a triparametric family of cardinality-based fuzzy similarity measures

Fuzzy Sets and Systems 158 (2007) 2466–2479www.elsevier.com/locate/fss

A triparametric family of cardinality-basedfuzzy similarity measuresKlaas Bosteels∗, Etienne E. Kerre

Fuzziness and Uncertainty Modelling Research Group, Department of Applied Mathematics and Computer Science,Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium

Received 13 February 2007; received in revised form 9 April 2007; accepted 11 May 2007Available online 18 May 2007

Abstract

Previously, we introduced a biparametric family of cardinality-based fuzzy similarity measures. In this paper, we generalize thisfamily by adding a third parameter. We also study the generalized family for specific values of the third parameter. More precisely,we show that for particular values of this parameter, certain properties can be ensured by imposing constraints on the two remainingparameters. To conclude, we examine some members of the presented family of fuzzy similarity measures.© 2007 Elsevier B.V. All rights reserved.

Keywords: Similarity measures; Fuzzy cardinality; Aggregation operators

1. Introduction

The task of measuring similarity occurs in many disciplines. Because the objects to be compared are often representedby sets, this task is frequently performed by means of measures that compare sets. Such measures are usually calledsimilarity measures. In many applications, however, fuzzy sets are more suitable than crisp sets for representing theobjects concerned. Consequently, there is a need for fuzzy similarity measures, i.e., measures that compare fuzzy sets.

Since the most commonly used similarity measures are based on cardinalities of the sets involved, a lot of researchhas been devoted to cardinality-based fuzzy similarity measures [3,4,7,6,9–14]. Particularly interesting is the work byDe Baets et al. [4,7,6,10] concerning fuzzification schemes for the class of crisp cardinality-based similarity measurespresented in [5]. More specifically, the development of meta-theorems that allow to ensure transitivity [7,6,10] is apromising recent advancement of this research, because such theorems can be used to systematically construct suitablemeasures for a specific application. However, this approach also has some drawbacks, namely, the fuzzified familyis limited to rational cardinality-based similarity measures and the reflexivity is generally only preserved when thefuzzification is based on the minimum operator. We feel strongly about the latter disadvantage, because reflexivity canbe regarded as a very natural and intrinsic property of similarity.

In [2], we presented a systematic way of constructing and analysing cardinality-based fuzzy similarity measures thatis not restricted to rational measures. This was achieved by introducing a general form that depends on two parameters.

∗ Corresponding author. Tel.: +32 9 264 9639; fax: +32 9 264 4995.E-mail addresses: [email protected] (K. Bosteels), [email protected] (E.E. Kerre).

0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2007.05.006

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K. Bosteels, E.E. Kerre / Fuzzy Sets and Systems 158 (2007) 2466–2479 2467

We showed that various existing properties of the generated measures can be ensured by imposing constraints on theparameters. Moreover, we introduced constraints for some novel properties, namely, several forms of restrictability. Aswe explained, these properties allow to reduce the computation time in practical implementations.

Our previously introduced biparametric family includes many existing cardinality-based fuzzy similarity measures.Nevertheless, the aim of this paper is to further generalize it. More precisely, we want to generalize it in a reflexivity-preserving way. We realize this intention by introducing a third parameter. For a specific choice of the third parameter,the triparametric family reduces to the biparametric one. Therefore, rewritten versions of the previously introducedconstraints are valid in this case. In addition to these rewritten constraints, we also provide constraints for some othervalues of the third parameter.

2. Preliminary notions

2.1. Fuzzy aggregation operators

In the context of fuzzy set theory, an aggregation operator H is an increasing [0, 1]n → [0, 1] mapping, withn ∈ N \ {0}, such that H(0, 0, . . . , 0) = 0 and H(1, 1, . . . , 1) = 1. The natural number n is the arity of H. Binaryaggregation operators are aggregation operators of arity 2. The dual H∗ of an aggregation operator H of arity n isdefined as

H∗(x1, x2, . . . , xn) = 1 − H(1 − x1, 1 − x2, . . . , 1 − xn)

for all x1, x2, . . . , xn ∈ [0, 1]. Also, note that we can naturally extend the usual order on R to a partial order onaggregation operators. Namely, for two aggregation operators H1 and H2 of the same arity n, we write H1 �H2 ifH1(x1, x2, . . . , xn)�H2(x1, x2, . . . , xn) holds for all x1, x2, . . . , xn ∈ [0, 1].

Triangular norms (t-norms) are well-known binary aggregation operators. An associative and commutative binaryaggregation operator T is called a t-norm if is satisfies T (x, 1) = x for all x ∈ [0, 1]. The minimum TM is the largestt-norm and the drastic product TD, defined as

TD(x, y) ={

min(x, y) if max(x, y) = 1,

0 otherwise

for all x, y ∈ [0, 1], is the smallest t-norm, i.e., TD �T �TM for every t-norm T . Other common t-norms are thealgebraic product TP and the Lukasiewicz t-norm TL : TP (x, y) = x · y and TL (x, y) = max(0, x + y − 1), for allx, y ∈ [0, 1]. It can be proven that TL �TP . Hence, TD �TL �TP �TM.

Every t-norm can be understood as a generalization of the logical conjunction from the two-valued set {0, 1} tothe whole unit interval [0, 1]. The logical disjunction can be generalized from {0, 1} to [0, 1] by means of a triangularconorm (t-conorm), i.e., an associative and commutative binary aggregation operator S that satisfies S(x, 0) = x for allx ∈ [0, 1]. The dual of a t-norm T is a t-conorm T ∗, and vice versa. One can easily verify that T ∗

M(x, y) = max(x, y),T ∗

P (x, y) = x + y − x · y, T ∗L (x, y) = min(1, x + y) and

T ∗D(x, y) =

{max(x, y) if min(x, y) = 0,

1 otherwise

for all x, y ∈ [0, 1]. The ordering is as follows: T ∗M �T ∗

P �T ∗L �T ∗

D. Note that t-norms and t-conorms, as a consequenceof their associativity, can easily be generalized to arity n > 2 by recursive application. For arity n = 1, we let eacht-norm and t-conorm correspond to the identity mapping.

