Database Reference
In-Depth Information
{µ
A
(u
1
)/
u
1
,µ
A
(u
2
)/
u
2
,... µ
An
(u
n
)/
u
n
}
(14)
where
u
i
, with
i
= 1, 2,…,
n
, are all instances of superclass
S
, and µ
A
(
u
1
) is the mem-
bership degree of
u
i
to subclass A.
2.
From the point of view of superclass instances
: Each superclass instance may belong
to some subclasses. This membership is measured with a fuzzy set. The underlying
domain of this fuzzy set is the set of all subclasses names. Let A
j
domain of this fuzzy set is the set of all subclasses names. Let A
j
domain of this fuzzy set is the set of all subclasses names. Let A, with
j
= 1,2,…
j
m
be the
m
subclasses of
S
. Then the fuzzy set of instance
u
i
is:
{µ
A1
(
u
1
)/A
1
,µ
A2
(
u
2
)/A
2
,... µ
An
(
u
n
)/A
n
}
(15)
where µ
Aj
(
u
1
), with
j
= 1,2,…
m
, is the membership degree of
u
i
to subclass A
j
to subclass A
j
to subclass A. Note
that in disjointed specializations the number of subclasses for a superclass instance is
one.
Observe that both points of view work with fuzzy sets with a different discrete un-
derlying domain.
Example 7.
Figure 8 indicates that our conceptual schema is also concerned with
storing to what extent each employee belongs to each of the subclasses. Thus, the system's
programmers set is a fuzzy set (an employee can belong to this set with a certain member-
ship degree), whereas the set of accountant is not a fuzzy set (an employee can or cannot
belong to this set). This is the fi rst point of view.
The second point of view starts with a particular employee: an employee who is an
expert at programming management applications, although he may also be skilled in other
types of applications and less skilled as an analyst, could be represented in the database by
the following fuzzy set: {1/Management Programmer, 0.8/Systems Programmer, 0.3/Ana-
lyst}. Note that the underlying domain is the set of all subclasses names.
This will allow us to make selections of the type: “Find the name of the best manage-
ment applications programmer amongst those who are not assigned to many projects and
who is at least a regular analyst.”
This constraint does not prevent the use of other fuzzy constraints (completeness or
cardinality). However, when they are mixed with a fuzzy disjointed or overlapping con-
straint, they must be studied in order to defi ne the method with which the DBMS ensures
the fulfi llment of these constraints:
If a
fuzzy completeness constraint
exists then the DBMS must compute whether each
superclass instance belongs to some subclass; for example, in order to decide if “almost
all” superclass instances belong to some subclass. The problem is that membership
is now fuzzy. Membership degree of an instance to the subclasses may be computed
in various manners: a) By using the maximum membership degree of this instance to
any subclass, i.e., the height (Pedrycz et al., 1998) of the second point of view fuzzy
set, b) By using the fuzzy set cardinality (Pedrycz et al., 1998) of the second point of
view fuzzy set (adding all membership degrees) or by using generalized measures,
like the fuzzy set energy (De Luca et al., 1974). We can, certainly, set a minimum
threshold in order to decide whether an instance belongs to some subclass.
1.
