Database Reference
In-Depth Information
respect to the total. We will study this in next sections. Of course, the quantifi er
Q
must be
previously defi ned in the data dictionary of the model.
In this context, we need a threshold γ∈ [0,1] indicating the minimum fulfi llment degree
that must be satisfi ed. This threshold will be written in square brackets: Q[γ]. For example,
we may use “almost_all [0.2]”, indicating that this fuzzy quantifi er must be satisfi ed in a
minimum degree of 0.2. Consequently, the underlining constraint requires that:
Q
(Φ) ≥γ
(5)
Every time the database is modifi ed, the DBMS (DataBase Management System) com-
putes Φ and checks whether Equation 5 is satisfi ed. The meaning of Φ will be defi ned in the
next sections because it depends on where the fuzzy quantifi er is used. In order to simplify
the expression, we can set a default value for γ at 0.5, for example. If
Q
is an increasing
function, then we can change Equation 5 because we obtain that:
Φ≥
Q
-1
(γ)
(6)
Similarly, if
Q
is a decreasing function, then what we get is:
Φ≤
Q
-1
(γ)
(7)
The last two equations may be useful because
Q
and γ are constants, whereas Φ is
a varying value. Value Φ may change with every DML sentence (INSERT, DELETE or
UPDATE).
In addition, another optional value δ can be established, which is greater than the
threshold γ of minimum fulfi llment degree, in the following way: Q[γ,δ] such that γ<δ. The
value δ is more restrictive than γ and it establishes that when the constraint is unfulfi lled
with this higher value, the DBMS will inform the user, but it will permit the modifi cation of
the database which is underway. Thus, if the quantifi er is unfulfi lled with a value between
γ and δ, then the DBMS must warn the user (or only the database administrator). Therefore
the warning message is generated when
δ≥
Q
(Φ) ≥γ
(8)
Figure 3 depicts a fuzzy quantifi er with the thresholds γ and δ. Note that these thresh-
olds divide the domain of Φ in three areas: the allowed area, the not allowed area and the
warning area. The warning area is included in the allowed area. Note that the not allowed
area is defi ned when Equation 5 is false.
Therefore, a fuzzy quantifi er can be written in three ways:
1.
Quantifi er without threshold γ: Default threshold is γ = 0.5. For example, approx_2.
For the purpose of simplicity we will use this form in the examples.
2.
Quantifi er with a threshold γ: For example, approx_8[0.25].
3.
Quantifi er with two thresholds γ and δ, with γ<δ: For example, approx_3[0.5,0.6].
Both values would be close, in order to avoid too much warnings by the DBMS.
