Complex object comparison in a fuzzy context

N. Marın*, J.M. Medina, O. Pons, D. Sanchez, M.A. Vila

Intelligent Databases and Information Systems Research Group, Department of Computer Science and Artificial Intelligence, University of Granada,

18071 Granada, Andalucıa, Spain

Abstract

The comparison concept plays a determining role in many problems related to object management in an Object-Oriented Database Model.

Object comparison is appropriately managed in a crisp object-oriented context by means of the concepts of identity and value equality.

However, when dealing with imprecise or imperfect objects, questions like ‘To which extent may two objects be the same one?’ or ‘How

similar are two objects?’ have not a clear answer, because the equality concept becomes fuzzy. In this paper we present a set of operators that

are useful when comparing objects in a fuzzy environment. In particular, we introduce a generalized resemblance degree between two fuzzy

sets of imprecise objects and a generalized resemblance degree to compare complex fuzzy objects within a given class.

Keywords: Object-oriented model; Fuzzy database; Objects resemblance; Inclusion degree

1. Introduction

Probably one of the most important data paradigms in

both the programming [1] and the databases world [2] is the

Object-Oriented Data Model. Modeling the reality which is

behind many software problems as a set of objects grouped

around classes has proved to be a suitable approach for

many developers, particularly when dealing with complex

and dynamic problems.

This well-deserved popularity has allowed the object-

oriented data model to become one of the active research

fields in the world of Computer Science. One of the major

areas where this research has occurred has been in the field

of fuzzy database modelling. As was the case with relational

database modelling, the object-oriented data model is being

widely studied in order to accept the representation of

imperfect (i.e. imprecise, uncertain, vague) information in

the database. As a result of this research many relevant

works have appeared in the literature [3].

Identity equality is a fundamental concept in the object-

oriented data model and constitutes the basic criterion used

to distinguish objects. This notion of equality between

objects states that two objects are identical if they are the

same object, i.e. if they have the same object identifier.

However, there exists situations where this criterion is

insufficient and we need to use the concept of value equality

i.e. two objects are equal if the values of all their attributes

are recursively equal.

Query management in object-oriented databases is an

example where this last criterion is commonly used. When

the user orders the execution of a query he or she writes a set

of value conditions that must be fulfilled by the objects that

will belong to the query result. It should be noted that

identity restrictions may also appear.

When we deal with perfect information the usual set of

relational operators (including the basic equality operator)

allows us to solve these kinds of problems. If the database is

affected by imperfection the classical concept of equality is

not valid. In some cases it could be replaced by a similarity

concept or more generally by a resemblance concept.

In the context of object-oriented knowledge bases there

are some approximations to this problem. For example

Yazici et al. propose in Ref. [4] an approach to calculate the

matching between an object and the conditions of a rule.

The aim of this paper is to propose a more general way of

managing fuzzy object comparisons in a fuzzy object-

oriented data model similar to that presented in Ref. [5]. The

paper is organized as follows. Section 2 presents the

comparison problem, while Section 3 is devoted to the study

of the generalization of the value equality concept when

dealing with basic objects. In Section 4 we explain how to

compare fuzzy sets of fuzzy objects. Based on the material

presented in the previous sections, Section 5 demonstrates

how to obtain a resemblance degree between two complex

doi:10.1016/S0950-5849(03)00014-4

Information and Software Technology 45 (2003) 431–444

www.elsevier.com/locate/infsof

* Corresponding author.

E-mail addresses: nicm@decsai.ugr.es (N. Marın), medina@decsai.

ugr.es (J.M. Medina), opc@decsai.ugr.es (O. Pons), daniel@decsai.ugr.es

(D. Sanchez), vila@decsai.ugr.es (M.A. Vila).

objects in a fuzzy object-oriented environment. An

explanatory example is presented in Section 6 and finally

the paper ends with some concluding remarks and a

discussion of future work in Section 7.

2. Value equality generalization: resemblance

relationships

Consider the example illustrated in Fig. 1. The

rectangles represent two rooms characterized by their

quality, their extension, the floor they are on, and the set

of students which attend their lessons in each room. The

three first attributes that characterize the class Room may

take imprecise values: the quality is expressed by an

imprecise label and the attributes extension and floor can

be expressed using a numerical value or an imprecise

label. The set of students is fuzzy, taking into account the

percentage of time each student spends in the room

receiving the lessons.

If we want to compare these two objects we need to solve

the following tasks:

† Firstly we have to handle resemblance in basic domains.

† Secondly we also have to be able to compare fuzzy

collections of imprecise objects.

† Finally we need to aggregate the resemblance infor-

mation that we have collected by studying the attributes

and calculate a general resemblance opinion for the

whole complex objects.

The following sections are devoted to the study of each

one of these problems.

3. Resemblance in basic domains

If we wish to compare complex objects, the first level

where resemblance must be studied corresponds to basic

objects. That is, simple objects that have no OID (object

identifier) and whose definition is not made up using

attributes. We will refer to these kinds of objects as values

of a given domain.

We are going to consider the following classification of

simple objects:

† Precise values. This category of values involves all the

classical basic classes that usually appear in an object-

oriented data model (e.g. numerical classes, string

classes, etc.). Values of this kind of domains are easily

compared using the classic set of relational operators

(e.g. equal, less than, greater than, etc.). In these

situations, resemblance is implemented using the model

proposed by the classical equality.

† Imprecise values. The case of imprecise values is a bit

more complex. Different types of imprecise values must

be considered according to the semantics of the

imprecise value. As we will see, the equality concept

generalization depends on the domain nature.

We are going to consider the different kinds of simple

imprecise domains defined in the fuzzy object-oriented

model proposed in Ref. [5]. In this work, we propose the use

of linguistic labels [6–8] in order to express vagueness.

Taking into account the model presented in Refs. [9,10],

we consider three different types of imprecise basic domains

(see Fig. 2).