An aggregation operator H is compensatory if it satisfies TM = min �H� max = T ∗M. The arithmetic mean AM

and the geometric mean GM are well-known compensatory aggregation operators. They are given by

AM(x1, x2, . . . , xn) = x1 + x2 + · · · + xn

n,

GM(x1, x2, . . . , xn) = n√

x1 · x2 · · · · · xn

2468 K. Bosteels, E.E. Kerre / Fuzzy Sets and Systems 158 (2007) 2466–2479

for all x1, x2, . . . , xn ∈ [0, 1]. More generally, we can define a family G�, with � ∈ R ∪{−∞, +∞}, of mean operatorsas follows [8]:

G�(x1, x2, . . . , xn) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(1

n

n∑i=1

x�i

)1/�

if � ∈ R \ {0},

limr→�

(1

n

n∑i=1

xri

)1/r

otherwise.

All members of this family are compensatory. For � = 0, 1, −∞ and +∞, G� equals GM, AM, TM and T ∗M, respectively.

It can be proven that, for all r, s ∈ R ∪ {−∞, +∞}, Gr �Gs if r �s [8]. Hence, TM �GM �AM �T ∗M.

2.2. Fuzzy sets

Recall that a fuzzy set A in a universe X is a X → [0, 1] mapping that associates with each element x from theuniverse X a degree of membership A(x). We use the notation F(X) for the class of fuzzy sets in X. For two fuzzy setsA and B in X, we write A ⊆ B iff A(x)�B(x) for all x ∈ X, and A = B iff A ⊆ B ∧ B ⊆ A. The support of a fuzzyset A in X is given by supp A = {x ∈ X | A(x) > 0}.

Obviously, crisp sets are special fuzzy sets, since they can be represented by a characteristic X → {0, 1} mapping.To avoid notational clutter, we reuse the name of a crisp set for its characteristic mapping. In particular, X denotes boththe universe and a mapping that associates 1 with each element of the universe. Also, ∅ denotes the empty set as wellas the X → [0, 1] mapping given by ∅(x) = 0, for all x ∈ X.

It is well known that the classical set-theoretic operations intersection and union can be generalized as follows:

(A ∩T B)(x) = T (A(x), B(x)),

(A ∪T ∗ B)(x) = T ∗(A(x), B(x))

for each x ∈ X, with A, B ∈ F(X) and T a t-norm. The crisp concept cardinality can be generalized to fuzzy sets bymeans of the sigma count, which is defined as |A| = ∑

x∈X A(x), for each fuzzy set A in a finite universe X. In theremaining of this paper, X always denotes a finite universe.

2.3. Fuzzy similarity measures

A fuzzy comparison measure is a binary fuzzy relation on F(X), i.e., a fuzzy set in F(X) × F(X). We call a fuzzycomparison measure M a fuzzy similarity measure if it is reflexive, i.e., M(A, A) = 1 for all A ∈ F(X). Hence, weconsider reflexivity to be an inherent property of a fuzzy similarity measure. In addition to reflexivity, we also considerthe following properties [2]:

M(A, B) = 1 ⇒ A = B (coreflexive),

M(A, B) = 1 ⇐ A ⊆ B ∨ B ⊆ A (strong reflexive),

M(A, B) = 1 ⇒ A ⊆ B ∨ B ⊆ A (weak coreflexive),

M(A, B) = 1 ⇐ A ⊆ B (inclusive),

M(A, B) = 1 ⇒ A ⊆ B (coinclusive),

M(A, B) = 0 ⇐ A ∩T B = ∅ (∩T -exclusive),

M(A, B) = 0 ⇒ A ∩T B = ∅ (∩T -coexclusive),

M(A, B) = M(B, A) (symmetric),

M(A, B) = M(A/supp A, B/supp A) (left-restrictable),

M(A, B) = M(A/supp B, B/supp B) (right-restrictable),

M(A, B)�M(A/supp A, B/supp A) (weak left-restrictable),

M(A, B)�M(A/supp B, B/supp B) (weak right-restrictable)


for all A, B ∈ F(X), with C/Y , for C ∈ F(X), the restriction of C to Y ⊆ X, i.e., C/Y is the Y → [0, 1] mapping thatassociates C(x) with each x ∈ Y . We use the term “fuzzy inclusion measure” for a fuzzy similarity measure that is bothinclusive and coinclusive. If a fuzzy similarity measure is symmetric, then we call it a fuzzy resemblance measure.

For most of the above-mentioned properties, it does not take a genius to see that they can be useful, or even desirable, inpractice. In case of the restrictability properties, however, the practical value is not immediately obvious. The advantageof restrictable fuzzy similarity measures lies in the fact that, e.g., M(A/supp A, B/supp A) often requires significantly lesscalculations than M(A, B). Since both of these expressions are equal when M is left-restrictable, the computation timecan then be reduced substantially for such an M by restricting the two fuzzy sets A and B to supp A before comparingthem. When M is weak left-restrictable, it can also be possible to omit some calculations. For instance, when searchingfor a fuzzy set B that is very similar to a certain reference fuzzy set A, we can first calculate M(A/supp A, B/supp A)

and then execute the extra computations required for M(A, B) only if M(A/supp A, B/supp A) is sufficiently large. Adetailed illustration, by means of two examples, of the practical use of the restrictability properties can be found in [2],and in [1] we provided some experimental proof.

