Database Reference
In-Depth Information
Figure 3: Thresholds
γ
and
δ
in a fuzzy quantifi er “approximately between a and b”, and
its generated areas
Formalization of Quantifi ers for Fuzzy Constraints
Fuzzy constraints may be used in two ways, and in both of them the constraint is de-
noted with an arc crossing the line where the constraint has effect. Besides, the arc is labeled
with the quantifi er or quantifi ers according to the constraint type:
1.
If the arc is labeled with the quantifi er
Q
, this constraint establishes Equation 5, with
Φ defi ned by Equation 4.
2.
If the arc is labeled with the fuzzy (min,max) notation (
Q
min
;
Q
max
) this constraint
establishes the minimum and maximum number or proportion of elements fulfi lling
a certain constraint. In other words, fuzzy (min,max) notation (
Q
min
;
Q
max
) establishes
that:
λ
min
≤Φ
min
∧ λ
max
≥ϕ
max
(9)
where
λ
min
= min{α : α = (
Q
min
)
-1
(γ
min
)}
(10)
λ
max
= max{β : β = (
Q
max
)
-1
(γ
max
)}
(11)
where, γ
min
and γ
max
are the minimum thresholds for
Q
min
y
Q
max
respectively, and
a
si
Q
min
is absolute
Φ
min
=
(12)
a
/
b
si
Q
min
is relative