† Domains made up by a set of labels whose semantics

cannot be expressed over an underlying base domain (e.g.

the domain used to express the quality of a room, ‘The

quality of the room is high’, or the prospects of a student,

‘Mary’s prospects are good’). The only alternative to

handle comparison of this type is to ask the designer of the

domain for the definition of the fuzzy relation that manages

the resemblance among the labels. As an example, Table 1

represents a similarity relation for the attribute quality.

† Domains where the labels can be expressed using a fuzzy

set defined over an underlying domain. There are two

possibilities:

(1) The interpretation of the fuzzy set that represents the

label is disjunctive. That is, it is a possibility distribution

and only represents one value among a set of possible values

(e.g. in the domain used to express the extension of a room,

‘The room is big’ or the age of a student, ‘Mary is young’,

Fig. 1. The problem.

N. Marın et al. / Information and Software Technology 45 (2003) 431–444432

and so on). In this case, the domain is made up of precise

values (those of the underlying domain) and imprecise

labels, and the comparison can be managed considering a

generalization of the classical equality that holds in the

underlying domain, taking into account the definitions of the

labels.

If B stands for the basic domain, L is the labels set, and

D ¼ B < L then ;x; y;[ D we can use the following

resemblance relation (^ stands for a t-norm):

mSðx;yÞ¼

1 ðx¼yÞ^ðx;y[BÞ

0 ðx–yÞ^ðx;y[BÞ

mlðzÞ ððx¼ l[LÞ^ðy¼z[BÞÞ

_ððy¼ l[LÞ^ðx¼z[BÞÞ

supz[BðmxðzÞ^myðzÞÞ otherwise

8>>>>>>>><>>>>>>>>:

Consider for example the attribute extension of the class

Room. The basic underlying domain of this attribute is the

interval [0,1)and we add to the domain the set of labels

{small, middel-size, big} whose definition is represented in

Fig. 3.

Taking into account the definitions of the labels and

Eq. (1) we have that mSð30; 30Þ ¼ 1; mSð30; 35Þ ¼ 0; mSð30;

bigÞ ¼ 0:9; and mSðbig;middle-sizedÞ ¼ 0:5:

Although Eq. (1) is an accepted resemblance approach

for these situations, there are several alternative proposals in

the literature that describe a calculus for similarity and

resemblance measures between fuzzy sets. A complete

study of them can be found in Ref. [11].

(2) If the interpretation of the fuzzy set that represents the

labels is conjunctive then we are not comparing imprecise

values but sets.

In this last case, whether labels are used to express the set

(e.g. academic years of a student, ‘Mary is in her final

academic years’) or not (e.g. fuzzy collections of objects—

the students of a room), the comparison process is more

complicated. Section 4 will analyze this problem in depth.

4. Resemblance in fuzzy sets of fuzzy objects

In Section 3, we have presented the way of comparing the

values of the two first attributes that characterize a room.

This section studies the comparison of fuzzy collections of

objects.

Fig. 4 shows the two sets of students that we have to

compare in order to evaluate the resemblance between our

two rooms. To compare the two fuzzy sets of students we

need to generalize the fuzzy set comparison operators,

taking into account that the objects of the set may be

imprecise.

4.1. Resemblance driven inclusion degree

Conjunctive fuzzy set comparison is often done by

means of the concept of inclusion:

A ¼ B if; and only if; ðA # BÞ ^ ðB # AÞ ð2Þ

Several proposals for the calculus of this inclusion degree

can be found in the literature. In Ref. [12] the inclusion of

A in B is calculated as follows:

NðBlAÞ ¼ minu[U

{IðmAðuÞ;mBðuÞÞ} ð3Þ

where I stands for an implication operator, and mX is the

membership function that describes the fuzzy set X: The

implication operator can be chosen in accordance to the

properties we want the inclusion degree to fulfill.

Nevertheless, independently of the chosen implication

operator, this formulation supposes that both A and B are

defined over a reference universe U made up of precise

elements, where the classical equality is the basis of

Fig. 2. Three different kinds of imprecise information.

Table 1

Domain for the quality attribute

High Regular Low

High 1 0.8 0

Regular 1 0.8

N. Marın et al. / Information and Software Technology 45 (2003) 431–444 433

the comparisons. That is, the implication operator compares

the degree with which each element of the universe Ubelongs to each one of the fuzzy sets (i.e. we compare the

membership degrees of the same object to both sets).

However in the context of fuzzy databases it frequently

happens that the elements of the universe U are imprecise

objects between which classical equality cannot be applied.

Instead a similarity or a resemblance relationship must be

used. That is, for a given element in the set A; it is not clear

which element of B has to be taken in order to compare the

membership degrees. In our example, the students may be

defined with imprecision and two different students (from

the identity equality point of view) may be the same one

(from the value equality point of view). Let us study a way

to solve this problem.

If the reference universe U is formed by imprecise

elements, Eq. (3) can be generalized as follows.

Definition 1. (Resemblance driven inclusion degree). Let A

and B be two fuzzy sets defined over a finite reference

universe U, S be a resemblance relation defined over the

elements of U, and ^ be a t-norm. The inclusion degree of A

in B driven by the resemblance relation S is calculated as

follows:

QSðBlAÞ ¼ minx[U

maxy[U

uA;B;Sðx; yÞ ð4Þ

uA;B;Sðx; yÞ ¼ ^ðIðmAðxÞ;mBðyÞÞ;mSðx; yÞÞ ð5Þ

In Eq. (3), the inclusion degree is calculated taking into

account the element of the reference universe U that fulfills

in a lower degree the implication condition between the

membership degrees of this element to both sets. In Eq. (4)

since the membership degrees of similar (but not necessarily

equal) elements can be compared, we restrict the impli-

cation degree using the resemblance degree of the two

elements. In summary, for each element that belongs with a

certain degree to the set A we look for a quite similar object

in U that belongs to the set B with a higher degree.