3. The parameter �

In [2], we introduced the following general form:

M�1�2

(A, B) = �1(‖A‖, ‖B‖, ‖A ∩TMB‖)

�2(‖A‖, ‖B‖, ‖A ∪T ∗M

B‖) (1)

for all A, B ∈ F(X), with �1 and �2 increasing mappings from [0, 1]3 to [0, 1], and ‖.‖ the relative sigma count, whichis given by ‖A‖ = ∑

x∈X A(x)/|X|, for each A in F(X). Now, it holds that ‖A ∩TMA‖ = ‖A‖ and ‖A‖ + ‖B‖ =

‖A ∩TMB‖ + ‖A ∪T ∗

MB‖, for all A, B ∈ F(X). Thus, if we define

MTM�1,�2

(A, B) = �1(‖A ∩TMA‖, ‖B ∩TM

B‖, ‖A ∩TMB‖)

�2(‖A ∩TMA‖, ‖B ∩TM

B‖, ‖A ∩TMB‖) (2)

for all A, B ∈ F(X), with �1(x, y, z) = �1(x, y, z) and �2(x, y, z) = �2(x, y, x + y − z) for all x, y, z ∈ [0, 1] that

satisfy 0�x + y − z�1, then MTM�1,�2

= M�1�2

. This observation leads us to the following family of measures:

M��1,�2

(A, B) = �1(‖A �� A‖, ‖B �� B‖, ‖A �� B‖)�2(‖A �� A‖, ‖B �� B‖, ‖A �� B‖) (3)

for all A, B ∈ F(X), with �1 and �2 two [0, 1]3 → R mappings that are increasing in their first and second argument,� a binary aggregation operator, and �� an infix notation for the pointwise extension of �, i.e., (A �� B)(x) =�(A(x), B(x)) for all x ∈ X, with A and B fuzzy sets in X. Henceforth, we often give in to the space limitations byusing the following abbreviations:

a� = ‖A �� A‖, a� = ‖A/supp A �� A/supp A‖, a� = ‖A/supp B �� A/supp B‖,b� = ‖B �� B‖, b� = ‖B/supp A �� B/supp A‖, b� = ‖B/supp B �� B/supp B‖,c� = ‖A �� B‖, c� = ‖A/supp A �� B/supp A‖, c� = ‖A/supp B �� B/supp B‖.

In general, M��1,�2

is a F(X) × F(X) → R mapping. Hence, we need to impose additional constraints on theparameters if we want the generated measures to be fuzzy similarity measures. Namely, we have to make sure thatM�

�1,�2is both [0, 1]-valued and reflexive. Since it can easily be proven that the implications

(∀ x, y, z ∈ [0, 1])(0��1(x, y, z)��2(x, y, z)) ⇒ M��1,�2

is [0, 1]-valued, (4)

(∀ x ∈ [0, 1])(�1(x, x, x) = �2(x, x, x)) ⇒ M��1,�2

is reflexive (5)

hold, we can achieve this by imposing additional constraints on �1 and �2. Now, for specific choices of �, we can(i) weaken the antecedent of (4) and (ii) introduce constraints that ensure other properties. Prior to introducing these


supplementary constraints, however, it is useful to take a closer look at the concept of a constraint. Implications (4) and(5) can be rewritten as follows:

(∀ x, y, z ∈ [0, 1])(0��1(x, y, z)��2(x, y, z)) ⇒ (∀ A, B ∈ F(X))(0�M��1,�2

(A, B)�1),

(∀ x, y, z ∈ [0, 1])(�1(x, x, x) = �2(x, x, x)) ⇒ (∀ A, B ∈ F(X))(A = B ⇒ M��1,�2

(A, B) = 1).

Hence, they are both of the form

(∀ x, y, z ∈ [0, 1])(P (x, y, z)) ⇒ (∀ A, B ∈ F(X))(Q(M��1,�2

(A, B)))

with P and Q predicates on [0, 1]3 and [0, 1], respectively. We call the antecedent of such an implication a constraint,and we say that this constraint ensures the consequent of the implication. By assuming implicit universal quantificationover x, y and z, we can also call P(x, y, z) a constraint. If a constraint involves parameters, then it is a constraint on thoseparameters. The power of these constraints lies in the fact that they are defined in terms of three simple [0, 1]-valuedvariables, which makes them easy to handle. In order to prove that a particular constraint ensures a certain property,however, we need to convert it to an expression in terms of fuzzy sets. This can be done by means of the followingproposition:

Proposition 1. Let P and Q be predicates on [0, 1]3 and [0, 1], respectively. If

(∀ A, B ∈ F(X))(P (a�, b�, c�) ⇒ Q(M��1,�2

(A, B))) (6)

then we have

(∀ x, y, z ∈ [0, 1])(P (x, y, z)) ⇒ (∀ A, B ∈ F(X))(Q(M��1,�2

(A, B))). (7)

Proof. From the antecedent of (7) follows that P(a�, b�, c�) for all A, B ∈ F(X). Combined with (6), this impliesthe consequent of (7). �

Note that the implications

0��1(a�, b�, c�)��2(a�, b�, c�) ⇒ 0�M��1,�2

(A, B)�1, (8)

�1(a�, a�, a�) = �2(a�, a�, a�) ⇒ M��1,�2

(A, A) = 1 (9)

are clearly satisfied for all A, B ∈ F(X) and hence, by Proposition 1, (4) and (5) hold. Also observe that, in general,(9) would not be valid for all A ∈ F(X) if M�

�1,�2(A, B) would be equal to �1(‖A‖, ‖B‖, ‖A �� B‖)/�2(‖A‖, ‖B‖,

‖A �� B‖). That is the main reason why we did not use this alternative generalization of M�1�2

. So, in other words, wechose the less obvious generalization given in (3) because we want to preserve the reflexivity.

