6. ([42]) s is called a similarity measure, if it fulfills G1*, G2, S1*, S2, S3* and S4*. Definition 7. ([17]) A possibilistic similarity measure should satisfy Axioms G1*, G2, G3, S1, S3 and S4. Furthermore, the following properties would be desirable: S1*-var, S2*, S5* and S6. Remark 3.3. The notion of possibilistic similarity measure introduced in [17] was originally restricted to normalized fuzzy sets, as they were interpreted as possibility distributions. In this section, we are just considering formal mathematical relationships between different measures of similarity, so we have not taken this restriction into account. 3.2. Formal relationships between different measures of “equality” To the best of our knowledge, there are no studies in the literature about the formal relationships among the axiomatic definitions considered above. Our aim in this section is to develop a systematic study about them.First, we will display Definitions 1 to 7 in a compact manner, in order to simplify our study. Those definitions can be summarized as follows: Now we will check some implications between axioms or groups of axioms listed in Subsections 2.1 and 2.2, that will be used later on to show the formal 8

Definition Def. 1 Def. 2 Def. 3 Def. 4 Def. 5 Def. 6 Def. 7 (nec.) Def. 7 (desir.)

G1*

G2 • • • • • • • •

• • • • • • •

G3

• •

Table 1: Definitions of similarity (Definitions from 1 to 7), first part

Definition Def. 1 Def. 2 Def. 3 Def. 4 Def. 5 Def. 6 Def. 7 (nec.) Def. 7 (desir.)

S1

S1* • • • • • •

• •

S1*-var

S2 • • • • • •

•

S2*

S3 •

S3*

S4 • •

• • • •

• • •

• •

S4*

• •

S5

S5*

S7

•

•

•

Table 2: Definitions of similarity (Definitions from 1 to 7), second part

relations between the different variants of the notion of “similarity” proposed in Definitions 1 to 7. With respect to the general properties (listed in Subsection 2.1), we straightforwardly see that G1* implies G1. Furthermore, by the commutativity of the sum, we also observe that G3* implies G3. Regarding the properties listed in Subsection 2.2 (axioms of “equality”), we easily observe that S1* implies S1, S3* implies S3 and S4* implies S4. Now we will check some other less evident implication relations: Proposition 1. If s : F(U ) × F(U ) → R satisfies properties S1*-var and S4, then it fulfills S1*. Proof: Let A ⊆ B ⊆ C be three arbitrary nested fuzzy sets, and suppose that s satisfies S1*-var and S4. If A ( B ( C, then s(A, C) ≤ s(A, B), according to S1*-var. Otherwise:

9

• If A = B, then, according to S4: s(A, B) =

max

s(D, E) ≥ s(A, C).

D,E∈F (U )

• If A 6= B then B must be equal to C, and therefore s(A, C) = s(A, B). Thus we have checked that s(A, C) ≤ s(A, B) in any case, and so s satisfies S1*.  Proposition 2. S2* implies S2. Proof: Let A ⊆ B ⊆ C be three arbitrary nested fuzzy sets and assume that s satisfies property S2∗ . Let k denote the number of indices i such that µA (xi ) is strictly lower than µB (xi ), and assume that k ≥ 1 (otherwise, the result is straightforward). Thus, let us suppose that • µA (xi(j) ) = µB (xi(j) ), ∀ j = 1, . . . , n − k and • µA (xi(j) ) < µB (xi(j) ), ∀ j = n − k + 1, . . . , n. Now, let us define the following finite sequence of fuzzy sets: • A0 = A • A1 is the fuzzy set with the membership function µA1 (xi(j) ) = µA0 (xi(j) ), ∀ j ≤ n − 1 and µA1 (xi(n) ) = µB (xi(n) ), and, in general: • µAr+1 (xi(j) ) = µAr (xi(j) ), ∀ j 6= n − r, µAr+1 (xi(n−r) ) = µB (xi(n−r) ), ∀ r = 0, . . . , k − 1. We can easily check that: • Ai ⊆ Ai+1 , ∀ i = 0, . . . , k − 1 • Ak−1 = B • Ai , Ai+1 and C satisfy the properties required in S2* to a triple of sets. Therefore, according to S2*, we obtain the following finite chain of inequalities: s(A, C) = s(A0 , C) ≤ s(A1 , C) ≤ . . . ≤ s(Ar−1 , C) = s(B, C). 10



Proposition 3. If s ≥ 0 satisfies S5*, then it fulfills S5. Proof: Let us consider a pair of fuzzy sets A, B ∈ F(U ) such that A∩B 6= ∅ and let s satisfy property S5*. To check the result, it suffices to find a different pair of fuzzy sets C and D such that s(C, D) < s(A, B). Let us denote by α the maximum α = maxx∈U µA∩B (x). According to the above hypotheses, it is strictly positive. Let us suppose that this membership value is attained in k elements xi1 , . . . , xik , with k ≥ 1. Let us now consider a pair of finite sequences of fuzzy sets A0 , . . . , Ak+1 and B0 , . . . , Bk+1 as follows: • A0 = A and B0 = B • µAj (xij ) = µAj−1 (xij ) − α, ∀ j ∈ {1, . . . , k} and µAj (xik ) = µAj−1 (xik ), ∀ k 6= j and all j ∈ {1, . . . , k} • µBj (xij ) = µBj−1 (xij ) − α, ∀ j ∈ {1, . . . , k} and µBj (xik ) = µBj−1 (xik ), ∀ k 6= j and all j ∈ {1, . . . , k} Then, according to property S5*, we observe that: • s(Aj , Bj ) = s(Aj−1 , Bj−1 ), ∀ j = 1, . . . , k − 1 • s(Ak , Bk ) < s(Ak−1 , Bk−1 ) Therefore, we obtain the following strict inequality: s(Ak , Bk ) < s(A, B).



Proposition 4. If s : F(U ) × F(U ) → R satisfies S3 and S5, then it fulfills S3*. Proof: Let us suppose that s satisfies S3 and S5. To check that it satisfies S3*, we just need to prove the implication s(D, Dc ) = 0 ⇒ D ∈ ℘(U ). According to S5, s(D, Dc ) = 0 implies that min{µD (x), µDc (x)} = min{µD (x), 1 − µD (x)} = 0, ∀ x ∈ U, and therefore µD (x) ∈ {0, 1}, ∀ x ∈ U.  Proposition 5. If s : F(U ) × F(U ) → R satisfies S2* and S4, then it satisfies S4*.

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Proof: Let us suppose that s satisfies S2* and S4. Let us consider a pair of different fuzzy sets A, C ∈ F(U ), A 6= C. To check the result given in this proposition, it would suffice to find a fuzzy set B such that s(B, C) > s(A, C). Suppose, without loss of generality, that there exists some x0 ∈ U such that µA (x0 ) < µC (x0 ), and let us define the membership function of fuzzy set B as follows: • µB (x) = µA (x), ∀ x 6= x0 . • µB (x0 ) = µC (x0 ) Therefore, according to S2*, s(B, C) is strictly higher than s(A, B).  According to the above results, we can complete Tables 1 and 2 as shown in Table 3. If a cell is filled, it means that any similarity measure satisfying the corresponding definition (row) also satisfies the corresponding property (column). If it is filled with a bullet, it means that the property is included in the original definition. If the content is SF, it means that the property is straightforward derived. If the content is Px, it means that the property is satisfied by virtue of Proposition x.