Consider the following example: let A ¼ 0:9=a þ 1=d and

B ¼ 1=a þ 0:7=b þ 0:9=c be two fuzzy sets, where {a; b; c; d}

is the reference universe U over which a resemblance

relation S is defined, such that mSða; bÞ ¼ mSða; cÞ ¼

mSða; dÞ ¼ mSðb; cÞ ¼ mSðb; dÞ ¼ 0 and mSðc; dÞ ¼ 0:7:

In this situation:

QSðBlAÞ ¼min{

max{^ðIðmAðaÞ;mBðaÞÞ;mSða;aÞÞ;…^ðIðmAðaÞ;

mBðdÞÞ;mSða;dÞÞ};…;

max{^ðIðmAðdÞ;mBðaÞÞ;mSðd;aÞÞ;…;^ðIðmAðdÞ;

mBðdÞÞ;mSðd;dÞÞ}}:

If we use the product as t-norm and Eq. (6) as implication

operator, then:

QSðBlAÞ ¼min{max{1;0;0;0};max{0;1;0;0};

max{0;0;1;0:7};max{0;0;0:63;0}}

¼min{1;1;1;0:63}¼ 0:63:

Iðx;yÞ ¼1; if x# y

y=x; otherwise

(ð6Þ

4.1.1. Properties of the resemblance driven inclusion degree

The first property that the resemblance driven inclusion

degree ðQÞ must fulfill is to be valid when the resemblance

relation is the classic equality.

Proposition 1. Let A and B be two fuzzy sets defined over a

finite reference universe U over which a resemblance

relation S is defined. Let S be the classic equality relation.

Then, Q¼ðBlAÞ ¼ NðBlAÞ:

Proof 1. Q¼ðBlAÞ ¼ minx[Umaxy[UuA;B;¼ðx; yÞ: Since

uA;B;¼ðx; yÞ ¼ ^ðIðmAðxÞ;mBðyÞÞ; m¼ðx; yÞÞ and taking into

account that;x – y;m¼ðx; yÞ ¼ 0 then;x – y;uA;B;¼ðx; yÞ ¼

^ðIðmAðxÞ;mBðyÞÞ; 0Þ ¼ 0 which yields ;x [ U;

maxy[UuA;B;¼ðx; yÞ ¼ uA;B;¼ðx; xÞ ¼ ^ðIðmAðxÞ;mBðxÞÞ; 1Þ ¼

IðmAðxÞ;mBðxÞÞ and thus Q¼ðBlAÞ ¼ minx[U{IðmAðxÞ;

mBðxÞÞ} ¼ NðBlAÞ (as we want to demonstrate). A

The inclusion degree must be monotonic with respect to

the inclusion relationship.

Fig. 3. Extension.

Fig. 4. Resemblance in fuzzy collections.

Proposition 2. Let A, B, and C be three fuzzy sets

defined over a finite reference universe U over which a

resemblance relation S is defined. Then, A # B )

QSðClAÞ $ QSðClBÞ:

Proof 2. A # B ) ;x [ U; mAðxÞ # mBðxÞ: That is, ;x [U; mAðxÞ þ 1 ¼ mBðxÞ; with 1 $ 0: Also SupportðAÞ #SupportðBÞ: By the properties of implications, QSðClAÞ ¼minx[Umaxy[UuA;C;Sðx; yÞ ¼ minx[Umaxy[U^ðIðmAðxÞ;

mCðyÞÞ;mSðx; yÞÞ $ minx[Umaxy[U^ðIððmAðxÞ þ 1Þ;mCðyÞÞ;

mSðx; yÞÞ; ;1 $ 0: Particularly, QSðClAÞ $ minx[U

maxy[U^ ðIðmBðxÞ;mCðyÞÞ;mSðx; yÞÞ ¼ QSðClBÞ: A

The following property also holds.

Proposition 3. Let A, B, and C be three fuzzy sets

defined over a finite reference universe U over which a

resemblance relation S is defined. Then, A # B )

QSðBlCÞ $ QSðAlCÞ:

Proof 3. A # B ) ;x [ U; mAðxÞ # mBðxÞ: That is, ;x [U; mAðxÞ þ 1 ¼ mBðxÞ; with 1 $ 0: Also SupportðAÞ #SupportðBÞ: By the properties of implications, QSðBlCÞ ¼

minx[Umaxy[UuC;B;Sðx; yÞ ¼ minx[Umaxy[U^ ðIðmCðxÞ;

mBðyÞÞ; mSðx; yÞÞ $ minx[Umaxy[U^ðIððmCðxÞÞ; mBðyÞ 2

1Þ; mSðx; yÞÞ; ;1 $ 0: Particularly, QSðBlCÞ $ minx[U

maxy[U^ ðIðmCðxÞ; mAðyÞÞ; mSðx; yÞÞ ¼ QSðAlCÞ: A

4.2. Matching resemblance opinions

When comparing two fuzzy sets of imperfect

elements we can consider both inclusion directions, as

shown in Eq. 2, by means of a resemblance driven

inclusion degree (Fig. 5).

However to take this approach we need to propose a way

to match both inclusion degrees in order to obtain the

resemblance degree between the two fuzzy sets.

Using Eq. (2) as a basis we can define a resemblance

degree operator between two fuzzy sets as follows.

Definition 2. (Generalized resemblance between fuzzy sets).

Let A and B be two fuzzy sets defined over a finite reference

universe U over which a resemblance relation S is defined,

and ^ be a t-norm. The generalized resemblance degree

between A and B restricted by ^ is calculated by means of

the following formulation:

S;^ðA;BÞ ¼ ^ðQSðBlAÞ;QSðAlBÞÞ ð7Þ

Since every t-norm is commutative and the inclusion

operator Q is idempotent, then the above measure is a

resemblance relation.