4. The case � = T

In this section we prove that, for an arbitrary t-norm T , the following implications hold:

(∀ x, y, z ∈ [0, 1])(min(x, y)�z� max(x, y) ⇒ �1(x, y, z) = �2(x, y, z))

⇒ MT�1,�2

is strong reflexive, (10)

(∀ x, y, z ∈ [0, 1])(x�z�y ⇒ �1(x, y, z) = �2(x, y, z)) ⇒ MT�1,�2

is inclusive, (11)

(∀ x, y ∈ [0, 1])(�1(x, y, 0) = 0) ⇒ MT�1,�2

is ∩T -exclusive, (12)

(∀ x, y, z ∈ [0, 1])(�1(x, y, z) = �1(y, x, z) ∧ �2(x, y, z) = �2(y, x, z))

⇒ MT�1,�2

is symmetric, (13)


(∀ x, z ∈ [0, 1])(∀ u, v ∈ [0, 1])(�1(x, u, z) = �1(x, v, z) ∧ �2(x, u, z) = �2(x, v, z))

⇒ MT�1,�2

is left-restrictable, (14)

(∀ y, z ∈ [0, 1])(∀ u, v ∈ [0, 1])(�1(u, y, z) = �1(v, y, z) ∧ �2(u, y, z) = �2(v, y, z))

⇒ MT�1,�2

is right-restrictable, (15)

(∀ x, z ∈ [0, 1])(∀ u, v ∈ [0, 1])(�1(x, u, z) = �1(x, v, z)) ⇒ MT�1,�2

is weak left-restrictable, (16)

(∀ y, z ∈ [0, 1])(∀ u, v ∈ [0, 1])(�1(u, y, z) = �1(v, y, z)) ⇒ MT�1,�2

is weak right-restrictable. (17)

Because of Proposition 1, we can achieve this by showing that for all A, B ∈ F(X):

(min(aT , bT )�cT � max(aT , bT ) ⇒ �1(aT , bT , cT ) = �2(aT , bT , cT ))

⇒ (A ⊆ B ∨ B ⊆ A ⇒ MT�1,�2

(A, B) = 1), (18)

(aT �cT �bT ⇒ �1(aT , bT , cT ) = �2(aT , bT , cT )) ⇒ (A ⊆ B ⇒ MT�1,�2

(A, B) = 1), (19)

�1(aT , bT , 0) = 0 ⇒ (A ∩T B = ∅ ⇒ MT�1,�2

(A, B) = 0), (20)

�1(aT , bT , cT ) = �1(bT , aT , cT ) ∧ �2(aT , bT , cT ) = �2(bT , aT , cT ))

⇒ MT�1,�2

(A, B) = MT�1,�2

(B, A), (21)

(∀ u, v ∈ [0, 1])(�1(aT , u, cT ) = �1(aT , v, cT ) ∧ �2(aT , u, cT ) = �2(aT , v, cT ))

⇒ MT�1,�2

(A, B) = MT�1,�2

(A/supp A, B/supp A), (22)

(∀ u, v ∈ [0, 1])(�1(u, bT , cT ) = �1(v, bT , cT ) ∧ �2(u, bT , cT ) = �2(v, bT , cT ))

⇒ MT�1,�2

(A, B) = MT�1,�2

(A/supp B, B/supp B), (23)

(∀ u, v ∈ [0, 1])(�1(aT , u, cT ) = �1(aT , v, cT )) ⇒ MT�1,�2

(A, B)�MT�1,�2

(A/supp A, B/supp A), (24)

(∀ u, v ∈ [0, 1])(�1(u, bT , cT ) = �1(v, bT , cT )) ⇒ MT�1,�2

(A, B)�MT�1,�2

(A/supp B, B/supp B). (25)

It is not hard to see that, for all A, B ∈ F(X),

A ⊆ B ⇒ ‖A ∩T A‖�‖A ∩T B‖, (26)

A ⊆ B ⇒ ‖A ∩T B‖�‖B ∩T B‖. (27)

These simple properties are the key for the proofs of (18) and (19):

Proof of (18). Because of (26) we have

A ⊆ B ∨ B ⊆ A ⇒ aT �cT ∨ bT �cT ⇔ min(aT , bT )�cT ,

and from (27) follows that

A ⊆ B ∨ B ⊆ A ⇒ cT �aT ∨ cT �bT ⇔ cT � max(aT , bT ).

Hence, A ⊆ B ∨ B ⊆ A implies that min(aT , bT )�cT � max(aT , bT ). By combining this with the antecedent of(18), we get �1(aT , bT , cT ) = �2(aT , bT , cT ), and thus MT

�1,�2(A, B) = 1. �


Proof of (19). From A ⊆ B follows, by (26) and (27), that aT �cT �bT . Combined with the antecedent of (19), thisimplies �1(aT , bT , cT ) = �2(aT , bT , cT ), and thus MT

�1,�2(A, B) = 1. �

From the definition of the relative sigma count follows immediately that

A = ∅ ⇐⇒ ‖A‖ = 0 (28)

for each A ∈ F(X), and the commutativity of T implies that

A ∩T B = B ∩T A (29)

for all A, B ∈ F(X). This is all we need to show (20) and (21):

Proof of (20). From A∩T B = ∅ follows, by (28), that cT = 0. Together with the antecedent of (20), this implies that�1(aT , bT , cT ) = 0, and thus MT

�1,�2(A, B) = 0. �

Proof of (21). Suppose that the antecedent of (21) holds, i.e., �i (‖A ∩T A‖, ‖B ∩T B‖, ‖A ∩T B‖) = �i (‖B ∩TB‖, ‖A∩T A‖, ‖A∩T B‖), for i ∈ {1, 2}. Because of (29), we can rewrite this as �i (‖A∩T A‖, ‖B∩T B‖, ‖A∩T B‖) =�i (‖B ∩T B‖, ‖A ∩T A‖, ‖B ∩T A‖). Hence, MT

�1,�2(A, B) = MT

�1,�2(B, A). �

The remaining implications can be proven by means of the following obvious properties:

‖A ∩T B‖ = ‖A/supp A ∩T B/supp A‖, (30)

‖B ∩T B‖�‖B/supp A ∩T B/supp A‖ (31)

for all A, B ∈ F(X). We only provide the proofs of (22) and (24), since the other two are analogous:

Proof of (22). If the antecedent of (22) holds, then �i (aT , bT , cT ) = �i (aT , bT , cT ), for i ∈ {1, 2}. Moreover, by(30) we have �i (aT , bT , cT ) = �i (aT , bT , cT ). Hence, MT