Definition Def. 1 Def. 2 Def. 3 Def. 4 Def. 5 Def. 6 Def. 7 (nec.) Def. 7 (des.)

S1 SF SF SF SF SF SF • •

S1* • • • • • •

S1*-var

P1

•

S2 • • • • • •

S2*

P2

•

S3 •

S3*

• SF • •

• P4

S4 • • SF SF SF SF • •

S4*

• • • • P5

S5

S5*

S7

•

•

•

P3

Table 3: Formal relationships among different variants of the notion of “similarity measure”.

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According to Table 3, we can easily check several implications among the definitions of similarity recalled in Section 3.1: Corollary 6. The following implications hold: • If s satisfies Definition 1, then it satisfies Definition 2, except for G1* (boundary conditions). Also, s satisfies the necessary properties of Definition 7 except for G3. • If s satisfies Definition 3, then it also satisfies Definition 2. • If s satisfies Definition 4, then it also satisfies Definitions 1 and 3, and therefore it satisfies Definition 2 and the necessary properties of Definition 7, except for G3. • If s satisfies Definition 5, then it also satisfies Definition 3, and therefore it satisfies Definition 2. • If s satisfies Definition 6, then it satisfies Definition 4, and therefore it satisfies Definitions 1, 2 and 3, and the “necessary” properties in Definition 7, except for G3. • If s satisfies all desirable properties listed in Definition 7, then it satisfies all definitions from Definition 1 to Definition 6. We will illustrate the above result in Figure 1, where the arrow means “is at least as restrictive as”. The converse results, in general, do not hold, as we show in the following examples. Example 1. Let f : [0, 1] → [0, 1] be an increasing mapping with f (0) > 0. If s satisfies Definitions 2 and 3, then s0 = f ◦ s also satisfies Definitions 2 or 3. But s0 (D, Dc ) > 0 for every D ∈ P(U ). So s0 does not satisfy Definitions 1 and 4. The same example allows us to check that a similarity measure satisfying Definition 5 does not necessarily satisfy Definition 1.

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Example 2. Let U be an arbitrary set, and let s be given by:   0 if A ∩ B = ∅ s(A, B) =  1 otherwise. It satisfies Definition 1, but not Definition 3. Consequently, s does not satisfy Definitions 4, 5 or 6, nor the desirable properties in Definition 7. Example 3. Let U be a finite set and s(A, B) = 1 − maxx∈U |A(x) − B(x)|, ∀ A, B ∈ F(U ). s satisfies S1*, S2, S3 and S4* (and therefore, Definition 4), but not S3*. Therefore, it does not fulfill Definition 6. Example 4. Let us consider a finite universe U and a function s : F(U ) × F(U ) → R, whose restriction to the family of pairs of crisp sets is given by:    0    s(A, B) = 0.5     1

if A 6= B and [A = ∅ or #A = #B = 1] if A 6= B and [#A ≥ 2 or #B ≥ 2] if A = B

and s(A, B) = d(CA , CB ), ∀ A, B ∈ F(U ) where CA represents the “nearest crisp set of A”, i.e, the crisp set given by CA = {x ∈ U : µA (x) > 0.5}, and # denotes the cardinal of a crisp set. s satisfies Definition 5, but not Axiom S3. Therefore, it does not satisfy Definitions 4, 6 and 7. Example 5. Let s : F(U ) × F(U ) → R be defined by: • s(A, A) = 1, ∀ A ∈ F(U ). • s(A, B) = 0 if A(x) ≤ 0.1 and B(x) = 1 ∀ x ∈ U . • s(A, B) = 0 if A ∈ P(U ) and B = Ac , • s(A, B) = 0.5 otherwise.

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It fulfills Definition 6 (and, in consequence, Definitions 3 and 4), but it does not satisfy Definition 5. Example 6. Let us consider a finite set U = {x1 , x2 , . . . , xn } and the following q
2 Pn 2 similarity measure given in [11]: Sdω (A, B) = 1 − i=1 ωi A(xi ) − B(xi ) Pn 2 where 0 ≤ ωi ≤ 1, i=1 ωi = 1. Sdω satisfies Definitions 1 and 6 but it does not fulfill G3. Consequently, it does not satisfy Definition 7. The information supported by these counterexamples is summarized in Figure 2. As we have pointed out in the Introduction, the notion of “similarity” is not the only concept introduced in the literature to express degree of “equality” between pairs of fuzzy sets. We will now recall the notions of “similitude” and “resemblance” introduced by Bouchon et al. in [5]. Definition 8. ([5]) An M-measure of comparison on U is a function c on F(U ) × F(U ) satisfying G1 and G4*. Remark 3.4. Regarding Definition 8, the following axiomatic definition for the notion of “difference” between fuzzy sets was considered in [5]: Diff1. If A ⊆ B then A \ B = ∅. Diff2. B \ A is monotone with respect to B (B ⊆ B 0 entails B \ A ⊆ B 0 \ A.) Definition 9. ([5]) An M-measure of similitude on U is an M -measure of comparison such that there exists a fuzzy measure M and a function Fs : R3 → R which is increasing on the first component and decreasing on the second and third components. Lemma 7. Any M-measure of similitude satisfies axiom S7. Proof: Let s = Fs ◦ M be an M-measure of similitude, and let A, B, C, D ∈ F(U ) satisfy the following relations A ∩ B ⊆ C ∩ D, A \ B ⊇ C \ D and B \ A ⊇ D \ C. It suffices to check that s(A, B) ≤ s(C, D). It is immediately derived from the monotonicity of M and the properties of the function Fs .  15

Definition 10. ([5], Bouchon-Meunier et al., 1996) An M - measure of resemblance is an M-measure of similitude satisfying G1*, G2 and S4.

In Tables 1 and 2, we have displayed the collection of properties fulfilled by each of the variants of the notion of “similarity” introduced in the literature and recalled in Definitions 1 to 7. Now we can complete it with the information about the notions of “similitude” and “resemblance” recalled in Definitions 9 and 10. Definition Def. 8 Def. 9 Def. 10 Definition Def. 9 Def. 10

S1

S1*

S1*-var

G1 • • • S2

G1*

G2

•

•

S2*

S3

G3

S3*

G4* • • • S4

S4*

S5

S5*

S6

•

Table 4: Definitions of “similitude” and “resemblance”.

Now we will show the formal connection between measures of similitude and resemblance and the different variants of the notion of similarity considered in Definitions 1 to 7. Proposition 8. If s satisfies properties G4* and S7, then it satisfies properties S1∗ and S2. Proof: Let us consider a similitude measure, s, and let us take three fuzzy sets A, B, C ∈ F(U ) satisfying the restrictions A ⊆ B ⊆ C. We can write s(A, C), s(A, B) and s(B, C) respectively as: • s(A, C) = Fs (M (A), M (∅), M (C \ A)), • s(A, B) = Fs (M (A), M (∅), M (B \ A)) and • s(B, C) = Fs (M (B), M (∅), M (C \ B)). Therefore, according to the monotonicity of M we can easily check that s satisfies properties S1∗ and S2.  16

S7 • •

According to the above result, we can complete Table 4 as shown in Table 5. Definition Def. 8 Def. 9 Def. 10 Definition Def. 9 Def. 10

S1 SF SF

S1* P8 P8

G1 • • •

S1*-var

G1*

G2

•

•

S2 P8 P8

S2*

S3

G3

S3*

G4* • • • S4

S4*

S5

S5*

S6

•

Table 5: Additional properties satisfied by “similitude” and “resemblance” measures.

