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tion. The expression arising from this notation will be placed next to the circle containing
the letter “o” (overlapping).
This fuzzy constraint has an effect on each superclass instance and must be satisfi ed
by each one. In general, both min and max should be absolute quantifi ers, although relative
quantifi ers will also be accepted (with regard to the total number of subclasses, value
b
).
Example 6.
Continuing with Example 5, we can establish a fuzzy cardinality constraint
on the overlapping specialization, such as: (approx_2, approx_half). This establishes the
constraint whereby each employee must appear in a minimum of “approximately 2” skills
and in a maximum of “approximately half” of existing skills (or subclasses).
This schema is depicted in Figure 7, too. Note that fuzzy quantifi er almost always is
a fuzzy completeness constraint and the (min,max) notation is used for a fuzzy cardinality
constraint.
Finally, observe that the quantifi ers can be of any type (absolute or relative) and each
quantifi er can also be followed, optionally of course, by one or two fulfi llment degrees in
square brackets [γ,δ], as explained previously.
The fuzzy cardinality constraint may be used also in the aggregation of entities. The
fuzzy quantifi er or the fuzzy (min,max) notation may label an arc crossing the line which
joins one entity with the rhombus in the aggregation. Notice that the aggregated entity may
be composed of some instances of one entity. For example, we can use fuzzy quantifi er ap-
prox_6 constraining the number of cylinders of a car (see Example 2 and Figure 4).
FUZZY DISJOINTED OR OVERLAPPING
CONSTRAINTS ON SPECIALIZATIONS
In specializations, the disjointed constraint specifi es that the subclasses of the spe-
cialization must be disjointed. This means that an entity can be a member of at most one
of the subclasses. If the subclasses are not constrained to be disjointed, it is an overlapping
specialization.
Thus, it can be interesting to include to what extent the superclass instance belongs to
each of the subclasses using linguistic labels (“a lot”, “a little”...) or, more simply, member-
ship degrees in the interval [0,1]. Note that it is to consider each subclass as a fuzzy subset
of the superclass. Just like all fuzzy sets, its elements are not clearly defi ned, since each
element can belong to the fuzzy set with a certain degree.
This extension can be applied on disjointed or overlapping specializations and such
specializations will be represented with letter “f” (fuzzy) before the other letter in the circle,
i.e., “fd” for fuzzy disjointed specializations and “fo” for fuzzy overlapping specializations.
However, it does not force all subclasses to be fuzzy subsets: fuzzy subclasses are repre-
sented with dashed rectangles.
This defi nition has two points of view:
1.
From the point of view of subclasses
: Subclasses are fuzzy sets and their underlying
domain is all superclass instances, i.e., each superclass instance has a membership
degree to each subclass (including the value zero). Let
A
be a subclass of
S
. Then
the fuzzy set of A is represented by the following equation (using the Equation 2
format):