For example, let A ¼ 1=a þ 1=b þ 1=c and B ¼ 1=d be

two fuzzy sets, such that {a; b; c; d} is a subset of a universe

U over which a resemblance relation S is defined, and

mSða; dÞ ¼ mSðb; dÞ ¼ mSðc; dÞ ¼ 0:5: In this situation:

QSðBlAÞ ¼ 0:5 ¼ QðAlBÞ: Using minimum as t-norm,

S;minðA;BÞ ¼ 0:5:

4.2.1. Cardinality ratio

In the last example, the generalized resemblance

operator ( ) has indicated that both fuzzy sets of objects

have a resemblance equal to 0.5. Now, suppose that C ¼

1=a is another fuzzy set defined over the same universe.

In this case, QSðBlCÞ ¼ 0:5 ¼ QSðClBÞ and S;minðC;BÞ ¼

The above example illustrates one drawback of the use of

the generalized resemblance operator ( ). The use of

resemblance relations instead of equality when calculating

inclusion degrees, may make the generalized resemblance

between two fuzzy sets be distinct to 0, even if there is a

great difference on terms of cardinality. In the above

example the cardinality of A and B is different whereas for B

and C it is the same. However this is not taken into account

when determining the resemblance degree.

In some situations cardinality may not be important.

Imagine that we are comparing two sets of tools: we are

probably looking for similar capabilities and thus the

number of tools is not relevant. However in the case of

students in a room the actual number of students may be

important when comparing the sets.

If we wish to distinguish between these kinds of

situations we need to weight the generalized resemblance

degree with a factor that takes into account the distance

between the cardinalities of the fuzzy sets that are being

compared (Fig. 6).

Definition 3. (Cardinality ratio). Let A and B be two fuzzy

sets. We define the cardinality ratio between A and B as a

measure of the relative resemblance between their cardin-

alities. This ratio can be calculated as follows:

FðA;BÞ ¼

1; if A ¼ B ^ B ¼ B

minðlAl; lBlÞmaxðlAl; lBlÞ

; otherwise

8><>: ð8Þ

The ratio reaches its maximum value (1) when both sets

present the same cardinality and approaches 0 when the

cardinality of one set is much greater than the other.Fig. 5. Two directions of inclusion.

The cardinality ratio F preserves commutativity and

idempotency. Consequently its use as a weighting factor for

does not prevent it being a resemblance measure.

Using the cardinality ratio in the example, we have that

S;minðA;BÞ ¼ 0:17 and S;minðC;BÞ ¼ 0:5:

4.2.2. Consistency degree between fuzzy sets

In Section 4.2.1 we have studied the way of matching

both directions of inclusion into the one resemblance

degree. To achieve this we have proposed combining the

degrees using a t-norm. The use of a t-norm as an

aggregation function may be a bit restrictive, but it is

suitable if we want to use resemblance to generalize the

concept of equality.

In some cases it is interesting to study the consistency

degree between the definition of two fuzzy sets instead of

studying the resemblance degree. Imagine we are interested

in the resemblance of two collections and we only want to

know to which extent one of the collections can be

completed to match the other collection. In this case a less

restrictive resemblance measure would be of interest.

Definition 4. (Consistency degree between fuzzy sets). Let A

and B be two fuzzy sets defined over a finite reference

universe U over which a resemblance relation S is defined,

and % a t-conorm. The consistency degree between A and B

restricted by % is calculated as follows:

S;%ðA;BÞ ¼ %ðQSðBlAÞ;QSðAlBÞÞ ð9Þ

With the consistency degree we intend to calculate to which

extent there exists an inclusion relation between the

contents of both sets (in at least one of the two directions).

This measure, as the reader can easily see, is also a

resemblance measure.

5. Calculating objects resemblance

The problem we want to solve is to establish the

resemblance degree between two objects that belong to the

same class. In the previous sections we have seen the way to

calculate the resemblance between basic objects and fuzzy

sets of objects. Let us now consider the comparison of more

general kind of objects.

An object can be viewed as a set of values that

correspond with the set of attributes that describe the type

of the class the object belongs to. Each of these values

can be in turn a basic object, another object, or a fuzzy set

of objects (basic or not). In general, an object is

considered to be complex if other objects take part in

its definition.

When comparing two objects, the starting point is to

study the resemblance between the attribute values that

describe their definition. The previous sections have

provided us with tools to face the attribute comparison. In

this section we are going to explain the way to aggregate the

resemblance degrees obtained from the comparison of the

attributes values in order to get a general resemblance

measure of the two objects.

5.1. Formal definition of the problem

Firstly, let us formally define the problem. Let o1 and o2

be two objects of the class C; characterized by the type TC;

whose structural component StrC is characterized by the

attributes set {a1; a2;…; an}: Our goal is to find the

resemblance between o1 and o2 from the aggregation of

the resemblance degrees that can be observed in the values

of their attributes.

We can outline the problem as a multi criteria

aggregation problem, where different resemblance opinions

(one for each pair of values of the same attribute) must be

aggregated to obtain a resemblance consensus. Let

Saiðo1; o2Þ stand for the resemblance degree observed

between the values of attribute ai in objects o1 and o2; and

Sðo1; o2Þ stand for the aggregated resemblance opinion we

want to calculate.

5.2. Attribute importance

It is natural to think that not all of the resemblance

degrees calculated for the attribute values of the objects

being compared have the same weight when computing the

resemblance between the objects. For example, if we are

comparing people, some attributes may be more determin-

ing (e.g. name, father, brother) while others may be less

determining (e.g. weight, age).

In order to reflect this fact, let us consider that every

attribute ai has associated a weight paithat points out the

importance that the resemblance in this attribute must have

when computing the resemblance degree between objects of

this class. Without sacrificing generality we are going to

consider that ;i; pai[ ½0; 1�:

5.3. Resemblance values aggregation

The calculus of the resemblance degree between two

objects of the same class is governed by the following

Fig. 6. Need for a cardinality ratio.

special vague sentence: Two objects are similar if: ‘Most of

the important attributes of the class present similar values in

the objects’.

Let us consider some interesting points with respect to

the previous sentence:

† The resemblance between the values that both objects

have for a given attribute can be measured by means of a

degree. In general, we will have a set of values

{Saiðo1; o2Þ [ ½0; 1�}:

† The importance of an attribute in determining

the resemblance between two objects must be

given by the class designer. In general, we will

have a set of values {pai}; also belonging to the [0,1]

interval.