We can easily deduce from Table 5, that the notion of resemblance is more restrictive than the notion of similarity considered in Definition 2, as we show in Corollary 9. Corollary 9. Any similitude measure satisfies properties S1∗ and S2 . Therefore, any resemblance measure satisfies Definition 2. Now, we can ask ourselves whether any measure of resemblance satisfies the rest of definitions of similarity listed in Section 3.1 or not. The following example shows that, in general, it does not. Example 7. Let U be a finite set and let s : P(U ) × P(U ) → R be the following M-measure of resemblance : s(A, B) = F (M (A ∩ B), M (B \ A), M (A \ B)) with M (A) =

#A #A

and F : [0, 1]3 → [0, 1] defined as

F (u, v, w) =

   u+(1−v)+(1−w)

if v 6= 0 6= w

 1

otherwise.

3

s does not satisfy S3 and S4*, and therefore, it does not satisfy Definitions 1, 3, 4, 5, 6 and 7.

17

S7 • •

We can finally summarize the results provided in this section concerning the formal relations among the different axiomatic variants of the notion of “equality between fuzzy sets” recalled in Definitions from 1 to 10. The left hand side of Figure 3 completes the information provided in Figure 1, and the right hand side completes the information provided in Figure 2. 4. Measures of “inequality” or “difference” 4.1. Axiomatic definitions in the literature We will start this section by recalling some definitions of “inequality” or “difference” from the literature. Definition 11. ([5]) d is an M-measure of dissimilarity if it satisfies G1, G4*, D3 and, furthermore, the associated function Fd : R3 → R does not depend on the first argument and is increasing on the second and the third arguments. Lemma 10. Let us suppose that a difference operator “ \” satisfies the following property: (A \ B) \ (B \ A) = A \ B, ∀ A, B ∈ F(U ).

(1)

Then, any function d satisfying Definition 11 fulfills Axioms D6 and D7. Proof: Let us consider an M-measure of dissimilarity, d = Fs ◦ M. • Axiom D6.- If A, B, C and D ∈ F(U ) satisfy D6, then, by the monotonicity of M and the properties of Fs , we straightforwardly see that d(A, B) ≥ d(C, D). • Axiom D7.- Taking into account the property considered in Equation 1, and the independence of Fs wrt the first argument, we can easily check that d(A, B) = d(A \ B, B \ A), for every pair A, B ∈ F(U ).  Every difference operator considered in the rest of the paper will be assumed to satisfy Equation 1. Let the reader notice that this equation is satisfied by many difference operators introduced in the literature, such as those introduced in [12] and [18], and also the T-norm-based difference operators considered in [10], for the minimum T-norm. 18

Definition 12. ([39]) d is called distance measure if it satisfies G2, D1, D2 and D3. Definition 13. ([42]) d is called distance measure, if it satisfies G1*, G2, D1, D2 and D3*. Remark 4.1. The term “distance measure” is used in [39, 42] without referring to the mathematical notion of metric. Definition 14. ([25]) d is a divergence measure if it satisfies G2, D3, D4 and D5. Definition 15. ([25]) d is a local divergence measure if it satisfies all the properties of divergence measures (G2, D3, D4 and D5) and, furthermore, G3*. 4.2. Formal relationships between different measures of “inequality” In this section, we will study the formal relationships among the axiomatic proposals recalled from Definition 11 to Definition 15). In order to provide as much organized information as possible, and to simplify our further analysis, we will summarize the information recalled in Subsection 4.1 in a compact way. The following table displays the list of properties fulfilled by each of the last definitions (from Definition 11 to Definition 15). Definition Def. 11 Def. 12 Def. 13 Def. 14 Def. 15

G1 •

G1*

•

G2 • • • •

G3*

G4* •

D1

D2

• •

• •

D3 • •

D4

D5

• •

• •

D6 •

• • •

•

D3*

Table 6: Definitions of “dissimilarity measure”, “distance measure” and “divergence measure” (Definitions from 11 to 15).

Now we will review the state-of-the-art about the formal connections between the different collections of axioms considered in those definitions. The literature about it is scarce and, in fact, it is reduced to some results provided in [25]. We summarize those results in the following proposition. 19

D7 •

Proposition 11. Let d denote a mapping defined on F(U ) × F(U ): • If d satisfies D4 and D5 then it satisfies D1. (cf.[25], Proposition 2.4) • If, furthermore, d satisfies G3* then it satisfies D2. (cf.[25], Proposition 3.11) Remark 4.2. We have recalled in Proposition 11 that any measure satisfying Axioms D4 and D5 also satisfies Axiom D1. More specifically, it can be checked that Axiom D4 implies D1a and Axiom D5 implies D1b. Conversely, none of them separately implies Axiom D1, as it is checked in [24]. According to the last results, we can complete Table 6 as follows. We will use the same kind of nomenclature as in Tables 3 and 5. Definition Def. 11 Def. 12 Def. 13 Def. 14 Def. 15

G1 •

G1*

SF

•

G2 • • • •

G3*

G4* •

•

D1

D2

• • P11 P11

• • P11

D3 • • SF • •

D3*

D4

D5

• •

• •

•

Table 7: Additional properties satisfied by “dissimilarity”, “distance” and “divergence measures”.

The information displayed in Table 7 allows us to state the following results. Corollary 12. The following implications hold: • If d satisfies Definition 13, then it satisfies Definition 12. • If d satisfies Definition 15, then it satisfies Definition 14. • If d satisfies Definition 15 then it satisfies Definition 12. We will illustrate the above result in Figure 4, where the arrow means “is at least as restrictive as”. Figure 4 shows that some implication relations exist between some of the definitions recalled in Subsection 4.1. In order to complete our study about the formal relationships between measures of “inequality” between fuzzy sets, we will proceed to show that the converse results, in general, do not hold. 20

D6 •

D7 •

Example 8. Let U be a singleton and let d : F(U ) × F(U ) → R be defined as follows: d(∅, U ) = d(U, ∅) = 5 and d(A, B) = 0 otherwise. We observe that d satisfies G2, D1, D2 and D3. However, it does not fulfill Axioms G1* and D3*. Thus, it fulfills Definition 12, but not Definition 13. Example 9. Let U be an arbitrary universe and let d : F(U ) × F(U ) → R be defined as follows:    1    d(A, B) = 0     0.5

if A 6= B and [A ⊂ B, B ⊂ A or A = B c ], if A = B otherwise.

It satisfies Definition 13. In fact, it takes values within the interval [0, 1], and it takes the value 1 at least for the pair of sets (∅, U ). Moreover, it satisfies G2, D2 and D3* by definition. Furthermore, it is easy to check that it fulfills D1. In fact, if we take three nested fuzzy sets A ⊆ B ⊆ C, then, according to the definition of d, we observe that d(A, C) =

  1

if A 6= C,

 0

if A = C.

