Database Reference
In-Depth Information
tions. In addition, they also provided an approach to mapping an FEER model to a fuzzy
object-oriented database schema.
All these works do not study how to relax the constraints expressed in the ER/EER
model so that they can be made more fl exible, because the constraints of the traditional
model are either too restrictive or too permissive. Perhaps the fi rst work relaxing constraint
was recently published in Galindo et al. (2001b), studying, for example, fuzzy participation
constraint and fuzzy cardinality constraint in a relationship.
Firstly, we will summarize the basic concepts of fuzzy logic, paying particular attention
to fuzzy quantifi ers. After it, we formalize how we will use fuzzy concepts in the following
sections. Then, we will study each of the FuzzyEER extensions separately: fuzzy aggrega-
tions, fuzzy degrees in specializations, fuzzy completeness constraint on specializations,
fuzzy cardinality constraint on overlapping specializations, fuzzy disjointed or overlapping
constraints on specializations, fuzzy attribute defi ned specializations, fuzzy constraints in
union types or categories and fuzzy constraints in shared subclasses. Finally, we outline
some conclusions and suggest some research lines for the future.
FUZZY SETS: FUZZY QUANTIFIERS
In 1965, Lotfi A. Zadeh defi ned the concept of fuzzy set based on the idea that there are
sets in which it is not totally clear whether an element belongs to the set or not. Sometimes
an element belongs to the set to a certain degree, which is called membership degree. For
example, the set of tall people is a fuzzy set because there is no height limit establishing the
minimum height for a person to be considered tall.
A fuzzy set A is defi ned as a function of belonging µµµ which connects or pairs up the
elements of a domain or discourse Universe
U
with elements of the interval [0,1]:
µµµ
(u): U →
[0, 1]
(1)
The closer µµµ (u) to the value 1, the greater the membership of the object
u
∈
U
to the
fuzzy set A. The values of membership vary between 0 (does not belong at all) and 1 (total
belonging). A fuzzy set A can be represented as a set of pairs of values: each element u with
its membership degree µµµ (u):
A
= {µµµ
(u) /u
:
u
∈
U
}
(2)
U
numbers R). Figure 1 shows the membership function of the fuzzy number “Approximately
n”. The margin value m indicates the limits of the fuzzy set. It is easy to observe that the
nearer a number is to the value n, the greater its membership to “approximately n”.
U
is
A fuzzy number is a fuzzy set, where
U
is a numerical domain (normally the real
U
called “underlying domain” of the fuzzy set. The underlying domain may be ordered or
non-ordered, and continuous or non-continuous (discrete).
From this simple concept a complete mathematical and computing theory has been
developed which facilitates the solution of certain problems (Pedrycz et al., 1998). Fuzzy
logic has been applied to a multitude of objectives such as: control systems, modeling,
simulation, patterns recognition, information or knowledge systems (databases, expert
systems...), computer vision, artifi cial intelligence, artifi cial life....