† The term most is a linguistic quantifier that denotes

the extent to which we want the whole set of

attributes to be taken into account when computing

the resemblance degree between two objects of the

class.

In the literature there are two main types of vague

sentences involving linguistic quantifiers [13,14]. They can

be represented as follows:

† Type I: Q of X are A (e.g. ‘Most of the students are tall’).

† Type II: Q of D are A (e.g. ‘Most of the tall students are

intelligent’).

where Q is a linguistic quantifier, X is a finite reference

universe, and A and D are vague properties, both defined as

fuzzy sets over X with membership functions mA and mD;

respectively.

The vague expression that begins this section can be

easily transformed in a Type II sentence. We only have to

consider the following:

† Q is the quantifier most.

† X ¼ {a1; a2;…; an} is the set of attributes that charac-

terize the type of the class.

† D is the set of the relevant attributes for the resemblance

calculus.

† A is the set of the attributes that have similar values in

both objects.

The membership function of these latter fuzzy sets are as

follows:

mDðaiÞ ¼ pai; ;ai [ X ð10Þ

mAðaiÞ ¼ Saiðo1; o2Þ; ;ai [ X ð11Þ

The next step in the resemblance calculation process is to

calculate the accomplishment degree of the Type II

sentence from the quantifier and the previously defined

membership functions. The literature offers very different

approaches to solve this kind of problems in the fuzzy

query environment:

† Zadeh’s approach [14] calculates the accomplish-

ment degree using the relative cardinal of A with

respect to D ðaA=DÞ; computing the compatibility

degree of this cardinal with the quantifier Q

definition:

ZQðA=DÞ¼mQðaA=DÞ¼mQ

ðmAðxÞ>mDðxÞÞXx[X

mDðxÞ

1CCA ð12Þ

† Yager’s approach [13] is founded on the use of the

Ordered Weighted Average (OWA) [15,16] aggrega-

tion operator. This operator involves the calculus of

two sequences of values and its ranking from the

highest to the lowest value:

·{ci¼mAðxiÞ_ð12mDðxiÞÞ}; with xi[X:

·{di¼mDðxiÞ}; with xi[X:

If {bi} and {ei} are the descendant ordered permutations

of {ci} and {di} (respectively), then the accomplishment

degree is given by the following formulation:

YQðA=DÞ¼Xn

wi·bi ð13Þ

where wi¼mQðei=Pn

1diÞ2mQððei=Pn

1diÞ21Þ

† Vila’s approach [17] is founded on the concept of

coherent family of quantifiers. Let us look at this

proposal in more detail.

Definition 5. (Coherent family of quantifiers). Let Q ¼

{Q1;…;Ql} be a set of linguistic quantifiers. Q is a

coherent family of quantifiers if it verifies the following

properties:

(i) The membership functions of the elements of Q are

non-decreasing.

(ii) A partial order relation X is defined in Q, and this

relation has as maximal element Q1 ¼ ’ and as

minimal element Ql ¼ ;: Besides, ;Qi; Qj [ Q;Qi # Qj ) Qj X Qi:

(iii) The membership function of the ’ quantifier is

mQ1ðxÞ ¼ 1 if x – 0 and mQ1

ð0Þ ¼ 0; and the

membership function of the ; quantifier is mQlðxÞ ¼

0 if x – 1 and mQ1ð1Þ ¼ 1:

The basic idea is to consider that the fulfillment degree of

the sentence is between the accomplishment degree of the

extreme sentences ‘There exists a D that is A’ and ‘All Ds

are A’:

† The accomplishment degree of the sentence ‘There exists

a D that is A’ is calculated as follows:

’x [ X; ðmDðxÞ ^ mAðxÞÞ ¼ maxx

ðmDðxÞ ^ mAðxÞÞ ð14Þ

† The accomplishment degree of the sentence ‘All Ds are

A’ is the next one:

;x[X;ðmDðxÞ!mAðxÞÞ¼minx

ðmAðxÞ_ð12mDðxÞÞ ð15Þ

From the above Vila’s approach for the calculus of the

accomplishment degree of a Type II sentence can be carried

out as follows.

Definition 6. (Accomplishment degree VQ). Let D and A be

two fuzzy sets with the same domain X and Q be a coherent

family of quantifiers such that every Qi has associated a

value gi in [0,1] verifying:

g’ ¼ 1; g; ¼ 0; ;Q; Q0 [ Q; Q X Q0Þ ) gQ $ gQ0 : ð16Þ

In these conditions, the accomplishment degree of the

sentence ‘Q Ds are A’ is calculated by means of the

following formulation:

VQðA=DÞ ¼ gQ maxx

ðmDðxÞ ^ mAðxÞÞ

þ ð1 2 gQÞminx

ðmAðxÞ _ ð1 2 mDðxÞÞÞ ð17Þ

As gi values, the authors propose the use of a value based on

the orness measure given by Yager in Ref. [15]. Such an

approximation is calculated as follows:

oQ ¼ð1

0mQðxÞdx ð18Þ

The final expression for the calculus of the accomplishment

degree is:

VQðA=DÞ ¼ oQ maxx

ðmDðxÞ ^ mAðxÞÞ

þ ð1 2 oQÞminx

ðmAðxÞ _ ð1 2 mDðxÞÞÞ ð19Þ

We propose this latter approach for the aggregation of

resemblance opinions. We do so because it has similar

behaviour to that of Yager when the quantifier is close to the

; quantifier (as is the case here) and is easier to calculate

(the calculus of oQ has to be performed only once for each

quantifier). With respect to Zadeh’s approach, this method is

less strict and has a similar calculus complexity. A

comparative study of the three approaches can be found in

Ref. [18].

Fig. 7 summarizes the process. In order to compare

objects o1 and o2; we first compare their attribute values,

obtaining the partial resemblance opinions Saiðo1; o2Þ: Then

using VQ we obtain a general resemblance measure between

the two objects.