We can easily check that max{d(A, B), d(B, C)} ≤ d(A, C), in both situations. Conversely, d does not satisfy Definition 14. To show it, let us take two different but non-disjoint fuzzy subsets of the universe, A, B ∈ F(U ), such that none of them is included in the other. Let C = A. Under these conditions d(A ∪ C, B ∪ C) = d(A, B ∪ A) = 1 and d(A ∩ C, B ∩ C) = d(A, A ∩ B) = 1. However, d(A, B) is equal to 0.5. According to this, we observe that d does not satisfy D4, nor does D5, and therefore, it is not a divergence measure. According to this example, Definition 13 does not imply Definition 14. Example 10. Let U be a finite universe and let d : F(U ) × F(U ) → R be defined as follows: 1

#{x : µA (x) 6= µB (x)} #(A∩B)+1 d(A, B) = , #U 21

where #A denotes the sigma-count [21] of the fuzzy set A: #A =

X

µA (x).

x∈U

It satisfies Definition 13: it is positive and symmetric (G2) by construction. It is increasing with respect to #{x : µA (x) 6= µB (x)} and decreasing wrt #(A ∩ B), and so we can easily check that it satisfies axiom D1. Furthermore, it takes the maximum value, d(A, B) = 1 when (A, B) = (∅, U ), and therefore it satisfies G1*. We also observe that d(D, Dc ) = 1, for any crisp set D, and thus it satisfies D2. Finally, d(A, B) is null if and only if A = B, by construction. On the other hand, d does not satisfy Definition 11: Let us consider for instance the universe U = {1, 2, 3, 4} and the crisp subsets A = {1, 2, 3} and B = {2, 3, 4}. Then, d(A, B) =

√ 3

2 4

is less than d(A \ B, B \ A) = 1.

According to this example, Definition 13 does not imply Definition 11. Example 11. Let U = {x1 , x2 } and let d : F(U ) × F(U ) → R be defined as follows: • The restriction of d to the set of pairs of crisp sets will be represented by means of the following double-entry table: ∅

{x1 }

{x2 }

{x1 , x2 }

∅

0

1

1

3

{x1 }

1

0

2

1

{x2 }

1

2

0

1

{x1 , x2 }

3

1

1

0

• d(A, B) := d(CA , CB ), ∀ A, B ∈ F(U ) where CA represents the nearest crisp set of A. Taking into account that CA∩B = CA ∩ CB and CA∪B = CA ∪ CB , we can easily check that d satisfies all the properties included in Definition 14. But, we observe that it does not satisfy property D2. In fact, if we consider the set A = {x1 }, we observe that the divergence between itself and its complement Ac = {x2 } does not reach the maximum value of d. 22

According to this, we observe that not every (non-local) divergence measure satisfies Definitions 12 and 13. Example 12. Let U = {x0 }, and let d : F(U ) × F(U ) be the comparison measure determined by the fuzzy measure M : F(U ) → R given by M (A) = µA (x0 ), ∀ A ∈ F(U ), and the function F : R3 → R defined as F (u, v, w) =

v+w 2 .

F does not depend on the first argument, u, and it is increasing on v and w, and therefore d is a dissimilarity measure. Now, we will show that it does not satisfy axiom D1. Let us consider the nested fuzzy sets A ⊆ B ⊆ C defined as follows: µA (x0 ) = 0.2, µB (x0 ) = 0.8, µC (x0 ) = 0.9. We observe that µA\B (x0 ) = 0.2, µA\C (x0 ) = 0.1, µB\A (x0 ) = 0.8 and µC\A (x0 ) = 0.8. Therefore d(A, B) = 0.5 > d(A, C) = 0.45. According to this example, Definition 11 does not imply Axiom D1, nor do Definitions 12, 13, 14 or 15. Example 13. Let U be a finite universe and let d : F(U ) × F(U ) → R be P defined as d(A, B) = x∈U h(µA (x), µB (x)), where   |a − b| if a < 0.5 or b < 0.5, h(a, b) =  0 otherwise. We can check that d is a local divergence measure, since h satisfies the following inequality: max{h(max{a, c}, max{b, c}), h(min{a, c}, min{b, c})} ≤ h(a, b), ∀ a, b, c ∈ [0, 1]. Conversely, it is not a dissimilarity measure. There does not exist any fuzzy measure M : F(U ) → R such that d(A, B) can be written as a function of M (A \ B) and M (B \ A). To check this, let us consider the fuzzy sets A, B, A0 and B 0 such that µA (x∗ ) = 0.1, µB (x∗ ) = 0.4, µA0 (x∗ ) = 0.6 and µB 0 (x∗ ) = 0.9 and µA (x) = µB (x) = µA0 (x) = µB 0 (x) = 0, ∀ x 6= x∗ . Then d(A, B) = 0.3 does not coincide with d(A0 , B 0 ) = 0. Conversely, the following equalities hold for every x ∈ U : • min{µA (x), 1 − µB (x)} = min{µA0 (x), 1 − µB 0 (x)} and 23

• min{µB (x), 1 − µA (x)} = min{µB 0 (x), 1 − µA0 (x)}. Thus, A \ B = A0 \ B 0 and B \ A = B 0 \ A0 . Therefore, d does not satisfy D7. According to this example, Definitions 14 and 15 do not imply Definition 11. As we have checked in Example 13, (local) divergence measures do not necessarily satisfy Axiom D7, i.e., the (local) divergence between two fuzzy sets cannot be written, in general, as a function of their respective differences. Thus, local divergences do not necessarily satisfy all the properties of dissimilarity functions. Notwithstanding this, there are some formal commonalities between both notions. In fact, any local divergence induces a fuzzy measure on F(U ) such that the divergence between any two disjoint fuzzy sets can be written as the sum of their measures (and therefore it satisfies Axiom G5), as we check in Proposition 14. The proof of Lemma 13 is straightforward. Lemma 13. If d satisfies D1a and D3 then the set function M (A) = d(∅, A), ∀ A ∈ F(U ) is a fuzzy measure. Proposition 14. Let d : F(U ) × F(U ) → R be a local divergence on U . Then: (a) The set function M (A) = d(A, ∅), ∀ A ∈ F(U ) is a fuzzy measure. (b) A ∩ B = ∅, d(A, B) = M (A) + M (B), ∀ A, B ∈ F(U ) with A ∩ B = ∅, and therefore it satisfies G5. Proof: (a) We have just taken into account Lemma 13 and the symmetry of d. (b) Let h denote the “generating” function of the local divergence d. According to the properties of d, h(0, 0) = 0. Furthermore, for any pair of disjoint fuzzy sets, A and B, µA (x) 6= ∅ implies that µB (x) = 0 and vice versa. Thus, we easily deduce that d(A, B) = d(A, ∅) + d(∅, B), or, equivalently, d(A, B) = M (A) + M (B) 

24

Moreover, any local divergence measure satisfies G4. This means that the divergence between two arbitrary fuzzy sets (not necessarily disjoint) can be written as a function of their intersection and their respective differences, as we check below. Proposition 15. Let us consider the difference operator associated to the minimum t-norm and the maximum t-conorm. If m satisfies G3*, then it satisfies G4. Proof: Let us assume that m satisfies G3*, i.e., that there exists a function P h : [0, 1] × [0, 1] → R such that m(A, B) = x∈U h(µA (x), µB (x)). Then, there P exists a mapping fh : R3 → R such that m(A, B) = x∈U fh (µA∩B (x), µA\B (x), µB\A (x)). Such a mapping can be expressed as follows:     h(1 − x3 , x1 ) if        h(x2 , x1 ) if    fh (x1 , x2 , x3 ) = h(1 − x3 , 1 − x2 ) if       h(1 − x3 , 0.5) if       h(x2 , x3 ) if

x1 < x2 , and x1 = 1 − x2 , x1 > x2 , and x1 = 1 − x3 , x1 > x2 , x1 = x2 = 0.5, x1 = x2 < 0.5.