�1,�2(A, B) = MT

�1,�2(A/supp A, B/supp A). �

Proof of (24). By combining the antecedent of (24) with (30), we get that �1(aT , bT , cT ) = �1(aT , bT , cT ). More-over, since �2 is increasing in its second argument, we have, by (31), that �2(aT , bT , cT )��2(aT , bT , cT ). Hence,MT

�1,�2(A, B)�MT

�1,�2(A/supp A, B/supp A). �

5. The case � = TP

The inequality ‖A∩TPB‖�

√‖A ∩TP

A‖ · ‖B ∩TPB‖, with A, B ∈ F(X), corresponds to the well-known Cauchy–

Schwartz inequality. Hence, it holds for all A, B ∈ F(X). Consequently, we have

(cTP�

√aTP

· bTP⇒ 0��1(aTP

, bTP, cTP

)��2(aTP, bTP

, cTP))

⇐⇒ 0��1(aTP, bTP

, cTP)��2(aTP

, bTP, cTP

)

and thus

(cTP�

√aTP

· bTP⇒ 0��1(aTP

, bTP, cTP

)��2(aTP, bTP

, cTP)) ⇒ 0�MTP

�1,�2(A, B)�1 (32)

since (8) holds. By Proposition 1, this implies that

(∀ x, y, z ∈ [0, 1])(z�√x · y ⇒ 0��1(x, y, z)��2(x, y, z)) ⇒ MTP

�1,�2is [0, 1]-valued. (33)

We could weaken the other previously introduced constraints in a similar way. However, this would not increase theirpractical value much, because the parts of these constraints that refer to �1 and �2 are equalities, and knowing that aninequality holds is usually not that helpful when we have to show that an equality holds.


For the case � = TP , we can also introduce a constraint for an additional property:

(∀ x, y, z ∈ ]0, 1])(z�√x · y ⇒ �1(x, y, z) > 0) ⇒ MTP

�1,�2is ∩TP

-coexclusive. (34)

Because of Proposition 1, this can be proven by demonstrating that

(aTP> 0 ∧ bTP

> 0 ∧ cTP> 0 ⇒ (cTP

�√

aTP· bTP

⇒ �1(aTP, bTP

, cTP) > 0))

⇒ (MTP�1,�2

(A, B) = 0 ⇒ A ∩TPB = ∅). (35)

We base the proof of this implication on the fact that

A = ∅ ⇐⇒ ‖A ∩TPA‖ = 0 (36)

for all A ∈ F(X), and also on the property

A ∩T ∅ = ∅ (37)

for each A ∈ F(X), with T an arbitrary t-norm:

Proof of (35). Since cTP�

√aTP

· bTP, we have (cTP

�√

aTP· bTP

⇒�1(aTP, bTP

, cTP)>0) ⇐⇒ �1(aTP

, bTP,

cTP)>0. Hence, �1(aTP

, bTP, cTP

) = 0 ⇒ aTP= 0 ∨ bTP

= 0 ∨ cTP= 0 if the antecedent of (35) holds. Now,

if MTP�1,�2

(A, B) = 0 then �1(aTP, bTP

, cTP) = 0. Consequently, we have aTP

= 0 ∨ bTP= 0 ∨ cTP

= 0 and hence,by (36), A = ∅ ∨ B = ∅ ∨ cTP

= 0. Each formula of this disjunction implies A ∩TPB = ∅ by (37) and (28). �

Note that (36) is not satisfied for an arbitrary t-norm. For instance, if X = {x1} and A = {(x1, 0.5)} then ‖A∩TLA‖ = 0

while A �= ∅. However, (36) does hold for each t-norm that has no zero divisors, and hence in particular for TM.

6. The case � = TM

One can easily verify that ‖A ∩TMB‖� min(‖A ∩TM

A‖, ‖B ∩TMB‖) holds for all A, B ∈ F(X). Thus, since (8),

(18) and (19) hold, we have

(cTM� min(aTM

, bTM) ⇒ 0��1(aTM

, bTM, cTM

)��2(aTM, bTM

, cTM)) ⇒ 0�MTM

�1,�2(A, B)�1, (38)

�1(aTM, bTM

, min(aTM, bTM

)) = �2(aTM, bTM

, min(aTM, bTM

))

⇒ (A ⊆ B ∨ B ⊆ A ⇒ MTM�1,�2

(A, B) = 1), (39)

(aTM�bTM

⇒ �1(aTM, bTM

, aTM) = �2(aTM

, bTM, aTM

)) ⇒ (A ⊆ B ⇒ MTM�1,�2

(A, B) = 1) (40)

for all A, B ∈ F(X). Moreover, we can easily convert the proof of (35) into a proof of

(aTM> 0 ∧ bTM

> 0 ∧ cTM> 0 ⇒ (cTM

� min(aTM, bTM

) ⇒ �1(aTM, bTM

, cTM) > 0))

⇒ (MTM�1,�2

(A, B) = 0 ⇒ A ∩TMB = ∅). (41)

By Proposition 1, this implies that the implications

(∀ x, y, z ∈ [0, 1])(z� min(x, y) ⇒ 0��1(x, y, z)��2(x, y, z)) ⇒ MTM�1,�2

is [0, 1]-valued, (42)

(∀ x, y ∈ [0, 1])(�1(x, y, min(x, y)) = �2(x, y, min(x, y))) ⇒ MTM�1,�2

is strong reflexive, (43)

(∀ x, y ∈ [0, 1])(x�y ⇒ �1(x, y, x) = �2(x, y, x)) ⇒ MTM�1,�2

is inclusive, (44)

(∀ x, y, z ∈ ]0, 1])(z� min(x, y) ⇒ �1(x, y, z) > 0) ⇒ MTM�1,�2

is ∩TM-coexclusive (45)


hold. Furthermore, we can introduce constraints for three additional properties when � = TM:

(∀ x, y, z ∈ [0, 1])(z < max(x, y) ∧ z� min(x, y) ⇒ �1(x, y, z) < �2(x, y, z))

⇒ MTM�1,�2

is coreflexive, (46)

(∀ x, y, z ∈ [0, 1])(z < min(x, y) ⇒ �1(x, y, z) < �2(x, y, z)) ⇒ MTM�1,�2

is weak coreflexive, (47)

(∀ x, y, z ∈ [0, 1])(z < x ∧ z�y ⇒ �1(x, y, z) < �2(x, y, z)) ⇒ MTM�1,�2

is coinclusive. (48)

Because of Proposition 1, we can prove these implications by showing that

(cTM< max(aTM

, bTM) ∧ cTM

� min(aTM, bTM

) ⇒ �1(aTM, bTM

, cTM) < �2(aTM

, bTM, cTM

))

⇒ (MTM�1,�2

(A, B) = 1 ⇒ A = B), (49)

(cTM< min(aTM

, bTM) ⇒ �1(aTM

, bTM, cTM

) < �2(aTM, bTM

, cTM))

⇒ (MTM�1,�2

(A, B) = 1 ⇒ A ⊆ B ∨ B ⊆ A), (50)

(cTM< aTM

∧ cTM�bTM

⇒ �1(aTM, bTM

, cTM) < �2(aTM

, bTM, cTM

))

⇒ (MTM�1,�2

(A, B) = 1 ⇒ A ⊆ B) (51)

for all A, B ∈ F(X). We first show that

A ⊆ B ⇐ ‖A ∩TMA‖�‖A ∩TM

B‖ (52)

holds for all A, B ∈ F(X) as a preliminary step:

Proof of (52). Suppose that ‖A ∩TMA‖�‖A ∩TM

B‖. If there exists a y ∈ X such that A(y) > B(y), then we have∑x∈X

min(A(x), A(x)) =∑x∈X

A(x)

� A(y) +∑

x∈X\{y}min(A(x), B(x))

> min(A(y), B(y)) +∑

x∈X\{y}min(A(x), B(x))

=∑x∈X

min(A(x), B(x)).

This implies that ‖A∩TMA‖ > ‖A∩TM

B‖, which contradicts with ‖A∩TMA‖�‖A∩TM

B‖. Consequently, A(x)�B(x)

holds for all x ∈ X, i.e., A ⊆ B. �

Note that (52) is not valid for an arbitrary t-norm. For example, if X = {x1, x2}, A = {(x1, 1), (x2, 0.2)} andB = {(x1, 0.9), (x2, 1)}, then we get:

‖A ∩TPA‖ = 1 + 0.04

2= 0.52,

‖A ∩TPB‖ = 0.9 + 0.2

2= 0.55


and hence ‖A ∩TPA‖�‖A ∩TP

B‖, while A(x1) > B(x1). Thus, we cannot replace TM by a general t-norm in thefollowing proofs:

Proof of (49). Suppose that MTM�1,�2

(A, B) = 1. If cTM< aTM

∨ cTM< bTM

then cTM< max(aTM

, bTM). Because

cTM� min(aTM

, bTM) holds, this implies that �1(aTM

, bTM, cTM

) < �2(aTM, bTM

, cTM) by the antecedent of (49), which

contradicts with MTM�1,�2

(A, B) = 1. Hence, cTM�aTM

∧ cTM�bTM

. By combining this with (52), we get A = B. �


(A, B) = 1. If cTM< aTM

∧ cTM< bTM

then cTM< min(aTM

, bTM). By the

antecedent of (50), this implies that �1(aTM, bTM

, cTM) < �2(aTM

, bTM, cTM

), which contradicts with MTM�1,�2

(A, B) =1. Hence, cTM

�aTM∨ cTM

�bTM. By combining this with (52), we get A ⊆ B ∨ B ⊆ A. �


(A, B) = 1. We have cTM�bTM

, since cTM� min(aTM

, bTM). Hence, if cTM

< aTM

then, by the antecedent of (51), �1(aTM, bTM

, cTM) < �2(aTM

, bTM, cTM

), which contradicts with MTM�1,�2

(A, B) = 1.Thus, cTM

�aTM. By combining this with (52), we get A ⊆ B. �

To conclude this section, we explain the connection between the above-mentioned constraints for MTM�1,�2

and the

constraints for M�1�2

presented in [2].

Proposition 2. Let P, Q and R be predicates on [0, 1]3, [0, 1] and [0, 1]4, respectively. If the formulas

(∀ A, B ∈ F(X))(P (aTM, bTM

, cTM) ⇒ Q(MTM

�1,�2(A, B))), (53)

(∀ x, y, z ∈ [0, 1])(0�x + y − z�1 ⇒ (�1(x, y, z) = �1(x, y, z)

∧ �2(x, y, z) = �2(x, y, x + y − z))), (54)

(∀ x, y, z ∈ [0, 1])(0�x + y − z�1 ⇒ (P (x, y, z) ⇐⇒ R(x, y, z, x + y − z))) (55)

are all valid, then we have

(∀ x, y, u, v ∈ [0, 1])(R(x, y, u, v)) ⇒ (∀ A, B ∈ F(X))(Q(M�1�2

(A, B))). (56)

Proof. Since 0� min(x, x) + min(y, y) − min(x, y)�1 for all x, y ∈ [0, 1], we have 0�aTM+ bTM

− cTM�1 for all

A, B ∈ F(X). Hence, we can rewrite (53) as follows:

(∀ A, B ∈ F(X))((0�aTM+ bTM

− cTM�1 ⇒ P(aTM

, bTM, cTM

)) ⇒ Q(MTM�1,�2

(A, B))).

By Proposition 1, this implies

(∀ x, y, z ∈ [0, 1])(0�x + y − z�1 ⇒ P(x, y, z)) ⇒ (∀ A, B ∈ F(X))(Q(MTM�1,�2

(A, B))).