In order to apply Vila’s approach to the problem of

resemblance aggregation we only need to determine the

semantics of the quantifier we want to use. Instead of

using the same quantifier to compute resemblance in all

the classes, it is reasonable to let the designer of the class

type choose the quantifier that must be used to compare

the objects of the class. That is, the designer will give the

value of gQ he wants to use to combine both extreme

cases. In this way, at the same time that we give freedom

to the designer, we simplify a lot the calculus of the

accomplishment degree.

5.4. Complex objects: recursivity

From the previous sections it may seem that the

problem of object resemblance is solved: couples of

attribute values are compared obtaining partial resem-

blance degrees, and then a final degree is obtained by

means of an aggregation process. However, the high

interrelation amongst data in the object-oriented data

model causes complexity in the calculus of object

resemblance in a given class. We refer to the complex

objects case and the cycle presence in the relationships

graph associated to a given schema.

In order to deal with the resemblance calculus

between complex objects, the only solution is to

propagate the problem by means of recursivity. Suppose

that we are comparing objects o1 and o2; and the values

of a given attribute in these objects are objects o3 and

o4: To compute the resemblance between o1 and o2; we

previously need to compute the resemblance between o3

and o4; and so on.

The use of recursivity is not a problem unless there are

cycles in the relationships graphs. For example: Let us

consider the graph in Fig. 8. Suppose that we have two

objects of class A; namely, oA1 and oA2; and we want to

calculate their resemblance degree. When we analyze

attribute a; we will need to study the resemblance degree

between two objects of the class B; for example oB1 and oB2:

When trying to solve this latter resemblance, we will need to

study the resemblance between two objects oC1 and oC2:

Finally, to know the resemblance between these objects, we

will need to compute the resemblance between two objects

of the initial class A:

The cycle in the above example introduces the following

problems:

† Firstly, we may have to solve a comparison problem in

the same class that was our starting point. This increases

the complexity of the problem and suggests the necessity

of exploring a wide data set in order to establish the

objects resemblance.

† Secondly, it may be that the resemblance degree of the

objects we want to compare is part of the recursive tree

generated in the computation. For example, if attribute c

led back to object oA1 and oA2 again, we would have an

infinite cycle.

The first problem is unavoidable because is due to the

high interrelation of data in the object-oriented model. The

second is far more dangerous because it prevent us from

ending the computation. We can solve it by means of the

following:

† Not propagating recursively, ignoring this in the general

calculus.

† Approximating the value in some way, making several

iterations until reaching the final value.

† Directly approximating the value with another semanti-

cally valid one.

The first alternative (not propagating recursively) is not

suitable because we may be ignoring important information.

Although the second alternative (the use of an initial

approximation and then iterate) is acceptable it requires a

considerable calculus effort and a higher algorithmic

complexity (more than one cycle may appear in a normal

recursivity process).

Our proposal is to use the third alternative. We can

unfold the objects resemblance in two different ways:

one that expresses the surface resemblance between the

objects and the other based on an object exploration in

depth. The first will be based on the object attributes

which will not involve the cycling problem and the other

will be a resemblance obtained taking into account the

attributes that need recursive monitoring. The surface

resemblance can be used as an approximation when, in

the calculus of the second one (the real resemblance),

a cycle is detected (see Fig. 9). Let us explore this in

more detail.

5.4.1. Surface resemblance

We are going to separate, into two different partitions,

the attributes that characterize the structural component of

the type of a class, in accordance with their possibility of

getting into a cycle within the application scheme where

they are defined.

Definition 7. (Superficial attributes). Let C be a class whose

type is made up by the set of attributes StrC ¼

{a1; a2;…; an}: We define the set SStrC of superficial

attributes of C as the subset of attributes of StrC that cannot

get into a cycle that returns to C in the schema where the

class is defined.

5.4.2. Deep resemblance: resemblance between two objects

Using Definition 7 and ideas presented in the previous

sections, we can define the calculus of the resemblance

between two objects of a given class as follows.

Definition 8. (Resemblance between two objects). Let C

be a class whose type is made up by the set of

attributes StrC ¼ {a1; a2;…; an}: Let o1 and o2 be two

objects of C: We define the resemblance (SR) between

the objects o1 and o2 as the value returned by

the following function:

SR :FC£OðFCÞ£OðFCÞ£PðOðFCÞ£OðFCÞÞ£{0;1}!½0;1�

Fig. 7. Dealing with complex object comparison.

Fig. 8. The problem of cycles.

where FC is the family of all the classes and OðFCÞ is

the set of all the objects that exist in the database. The

calculus of SRðC;o1;o2;V;tÞ involves the following case

selections:

If o1¼o2; then:

SRðC;o1;o2;V;tÞ¼1 ð20aÞ

If exists a resemblance relation S defined in C; then:

SRðC;o1;o2;V;tÞ¼mSðo1;o2Þ ð20bÞ

If o1 and o2 are fuzzy sets then:

SRðC;o1;o2;V;tÞ¼ SRV;^ðo1;o2Þ¼

^ðQSRVðo2lo1Þ;QSRV

ðo1lo2ÞÞð20cÞ

If ðo1;o2Þ[V^ t–1; then:

SRðC;o1;o2;V;1Þ ð20dÞ

Otherwise:

oQ maxi:ai[A

ðpai^SRðCai

;o1·ai;o2·ai;V<ðo1;o2Þ;0ÞÞþ

ð12oQÞ mini:ai[A

ðSRðCai;o1·ai;o2·ai;V<ðo1;o2Þ;0Þ_ð12pai

ð20eÞ

QSRVðolo0Þ¼ min

x[Supportðo0Þmax

y[SupportðoÞ{Iðmo0 ðxÞ;moðyÞÞ;

SRðCD;x;y;V;0Þ}

where CD stands for the class that is the reference

universe of the sets,

A¼SStrC if t¼1

StrC if t¼0

and Caiis the domain class of the attribute ai:

The latter function, in spite of the complexity of its

definition, is just a case selection that proposes the different

tools that can be used to compare both objects:

† There are two basic cases:

·When an identity equality holds between the objects.