 According to Propositions 14 and 15, we can complete Table 7 as follows: Definition Def. 11 Def. 12 Def. 13 Def. 14 Def. 15

G1 •

G1*

SF

•

G2 • • • •

G3*

•

G4

P15

G4* •

G5

D1

D2 • •

P14

• • P11 P11

P11

D3 • • SF • •

D3*

D5

• •

• •

•

Table 8: Additional properties satisfied by “dissimilarity”, “distance” and “divergence measures” (II)

To summarize, local divergences do not satisfy, in general, all the properties of dissimilarity measures, as we have illustrated in Example 13. However, a similar “intuition” seems to be behind both notions. To complete this subsection, 25

D4

D6 •

D7 •

we will show that local divergences do not fulfill, in general, all the properties required in Definition 13. Example 14. Let us consider an arbitrary finite universe U and the mapping P d : F(U ) × F(U ) → R defined as d(A, B) = x∈U h(µA (x), µB (x)), where   1 if a = 0 or a = 1 and b = 1 − a, h(a, b) =  0 otherwise We can check that d is a local divergence measure, as h satisfies the following inequality: max{h(max{a, c}, max{b, c}), h(min{a, c}, min{b, c})} ≤ h(a, b), ∀ a, b, c ∈ [0, 1]. Conversely d does not satisfy property D3*. According to this fact, Definition 15 does not imply Definition 13. The information reported by these counterexamples is summarized in Figure 5. 4.3. The particular case of crisp sets According to the above counterexamples, there are no additional implication relationships among Definitions 11 to 15, apart from those implications displayed in Figure 4. But, we can find further formal relationships if we restrict ourselves to the family of crisp sets of the universe, as we check below. Proposition 16. If d satisfies D6 then it satisfies D1, when restricted to the family of pairs of crisp sets. Proof: Consider three arbitrary nested crisp subsets of the universe, A ⊆ B ⊆ C. a) A ∩ B = A ∩ C = A, A \ B = ∅ = A \ C and B \ A ⊆ C \ A, and therefore, according to D6, d(A, B) ≤ d(A, C). b) A∩C ⊆ B ∩C, A\C = B \C = ∅ and C \B ⊆ C \A. Therefore, according to D6, d(B, C) ≤ d(A, C). 26

Proposition 17. If d satisfies D6 and D7, then it fulfills D4 and D5, when restricted to the family of crisp sets. Proof: Let us assume that d satisfies D6 and D7, and let us consider three arbitrary crisp sets A, B and C ∈ P(U ). • In order to check D4, we will compare d(A, B) and d(A ∪ C, B ∪ C). According to D7, the following equalities hold: d(A, B) = d(A \ B, B \ A) and d(A ∪ C, B ∪ C) = d((A ∪ C) \ (B ∪ C), (B ∪ C) \ (A ∪ C)). Furthermore, let us notice that (A ∪ C) \ (B ∪ C) = A ∩ B ∩ C and (B ∪ C) \ (A ∪ C) = B ∩ A ∩ C. Thus, according to D6, we observe that d(A \ B, B \ A) ≥ d(A ∩ B ∩ C, B ∩ A ∩ C), and therefore the inequality d(A, B) ≥ d(A ∪ C, B ∪ C) holds. • To check D5, we will compare d(A, B) and d(A ∩ C, B ∩ C). According to D7, d satisfies the following equalities: d(A, B) = d(A \ B, B \ A) and d(A ∩ C, B ∩ C) = d((A ∩ C) \ (B ∩ C), (B ∩ C) \ (A ∩ C)). Furthermore, we can easily check the equalities (A∩C)\(B∩C) = A∩B∩C and (B ∩ C) \ (A ∩ C) = B ∩ A ∩ C. According to D6, we observe that d(A \ B, B \ A) ≥ d(A ∩ B ∩ C, B ∩ A ∩ C). Thus d(A, B) ≥ d(A ∩ C, B ∩ C).  Proposition 18. Consider the function d : F(U ) × F(U ) → R. (a) If d satisfies D5, then d(A, B) ≤ d(A \ B, B \ A), ∀ A, B ∈ P(U ). (b) If d satisfies D4, then d(A, B) ≥ d(A \ B, B \ A), ∀ A, B ∈ P(U ). (c) If d satisfies D4, then d(A, B) ≤ d(C, B), for all A, B, C ∈ P(U ) such that C ∩ B = ∅ and A ⊆ C.

27

(d) If d satisfies D4 and D5 then d(A, B) ≤ d(B c , B), ∀ A, B ∈ P(U ). (e) If d satisfies G2, D3, D4, D5 and G5, then it satisfies G4*on P(U ) and therefore it can be written as a composition Fd ◦ M . Furthermore, the mapping Fd does not depend on the first argument, and it is increasing on the second and the third ones. Proof: (a) It suffices to apply D5 to the sets A0 = (A\B), B 0 = (B\A) and C = A∩B. (b) It is enough to apply D4 to the sets A, B and C = Ac ∪ B c . (c) It is enough to apply D4 to the sets C, B and B ∪ A. (d) According to D5, d(A, B) = d((A\B)∪(B ∩A), B ∪(B ∩A)) ≤ d(A\B, B). Now, taking into account that A \ B ⊆ B c and that B c ∩ B = ∅, and according to paragraph (c), we obtain that d(A \ B, B) ≤ d(B c , B). (e) According to (a) and (b), d(A, B) = d(A \ B, B \ A), ∀ A, B ∈ P(U ). Furthermore, according to D4, the mapping M (A) = d(∅, A), ∀ A ∈ P(U ) is increasing wrt the inclusion of sets. Furthermore, according to D2, M (∅) = 0. Now, according to G5, there exists a well-defined increasing function g : R2 → R such that d(A − B, B − A) = g(d(A − B, ∅), d(∅, B − A)), ∀ A, B ∈ P(U ). Since d is assumed to be symmetric (Axiom G2), the last equality can be equivalently written as follows: d(A − B, B − A) = g(M (A − B), M (B − A)). Thus we can write: d(A, B) = g(M (A − B), M (B − A)), ∀ A, B ∈ P(U ), where g is increasing on both arguments and M is a fuzzy measure. This completes the proof.