As explained in Section 3, MTM�1,�2

= M�1�2

follows from (54). Hence, a proof of

(∀ x, y, u, v ∈ [0, 1])(R(x, y, u, v)) ⇒ (∀ x, y, z ∈ [0, 1])(0�x + y − z�1 ⇒ P(x, y, z)) (57)

is sufficient to establish (56). To prove (57), suppose that

(∀ x, y, u, v ∈ [0, 1])(R(x, y, u, v))


holds. Putting u = z and v = x + y − z, we then obtain

(∀ x, y, z ∈ [0, 1])(0�x + y − z�1 ⇒ R(x, y, z, x + y − z)).

By (55), this is equivalent with

(∀ x, y, z ∈ [0, 1])(0�x + y − z�1 ⇒ P(x, y, z)). �

As an illustration of Proposition 2, consider the following example. Let P and R be predicates on [0, 1]3 and [0, 1]4,respectively, such that

(z� min(x, y) ⇒ 0��1(x, y, z)��2(x, y, z)) ⇐⇒ P(x, y, z),

(u� min(x, y) ∧ v� max(x, y) ⇒ 0��1(x, y, u)��2(x, y, v)) ⇐⇒ R(x, y, u, v)

for all x, y, u, v ∈ [0, 1]. We then have (55) if we choose �1 and �2 such that (54) holds, since

x + y − z� max(x, y) ⇐⇒ x + y − max(x, y)�z

⇐⇒ x + y + min(−x, −y)�z

⇐⇒ min(x + y − x, x + y − y)�z

⇐⇒ min(y, x)�z

for all x, y, z ∈ [0, 1], and hence z� min(x, y) is equivalent with z� min(x, y) ∧ x + y − z� max(x, y) for allx, y, z ∈ [0, 1]. Because (54) and (55) are satisfied and (38) holds for all A, B ∈ F(X), Proposition 2 implies that

(∀ x, y, u, v ∈ [0, 1])(u� min(x, y) ∧ v� max(x, y) ⇒ 0��1(x, y, u)��2(x, y, v))

⇒ M�1�2

is [0, 1]-valued (58)

is valid. The antecedent of (58) corresponds with one of the constraints that we presented in [2].

7. Examples

We consider the measures listed in Table 1. As indicated in the first two columns, all of these measures are members

of the presented family. It is not hard to see that the antecedent of (33) is not satisfied for the parameters of MTP1 , M

TP2 ,

MTP3 and M

TP11 . Therefore, we omitted the expressions for these measures. Furthermore, note that we used the equality

|A ∪T ∗M

B| = |A ∩TMA| + |B ∩TM

B| − |A ∩TMB| to shorten some of the expressions.

Most of the fuzzy similarity measures shown in the third column of Table 1 are well-known fuzzifications of existingcrisp similarity measures. The measures in the last column of the table, however, differ from the usual fuzzificationsby means of TP . For instance,

|A ∩TPB|

|A ∪T ∗PB| = |A ∩TP

B||A| + |B| − |A ∩TP

B|is normally used instead of

|A ∩TPB|

|A ∩TPA| + |B ∩TP

B| − |A ∩TPB|

as TP -based fuzzification of the Jaccard coefficient (e.g., [4]). The reason for this difference becomes clear whenwe examine the reflexivity of both fuzzifications. One can easily verify that the latter expression leads to a reflexivefuzzy similarity measure, while the former fuzzification is definitely not reflexive. For example, if X = {x1} and

A = {(x1, 0.1)} then |A ∩TPA|/|A ∪T ∗

PA| = 0.01/0.19 �= 1. Furthermore, it is also interesting to note that M

TP4

corresponds to the popular cosine similarity measure. The fact that this often-used measure is a full member of theproposed family can certainly be regarded as an advantage.


Table 1Some cardinality-based fuzzy similarity measures

�1(x, y, z) �2(x, y, z) � = TM � = TP

M�1 z x

|A ∩TMB|

|A| n/a

M�2 z y

|A ∩TMB|

|B| n/a

M�3 z min(x, y)

|A ∩TMB|

min(|A|, |B|) n/a

M�4 z

√x · y

|A ∩TMB|√|A| · |B|

|A ∩TPB|√

|A ∩TPA| · |B ∩TP

B|

M�5 z

x + y

2

2 |A ∩TMB|

|A| + |B|2 |A ∩TP

B||A ∩TP

A| + |B ∩TPB|

M�6 z max(x, y)

|A ∩TMB|

max(|A|, |B|)|A ∩TP

B|max(|A ∩TP

A|, |B ∩TPB|)

M�7 z x + y − z

|A ∩TMB|

|A ∪T ∗M

B||A ∩TP

B||A ∩TP

A| + |B ∩TPB| − |A ∩TP

B|

M�8 min(x, y) x + y − z

min(|A|, |B|)|A ∪T ∗

MB|

min(|A ∩TPA|, |B ∩TP

B|)|A ∩TP

A| + |B ∩TPB| − |A ∩TP

B|

M�9

√x · y x + y − z

√|A| · |B||A ∪T ∗

MB|

√|A ∩TP

A| · |B ∩TPB|


B| − |A ∩TPB|

M�10

x + y

2x + y − z

|A| + |B|2 |A ∪T ∗

MB|


B|2 (|A ∩TP

A| + |B ∩TPB| − |A ∩TP

B|)

M�11 max(x, y) x + y − z

max(|A|, |B|)|A ∪T ∗

MB| n/a

M�12 min(x, y) max(x, y)

min(|A|, |B|)max(|A|, |B|)

min(|A ∩TPA|, |B ∩TP

B|)max(|A ∩TP

A|, |B ∩TPB|)

Using the implications that we presented in Sections 5 and 6, we can prove properties of the considered measures.

Table 2 indicates which properties can be proven in this way. As an example, we prove that MTP10 is [0, 1]-valued.