·When a known defined resemblance relation exists in the

class.

† The third case uses the generalized resemblance degree

to compare two fuzzy sets of objects, using recursivity

in order to compare the elements of the sets.

† The fourth case uses the variable V to determine the

existence of a cycle. If this is the case, a recursive

call is made that only focuses on the superficial

attributes.

† The last case is a general recursivity model, in

which, according to the parameter t; the aggregation

operator VQ is applied over the recursive calls using

couples of attribute values. If t is equal to 1, only

the surface is explored. Otherwise all attributes are

studied in depth.

Because the properties of the operators used in each of

the basic cases are those of a resemblance relation, the

operator SR is also a resemblance measure amongst objects

of a given class.

5.5. Organization in an object oriented model

The representation in the object oriented data model

of this approach for the calculus of objects resemblance

can be established implementing a method for the

resemblance calculus which satisfies the necessities of

every defined class, taking into account the structure of

these classes.

The designer will have to consider the importance of

every attribute, he or she will point out the superficial

attributes and will have to determine the policy that must be

applied when comparing null values.1 Besides, the designer

will have to consider the way in which fuzzy sets of

imprecise objects must be compared in case that this kind of

complex object may appear in the class definition.

A formal parameter of the comparison method will

perform the task of a stack, representing the history of

the comparison computation (i.e. the parameter V of the

latter function SR).

If the programming language that describes the classes

allows the use of meta-data (i.e. data about data), the

effort of implementing the comparison function for every

class may be avoided. For example, in Java [19], the

Reflection API can be used to elaborate reusable code that

implements the comparison method independently of the

class.

Fig. 9. Two ways of recursive call to deal with cycles.

1 When two null values are compared, a degree equal to 1 would be

an acceptable optimistic consideration, unless the null values had

different nature; in this case a degree equal to 0 would be more

appropriate.

6. An example

Now let us compare the two rooms introduced at the

beginning of the paper.

We will assume that the following two classes are

defined: a class Room, described by the set of attributes

{quality, extension, floor, students} and a class, Student,

described by the attributes {name, age, height}.

To compare objects of class Rooms, the importance

weights (Section 5.2) of the attributes are pfloor ¼ pstudents ¼

1; pextension ¼ 0:8; and pquality ¼ 0:5: On the other hand, to

compare objects of class Students, we have that pname ¼ 1

and page ¼ pheight ¼ 0:75:

The quality of a given room can be expressed using

values of the labels domain described in Table 1. The

extension will be described by the disjunctive imprecise

domain shown in Table 2 and Fig. 3. Although this table

describes the labels that belong to the domain, numerical

values can be used as well.2 The floor is an imprecise

attribute whose semantics is also disjunctive, described in

Table 3. Every room will have a fuzzy set of students, whose

membership degree will be determined by the percentage of

the day time they spend in the room attending their lessons.

The student age is expressed by the disjunctive imprecise

domain shown in Table 4, as well as by the use of numerical

values. Height is expressed using the set of values shown in

Table 5, as well as numerical values. As was the case for the

attribute extension of the class Room, we again omit the

semantics of the labels.

In the database there are two room and six student objects

(Fig. 10).

To compute the resemblance between the two rooms we

have to apply Eqs. (20a)–(20e) as follows:

SR(Room, room1, room2, B, 0)

This computation involves the following calculus

(Eq. (20e)):

SR(Quality, room1·quality, room2·quality, {(room1,

room2)},0) ¼ mS (high, regular) ¼ 0.8 (from Eq. (20b)

and Table 1).

SR(Extension, room1·extension, room2·extension,

{(room1, room2)}, 0) ¼ mbig(30) ¼ 0.9 (from Eqs.

(20b) and (1)).

SR(Floor, room1·floor, room2·floor, {(room1, room2)},

0) ¼ mS(4, high) ¼ 1 (from Eq. (20b) and Table 3).

To determine the resemblance between the fuzzy sets of

students of the two classes we apply the generalized

resemblance operator (Eq. (20c)):

SR(Student, room1·students, room2·students, {(room1,

room2)}, 0) ¼ SR{room1 ;room2}(room1·students, room2·-

students).

Taking into account that the attribute name is managed

by the classic equality, we can clearly see that only the

following pairs of students may have a resemblance far from

0: Pairs of the same student (SR(Student, stdnti, stdnti, V;

t) ¼ 1), students 2 and 5, and students 4 and 6.

To compute the resemblance between stdnt2 and stdnt5we have to apply Eqs. (20a)–(20e) as follows:

SR(Student, stdnt2, stdnt5, {(room1, room2)}, 0)

This computation involves the following calculus (Eq.

(20e)):

SR(Name, stdnt2·name, stdnt5·name, {(room1, room2),

(stdnt2, stdnt5)}, 0) ¼ 1 (from Eqs. (20b) and (1)).

SR(Height, stdnt2·height, stdnt5·height, {(room1, room2),

(stdnt2, stdnt5)}, 0) ¼ mmed(1.7) ¼ 1 (from Eqs. (20b)

and (1)).

SR(Age, stdnt2·age, stdnt5·age, {(room1, room2), (stdnt2,

stdnt5)}, 0) ¼ myoung(25) ¼ 0.8 (from Eqs. (20b) and

Taking into account Eq. (20e) with ai [ {name, age,

height} and applying the aggregation with oQ ¼ 0:2;we have:

SR(Student, stdnt2, stdnt5, {(room1, room2)},

Table 3

Domain for floor attribute

1 2 3 4 Low Intermediate High

1 1 0 0 0 1 0 0

2 1 0 0 0.8 1 0

3 1 0 0 1 0.7

4 1 0 0 1

Low 1 0.8 0

Intermediate 1 0.7

High 1

Table 4

Domain for the age attribute

Old Middle-aged Young

Old 1 0.6 0

Middle-aged 1 0.6

Young 1

Table 2

Domain for the extension attribute

Big Middle-sized Small

Big 1 0.5 0

Middle-sized 1 0.5

Small 1

2 To simplify, we omit the semantic definition of the labels. When

necessary, we will give the membership degree of a given numerical value.