Remark 4.3. According to Proposition 18, any mapping d satisfying D4 and D5, also satisfies D6 and D7, when restricted to crisp sets. Furthermore, it fulfills the following property, which is less restrictive than D2: d(A, B) ≤ d(B c , B), ∀ A, B ∈ P(U ). 28

From now on, we will denote it D2− . Let us notice that, not every (non-local) divergence measure satisfies Axiom D2, because it does not necessarily satisfy the equalities d(Ac , A) = d(B c , B), ∀ A, B ∈ P(U ). We illustrate this point in Example 11, where we consider a non-local divergence measure that does not satisfy Axiom D2, even if we restrict to the family of crisp sets. According to the last results, if we restrict ourselves to the family of crisp sets we can complete Table 8 as shown in Table 9. Definition Def. 11 Def. 12 Def. 13 Def. 14 Def. 15

G1 •

G1*

SF

•

G2 • • • •

G3*

•

G4* •

P18

D1 P11 • • P11 P11

D2− C12+P18 SF SF P18 SF

D2 • • P11

D3 • • SF • •

D3*

D4 P12

D5 P12

D6 •

D7 •

• •

• •

P18 P18

P18 P18

•

Table 9: Formal relationships among the notions of “dissimilarity”, “distance” and “divergence measure” when restricted to crisp sets.

According to the information contained in Table 9 and the results provided in Propositions 14 and 18(e), we can easily check several implications among the definitions of dissimilarity, (local/non local) divergence and distance recalled in Subsection 4.1, when we restrict ourselves to the family of crisp sets: Corollary 19. Let us consider an arbitrary universe U . • Any symmetric dissimilarity measure is a divergence measure. • Any symmetric dissimilarity measure is a distance measure according to Definition 12, except for the property d(A, Ac ) = d(B, B c ), ∀ A, B ∈ P(U ). • Any divergence measure satisfying G5 is a symmetric dissimilarity measure, except for Axiom G1 (boundary conditions).

29

• Any local divergence measure d is a symmetric dissimilarity measure, except for Axiom G1. • Any divergence measure is a distance measure according to Definition 12, except for the property d(A, Ac ) = d(B, B c ), ∀ A, B ∈ P(U ). We summarize all this information in Figure 6. According to this, the notions of “divergence” and “symmetric dissimilarity” are very similar. Any symmetric dissimilarity, when restricted to crisp sets, satisfies the axioms of divergence measures. Conversely, any divergence measure satisfying G5 and the equality d(∅, U ) = 1 is a dissimilarity measure. Therefore, the restriction to the family of crisp sets of any local divergence measure satisfies all the properties of symmetric dissimilarity measures, provided it satisfies G1. If not, we can easily generate the following symmetric dissimilarity measure from d: disd (A, B) =

d(A,B) d(∅,U ) ,

∀ A, B ∈ P(U ).

5. Duality between the notions of equality and inequality We must note that some of the papers reviewed in Subsection 3.1 propose pairs of dual notions of measures of “equality” and “inequality”, while some others do not. For instance, the notions of “similarity” and “distance” proposed in [42] are dual to each other, in the sense that, for any similarity measure, s, the function d = 1 − s is a distance measure, and vice versa. A similar relation can be found between those concepts in [39]. In this case, we observe that any decreasing function of a similarity measure satisfies the properties of a distance measure and vice versa. On the other hand, Montes et al. introduced the notion of divergence in [25], but they did not introduce the dual notion, and therefore we have not listed in Subsection 2.2 those axioms dual to Axioms D4 and D5. Bouchon et al. introduced the definitions of similitude and resemblance, corresponding to the first block and the non-dual notion of dissimilarity, corresponding to the second block. The following proposition compiles several results concerning the formal relation existing between some of the axioms of “equality” listed in Subsection 30

2.2 and their dual axioms listed in Subsection 2.3. The proof is immediate. Proposition 20. Let us consider an arbitrary referential set or universe U . Let F(U ) denote the family of fuzzy subsets of U. 1. d : F(U ) × F(U ) → R satisfies D1a) if and only if s = g ◦ d satisfies S1∗ for any strictly decreasing function g : R → R. Furthermore: • If s satisfies Axiom S1∗ and h is an arbitrary (non necessarily strictly-) decreasing function then h ◦ s satisfies D1a). • If d satisfies D1a) and g is an arbitrary (non necessarily strictly-) decreasing function then g ◦ d satisfies Axiom S1∗ . 2. d : F(U ) × F(U ) → R satisfies D1b) if and only if s = g ◦ d satisfies S2 for any strictly decreasing function g : R → R. Furthermore: • If s satisfies Axiom S2 and h is an arbitrary (non necessarily strictly-) decreasing function then h ◦ s satisfies D1b). • If d satisfies D1b) and g is an arbitrary (non necessarily strictly-) decreasing function then g ◦ d satisfies Axiom S2. 3. d : F(U ) × F(U ) → R satisfies D2 if and only if s = g ◦ d satisfies S3 and takes non-negative values for any strictly decreasing function g : R → R such that g(max Im(d)) = 0. Furthermore: • If d satisfies D2 and g : R → R satisfies the restriction g(max Im(d)) = 0, then g ◦ d satisfies S3. • If s satisfies S3 and h : R → R satisfies the restriction h(0) = max Im(h ◦ s), then h ◦ s satisfies D2. 4. s : F(U ) × F(U ) → R satisfies S4 if and only if d = g ◦ s satisfies D3 and takes non-negative values for any strictly decreasing function g : R → R such that g(max Im(s)) = 0. Furthermore: • If s satisfies S4 and g : R → R satisfies the restriction g(max Im(s)) = 0, then g ◦ d satisfies D3.

31

• If d satisfies D3 and h : R → R satisfies the restriction h(0) = max Im(h ◦ d), then h ◦ d satisfies S4. 5. s : F(U )×F(U ) → R satisfies S4* if and only if d = g ◦s satisfies D3* and takes non-negative values for any strictly decreasing function g : R → R such that g(max Im(s)) = 0. Furthermore: • If s satisfies S4* and g : R → R satisfies the restriction g(x) = 0 ⇔ x = max Im(s) then g ◦ d satisfies D3. • If d satisfies D3* and h : R → R satisfies the restriction h(y) = max Im(h ◦ d) ⇔ y = 0, then h ◦ d satisfies S4*. 6. d : F(U ) × F(U ) → R satisfies D6 if and only if s = g ◦ d satisfies S7 for any strictly decreasing function g : R → R. Furthermore: • If s satisfies Axiom S7 and h is an arbitrary (non necessarily strictly-) decreasing function then h ◦ s satisfies Axiom D6. • If d satisfies D6 and g is an arbitrary (non necessarily strictly-) decreasing function then g ◦ d satisfies Axiom S7. Table 10 summarizes the information provided in Proposition 20. Equality S1∗ S2 S3 S4 S4∗ S7