Because of (33), it is sufficient to prove that

z�√x · y ⇒ 0� x + y

2�x + y − z (59)

for all x, y, z ∈ [0, 1]. We do this as follows:

Proof of (59). Obviously, (x + y)/2�0, and when z�√x · y we have

x + y

2= x + y − x + y

2� x + y − √

x · y (by GM �AM)

� x + y − z (by z�√x · y). �

As illustrated by this proof of (59), knowledge of the ordering of common fuzzy aggregation operators can be useful

for demonstrating that a constraint is satisfied. Also, since TM �GM �AM, the above proof suffices to show that MTP8

and MTP9 are [0, 1]-valued. Moreover, together with GM �AM �T ∗

M this proof implies that MTP4 , M

TP5 , M

TP6 and M

TP7

are [0, 1]-valued if z�√x · y ⇒ z�√

x · y for all x, y, z ∈ [0, 1], which trivially holds. Thus, thanks to the ordering

of the fuzzy similarity measures, one proof is sufficient to show that MTP4 , M

TP5 , M

TP6 , M

TP7 , M

TP8 , M

TP9 and M

TP10 are


Table 2Properties of the considered measures that can be proven using the presented constraints

MTM1 M

TM2 M

TM3 M

TM4 M

TP4 M

TM5 M

TP5 M

TM6 M

TP6 M

TM7 M

TP7 M

TM8 M

TP8 M

TM9 M

TP9 M

TM10 M

TP10 M

TM11 M

TM12 M

TP12

(UV) C C C C C C C C C C C C C C C C C C C C

(RE) C C C C C C C C C C C C C C C C C C C C

(CR) C C C C C C

(SR) C C

(WC) C C C C C C C C C C C

(IN) C C C

(CI) C C C C C C C

(EX) C C C C C C C C C C C

(CE) C C C C C C C C C C C C C C C C C C C C

(SY) C C C C C C C C C C C C C C C C C C

(LR) C

(RR) C

(WL) C C C C C C C C C C C

(WR) C C C C C C C C C C C

(UV) [0, 1]-valued, (RE) reflexive, (CR) coreflexive, (SR) strong reflexive, (WC) weak coreflexive, (IN) inclusive, (CI) coinclusive, (EX) ��-exclusive, (CE) ��-coexclusive, (SY) symmetric, (LR) left-restrictable, (RR) right-restrictable, (WL) weak left-restrictable and (WR) weak right-restrictable.

[0, 1]-valued. In addition, this also proves that MTM4 , M

TM5 , M

TM6 , M

TM7 , M

TM8 , M

TM9 and M

TM10 are [0, 1]-valued, since

the antecedent of (42) is weaker than the antecedent of (33).The proofs for the other checkmarks in Table 2 are also straightforward. We provide three additional examples to

illustrate this. Because of (46), we can show that MTM5 , M

TM7 and M

TM10 are coreflexive by proving that the implications

z < max(x, y) ∧ z� min(x, y) ⇒ z <x + y

2, (60)

z < max(x, y) ∧ z� min(x, y) ⇒ z < x + y − z, (61)

z < max(x, y) ∧ z� min(x, y) ⇒ x + y

2< x + y − z (62)

hold for all x, y, z ∈ [0, 1]. Moreover, we also have that MTM6 , M

TM8 and M

TM9 are coreflexive if these implications

hold, since TM �GM �AM �T ∗M. We establish the proofs of (60)–(62) as follows:

Proof of (60). If z < max(x, y)∧z� min(x, y) then z < x∧z�y or z < y∧z�x. In both cases we have z+z < x+y,and hence z < (x + y)/2. �

Proof of (61). Since max(x, y) = x + y − min(x, y) we can write z < max(x, y) as z < x + y − min(x, y), andfrom z� min(x, y) follows that x + y − min(x, y)�x + y − z. Hence, z < x + y − z. �

Proof of (62). If z < max(x, y)∧z� min(x, y) then z < x∧z�y or z < y∧z�x. In both cases we have z+z < x+y,and thus 0 < x + y − 2z. By adding x + y to each side of this inequality, we get x + y < 2x + 2y − 2z, which impliesthat (x + y)/2 < x + y − z. �

Note that z < max(x, y) ∧ z� min(x, y) ⇒ z <√

x · y is not satisfied for all x, y, z ∈ [0, 1]. For example,0 < max(1, 0) ∧ 0� min(1, 0) holds and 0 �< √

1 · 0. In fact, this implication cannot be valid for all x, y, z ∈ [0, 1],


since (46) would then imply that MTM4 is coreflexive, while M

TM4 (A, B) = 1 for X = {x1}, A = X and B = ∅ if we

assume 00 = 1, which we did implicitly throughout this paper. Hence, the constraint that warrants coreflexivity does

not hold for MTM4 , but this is to be expected since M

TM4 is not coreflexive. More generally, our examples illustrate that it

is often the case that a property does not hold when the corresponding constraint is not satisfied. For instance, it is not

hard to see that the measures MTM1 , M

TM2 , M

TM3 , M

TM11 and M

TM12 , for which the antecedent of (46) clearly does not hold,

are not coreflexive. Obviously, this increases the practical use of the constraints. The search for necessary conditions,however, still remains an interesting direction for future research.

8. Conclusion

We have introduced a triparametric family of cardinality-based fuzzy similarity measures, together with severalconstraints on its parameters that ensure certain properties of the generated measures. As illustrated by the examplesin the previous section, this leads to a convenient framework for constructing and analysing cardinality-based fuzzysimilarity measures. This framework has two important advantages in comparison with already existing approaches,namely, it is not limited to rational measures and, even more importantly, the generated fuzzy similarity measures arealways reflexive. Another positive feature of the proposed framework is that it provides the possibility to ensure severalforms of restrictability, which allows to reduce the computation time in practical implementations. The fact that manyof the considered examples were found to satisfy some form of restrictability, illustrates that it can be worthwhile toanalyse the properties of existing fuzzy similarity measures by means of the presented framework.

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a triparametric family of cardinality-based fuzzy similarity measures

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