0) ¼ oQ maxi ðpai^ SR(Cai

; stdnt2·ai; stdnt5·ai; {(room1,

room2), (stdnt2, stdnt5)}, 0)) þ ð1 2 oQÞmini (SR(Cai;

stdnt2·ai; stdnt5·ai; {(room1, room2), (stdnt2, stdnt5)},0) _

ð1 2 paiÞ) ¼ (0.2) p (1) þ (1 2 0.2) p (0.8) ¼ 0.84.

To compute the resemblance between stdnt4 and stdnt6 we

have to apply Eqs. (20a)–(20e) as follows:

SR(Student, stdnt4, stdnt6, {(room1, room2)}, 0)

This computation involves the following calculus

(Eq. (20e)):

SR(Name, stdnt4·name, stdnt6·name, {(room1, room2),

(stdnt4, stdnt6)}, 0) ¼ 1 (from Eqs. (20b) and (1)).

SR(Height, stdnt4·height, stdnt6·height, {(room1, room2),

(stdnt4, stdnt6)}, 0) ¼ mhigh(1.9) ¼ 1 (from Eqs. (20b)

and (1)).

SR(Age, stdnt4·age, stdnt6·age, {(room1, room2), (stdnt2,

stdnt5)}, 0) ¼ myoung(24) ¼ 0.9 (from Eqs. (20b) and

Taking into account Eq. (20e) with ai [ {name, age,

height} and applying the aggregation with oQ ¼ 0:2;we have:

SR(Student, stdnt4, stdnt6, {(room1, room2)},

0) ¼ oQ maxi ðpai^ SR(Cai

; stdnt4·ai; stdnt6·ai; {(room1,

room2), (stdnt4, stdnt6)}, 0)) þ ð1 2 oQÞmini (SR(Cai;

stdnt4·ai; stdnt6·ai; {(room1, room2), (stdnt4, stdnt6)},

0) _ ð1 2 piÞ) ¼ (0.2) p (1) þ (1 2 0.2) p (0.9) ¼ 0.92.

Taking these latter calcula into account we can compute

the resemblance between both sets of students using Eq. (7)

(we will use the implication operator from Eq. (6)):

SR{room1 ;room2}(room1·students, room2·students) ¼ ^(

QSR{room1 ;room2}(room2·studentslroom1·students),

QSR{room1 ;room2}(room1·studentslroom2·students)) ¼ ^(

min{1, 0.84, 0.94, 0.92}, min{1, 0.84, 1, 0.77}) ¼ 0.77

If we weigh this resemblance using the cardinality ratio

(Definition 3), then we have that:

SR(Student, room1·students, room2·students, {(room1,

room2)}, 0) ¼ F(room1·students, room2,

students) p 0.77 ¼ 0.98 p 0.77 ¼ 0.76.

Finally, taking into account Eq. (20e) with ai [ {quality,

extension, floor, students} and applying the aggregation

with oQ ¼ 0:2; we have:

SR(Room, room1, room2,B, 0) ¼ oQ maxi ðpai^ SR(Cai

room1·ai; room2·ai; {(room1, room2)},

0)) þ ð1 2 oQÞmini (SR(Cai; room1·ai; room2·ai;

{(room1, room2)}, 0) _ ð1 2 piÞ) ¼ (0.2) p (1) þ (1 2

0.2) p (0.76) ¼ 0.81.

Table 5

Domain for height attribute

Tall Medium Short

Tall 1 0.5 0

Medium 1 0.5

Short 1

Fig. 10. Rooms example’s data.

7. Conclusion and further work

We have generalized the notion of state equality in order

to compare fuzzy objects. As a result we can compute a

resemblance between two objects of a given class. We have

proposed:

1. A policy to handle resemblance between basic objects.

2. A set of operators for computing the resemblance

between fuzzy sets of fuzzy objects.

3. A recursive way of comparing two complex objects of a

given class, handling cycles using two definitions of

resemblance.

To accomplish this we have used the concept of

similarity and resemblance from the theory of the fuzzy

subsets. In particular:

† Resemblance relations are the basis that allows us to

manage the different labels domains that can be defined

in order to manage imprecision.

† The resemblance driven inclusion degree (Definition

1) allows us to build different operators to

compute resemblance relations between fuzzy

sets of imprecise objects. The generalized

resemblance degree (Definition 2), the cardinality

ratio (Definition 3) and the consistency

degree (Definition 4) are founded on this important

concept.

† The resemblance calculus between two objects of a

same class is based on the use of both fuzzy

resemblance relations and the latter operators. This

calculus is a valuable tool in classes where, due to

their ability to store fuzzy information, the classic

equality cannot be applied.

These tools are the foundation over which

the information retrieval must be built in the object-

oriented database. Future and ongoing work includes the

following:

† The value equality generalization we have performed

here allows resemblance comparisons. However,

the retrieval of the database information

may require the use of order relationships (partial

or total) defined among the imprecise domains

elements.

† Although the literature contains proposals that allow

the management of both imprecision and uncertainty

when dealing with fuzzy information [20], these

propositions are insufficient in some cases. The set

of operators proposed in this paper are focused on

imprecision management. Now we need to extend

them to deal with uncertainty in data.

† The operators studied here are resemblance measures

whatever the possibility chosen by the designer of the

class to configure them. As future work we will also

study different configurations, pointing out which of

them are similarity measures (studying transitivity

properties).

† We are currently incorporating this comparison

capability into our FOODBI (Fuzzy Object Oriented

Data Base Interface) prototype.

Acknowledgements

Supported in part by the Spanish R&D project TIC99-

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Complex object comparison in a fuzzy context

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