Inequality D1(a) D1(b) D2 D3 D3∗ D6

Table 10: Duality relations between axioms

We have divided the axiomatic definitions surveyed in the paper into two main blocks: measures of “equality” and measures of “inequality” between fuzzy sets. If we compose any of those measures with a strictly increasing function, the resulting measure will remain in the same block or family of measures. If, on the contrary, we compose it with an arbitrary strictly decreasing function then the resulting measure will belong to the other block. This fact is related to 32

the notion of “equivalence in order” between two similarity measures considered by Omhover et al. in [28] and by Bouchon-Meunier et al in [4]. Two similarity measures are said to be equivalent [28] when they are co-monotone, i.e., when s(A, B) ≤ s(C, D) ⇔ s0 (A, B) ≤ s0 (C, D), ∀ A, B, C, D ∈ F(U ). In other words, both of them establish the same ranking when comparing any fuzzy set with any fixed prototype. We can easily check ([28]) that two similarity measures are equivalent in order if and only if there exists a strictly increasing function f : R → R connecting both of them. In our context, any increasing function of a fixed similarity measure will satisfy the same axiomatic definition as the initial one, except for some boundary conditions required in some axiomatic definitions (s(A, A) = 1, etc.) If we want to furthermore guarantee that those boundary conditions are satisfied, we will need to require f to fulfill some additional boundary conditions. In a similar way, we can introduce the notion of “duality”. From now on, we will say that a measure d is dual to some s if and only if there exists a strictly decreasing function g such that d = g ◦ s. This is obviously equivalent to saying that both measures are “antitone” to each other, i.e. s(A, B) ≤ s(C, D) ⇔ d(A, B) ≥ d(C, D), ∀ A, B, C, D ∈ F(U ). If any measure satisfies any specific property from the left column in Table 10, then its dual will satisfy the corresponding property from the right column. 6. Conclusion and perspectives We have reviewed and organized some remarkable definitions of measures of (in)equality between fuzzy sets, including all of them in a common framework, and we have analyzed and displayed their formal relationships. The reader can take advantage of the formal analysis provided in this paper, in order to choose the axiomatic definition that best suits each specific application problem. We have laid bare some close relations between some apparently different axiomatic definitions. As an example, the definitions of dissimilarity [5] and divergence 33

[25] are very close to each other when we restrict ourselves to the family of crisp sets of the universe. In fact, both of them focus on the differences between the compared sets, but not on their commonalities. Let us consider, for instance, the set of languages U = {Chinese (c), English (e), Dutch (d), French (f), Italian (i), Russian (r)}. Let the crisp subsets E = {d, e, f, i, s}, G = {d, e, f, r, s}, A = {i} and V = {r} denote the respective communication skills of four people called Enrique, Gert, Angelo and Vladimir. Enrique and Gert share much more language skills than Angelo and Vladimir, but those commonalities cannot be detected by means of the above measures (divergences and dissimilarities). If we just wanted to focus on the differences, those measures would be useful. But, if we also want to take into account their common skills, we should use different comparison measures. We have reviewed some other axiomatic definitions of inequality measures that may take into account the commonalities between the compared sets (see Definitions 12 and 13, for instance). There are several extensions of our study that deserve future research: We have considered here the intersection and union of fuzzy sets based on the “min” and the “max” operators. In some recent publications, such as [8, 9, 10], a more general framework, where other T-norms and T-conorms are also taken into account, is considered. It would be worth extending our formal relational study to those more general situations. In this general framework, the study between some T-transitivities (such as those based on Lukasiewicz and the product Tnorms) and the generalizations of some of the axioms reviewed in this paper would be of much interest in the field. Furthermore, some recent papers establish classes of equivalence of measures of similarity and propose different ways to sort according to their discriminant power (see [4, 27], for instance). Taking into account the results obtained in this paper, we could try to extend these studies to more general kinds of comparison measures. Conversely, in recent literature, many different non-axiomatic definitions of similarity and dissimilarity/distance measures between (interval-valued) fuzzy sets to be used in some specific applications have been introduced. In the near future, we aim to develop a detailed bibliographic study of those works, relating 34

them to this more theoretical work. In our opinion, this connection would allow us to detect what axioms each parametric family of measures satisfies, and furthermore, which of those measures are co-monotonic to each other and, therefore, equivalent. Acknowledgments We thank two anonymous reviewers for their very helpful comments and suggestions. This work has been jointly supported by the Spanish Ministry of Education and Science and the European Regional Development Fund (FEDER), under projects and TIN2011-24302 and MTM2010-17844. [1] V. Balopoulos, A. G. Hatzimichailidis, B. K. Papadopoulos, Distance and similarity measures for fuzzy operators, Information Sciences 177 (2007) 2336-2348. [2] K. R. Bhutani, A. Rosenfeld, Dissimilarity measures between fuzzy sets or fuzzy structures, Information Sciences 152 (2003) 313-318. [3] K. Bosteels, E. E. Kerre, A triparametric family of cardinality-based fuzzy similarity measures, Fuzzy Sets and Systems 158 (2007) 2466-2479. [4] B. Bouchon-Meunier, G.Coletti, M.-J. Lesot, M. Rifqi, Towards a conscious choice of a fuzzy similarity measure: a qualitative point of view, in: E. Hullermeier, R. Kruse, and F. Hoffmann (Eds.), IPMU 2010, LNAI 6178, Springer- Verlag, Berlin Heidelberg (2010) pp.1-10. [5] B. Bouchon-Meunier, M. Rifqi, S. Bothorel, Towards general measures of comparison of objects, Fuzzy Sets and Systems 84 (1996) 143-153. [6] S-M. Chen, M-S. Yeh, P-Y. Hsiao, A comparison of similarity measures of fuzzy values, Fuzzy Sets and Systems 72 (1995) 79-89. [7] I. Couso, S. Montes, An axiomatic definition of fuzzy divergence measures, Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 16 (2008) 1-17. 35

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[17] I. Jenhani, S. Benferhat, Z. Elouedi, Possibilistic Similarity Measures, Foundations of Reasoning under Uncertainty, Studies in Fuzziness and Soft Computing 249 (2010) 99-123. [18] A. Kaufmann, Introduction `a la Th´eorie des Sous-Ensembles Flous, Masson, Paris, 1973. [19] J. Li, Guannan Deng, Hongxing Li, Wenyi Zeng, The relationship between similarity measure and entropy of intuitionistic fuzzy set, Information Sciences 188 (2012) 314-321. [20] Z. Liang, P. Shi, Similarity measures on intuitionistic fuzzy sets, Pattern Recognition Letters 24 (2003) 2687-2693. [21] A. De Luca, S. Termini (1972) A definition of non-probabilistic entropy in the setting of fuzzy sets theory, Inform. Control 20, 301-312. [22] H.B. Mitchell, On the Dengfeng-Chuntian similarity measure and its application to pattern recognition, Pattern Recognition Letters 24 (2003) 31013104. [23] B. Mondal, D. Mazumdar, S. Raha, Similarity in approximate reasoning, International Journal of Computational Cognition, 4 (2006) 46-56. [24] S. Montes, Particiones y medidas de divergencia en modelos difusos, PhD Thesis, University of Oviedo (In Spanish). [25] S. Montes, I. Couso, P. Gil, C. Bertoluzza, Divergence measure between fuzzy sets, International Journal of Approximate Reasoning 30 (2002) 91105. [26] C. P. Pappis, N. I. Karacapilidis, A comparative assessment of measures of similarity of fuzzy values, Fuzzy Sets and Systems 56 (1993) 171-174. [27] M. Rifqi, V. Berger , B. Bouchon-Meunier, Discrimination power of measures of comparison, Fuzzy Sets and Systems 110 (2000) 189–196.

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[28] J.F. Omhover, M. Rifqi, M. Detyniecki, Ranking invariance based on similarity measures in document retrieval, in Adaptive Multimedia Retrieval AMR’05. Springer LNCS, 2006, pp. 55-64. [29] S. Santini, R. Jain, Similarity measures, IEEE Transactions on Pattern Analysis and Machine Intelligence 21 (1999) 871-883. [30] Z. Shi, Z. Gong, The further investigation of covering-based rough sets: Uncertainty characterization, similarity measure and generalized models, Information Sciences 180 (2010) 3745-3763. [31] P.Z. Wang, Fuzzy Sets and Its Applications, Shanghai Science and Technology Press, Shanghai, 1983, in Chinese. [32] X. Wang, B. De Baets, E. Kerre, A comparative study of similarity measures, Fuzzy Sets and Systems 73 (1995) 259-268. [33] C-P. Wei, P. Wang, Y-Z. Zhang, Entropy, similarity measure of intervalvalued intuitionistic fuzzy sets and their applications, Information Sciences 181 (2011) 4273-4286. [34] D. Wu, J. M. Mendel, A vector similarity measure for linguistic approximation: Interval type-2 and type-1 fuzzy sets, Information Sciences 178 (2008) 381-402. [35] D. Wu, J. M. Mendel, A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets, Information Sciences 179 (2009) 1169-1192. [36] Z. S. Xu, An overview of distance and similarity measures of intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems 16 (2008) 529-555. [37] Z. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181 (2011) 2128-2138.

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[38] L. Xudong, C. Zhang, An axiom foundation for uncertain reasonings in rule based expert systems: NT-Algebra, Knowledge and Information Systems, 1 (1999) 415-433. [39] L. Xuecheng, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems 52 (1992) 305-318. [40] J. Ye, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Mathematical and Computer Modelling 53 (2011) 91-97. [41] W. Zeng, H. Li, Relationship between similarity measure and entropy of interval valued fuzzy sets, Fuzzy Sets and Systems 157 (2006) 1477-1484. [42] H. Zhang, W. Zhang, C. Mei, Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure, Knowledge-Based Systems 22 (2009) 449-454. [43] Q-S. Zhang, S. Jiang, B. Jia, S. Luo, Some information measures for interval-valued intuitionistic fuzzy sets, Information Sciences 180 (2010) 5130-5145. [44] R. Zwick, E. Carlstein, D.V. Budescu, Measures of similarity among fuzzy concepts: a comparative analysis, International Journal of Approximate Reasoning 1 (1987) 221-242.

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I. Jenhani et al., 2010 (Def.7, necessary prop.) (SF)

(P1, P2, P3, P5)

(P1, P2, P4, P5)

I. Jenhani et al., 2010 (Def.7, desirable prop.)

H. Zhang et al., 2009

B. Mondal et al., 2006

(except for G3)

L. Xudong, C. Zhang, 1999

(SF)

(Def.6)

(Def.5)

(Def.4)

Z. Liang & P.Shi, 2003

(SF)

(SF)

W. Zeng & H. Li, 2006

(SF)

(SF)

H.B. Mitchel, 2003 (Def.3)

(except for G1*)

L. Dengfeng & C. Chuntian, 2002

(Def.2)

L. Xuecheng, 1992 (Def.1)

Figure 1: Summary of implication relations between different variants of the notion of “similarity measure”.

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I. Jenhani et al., 2010 (Def.7, necessary prop.)

E6

I. Jenhani et al., 2010 (Def.7, desirable prop.)

E4

H. Zhang et al., 2009 (Def.6) E5

B. Mondal et al., 2006 E4

(Def.5)

E3

L. Xudong, C. Zhang, 1999

W. Zeng & H. Li, 2006

Z. Liang & P.Shi, 2003

E1

E1

(Def.4)

E2

L. Dengfeng & C. Chuntian, 2002

(Def.2) 1E

E1

E1

H.B. Mitchel, 2003 (Def.3)

L. Xuecheng, 1992 (Def.1)

Figure 2: Counterexamples concerning implication relations between different variants of the notion of “similarity measure”.

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Bouchon-Meunier et al., 1996 (Def.9)

I. Jenhani et al., 2010 (Def.7, desirable prop.)

I. Jenhani et al., 2010 (Def.7, desirable prop.)

I. Jenhani et al., 2010 (Def.7, necessary prop.)

I. Jenhani et al., 2010 (Def.7, necessary prop.)

(SF)

H. Zhang et al., 2009

H. Zhang et al., 2009

(Def.6)

(Def.6)

E4

(P1, P2, P4, P5)

(P1, P2, P3, P5)

Bouchon-Meunier et al., 1996 (Def.9)

E6

Bouchon-Meunier et al., 1996 (Def.10)

Bouchon-Meunier et al., 1996 (Def.10)

(SF)

W. Zeng & H. Li, 2006

H.B. Mitchel, 2003 (Def.3)

E1

(Def.2)

L. Dengfeng & C. Chuntian, 2002

(Def.2)

L. Xuecheng, 1992

L. Xuecheng, 1992

(Def.1)

(Def.1)

COUNTEREXAMPLES

IMPLICATIONS

Figure 3: Implication relations and counterexamples concerning Definitions 1 to 10.

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E2

(SF)

(SF) (except for G1*)

L. Dengfeng & C. Chuntian, 2002

E1

Z. Liang & P.Shi, 2003

H.B. Mitchel, 2003 (Def.3)

E1

(Def.4)

(SF)

(SF)

(Def.4)

Z. Liang & P.Shi, 2003

(Def.5)

E3

(Def.5)

W. Zeng & H. Li, 2006

B. Mondal et al., 2006

E7

B. Mondal et al., 2006

(SF)

L. Xudong, C. Zhang, 1999

E7

(P7)

E5

L. Xudong, C. Zhang, 1999

(P9)

(SF)

S. Montes et al., 2002 (Def.15, local divergence)

S. Montes et al., 2002 (Def.14, divergence)

H. Zhang et al., 2009

L. Xuecheng, 1992

(SF)

(Def.13, distance)

(Def.12, distance)

Bouchon et al., 1996 (Def.11, dissimilarity)

Figure 4: Summary of implication relations between different axiomatic definitions for the notion of “inequality”.

43

S. Montes et al., 2002 (Def.15, local divergence) E13

E9

E12

S. Montes et al., 2002 (Def.14, divergence)

H. Zhang et al., 2009

E8

(Def.13, distance)

E10

L. Xuecheng, 1992

E12

(Def.12, distance)

Bouchon et al., 1996 (Def.11, dissimilarity)

Figure 5: Counterexamples concerning implication relations between different variants of the notions of “dissimilarity”, “distance” and “divergence”.

44

S. Montes et al., 2002 (Def.15, local divergence) SF

C19

C19

(Def.13, distance) SF

L. Xuecheng, 1992 (Def.12, distance)

(except for G1) C19

H. Zhang et al., 2009

(except for G1 and G5) C19

d(A, Ac ) = d(B, B c ), ∀ A, B ∈ P(U)

except for

S. Montes et al., 2002 (Def.14, divergence)

Def.11+ G2 (symmetric dissimilarity)

SF

Bouchon et al., 1996 (Def.11, dissimilarity)

Figure 6: Summary of implication relations between different variants of the notion of “dissimilarity”, “distance” and “divergence”. The particular case of crisp sets.

45