![](bg4.png)
metrical representation for fuzzy numbers, we need to
store four values for the VST and four values for the
VET, as shown in figure 3. Since the meaning of the
attributes VST and VET is the period of time during
which the values of a tuple are valid, it is more conve-
nient to summarize the information given by the two
fuzzy attributes in an only but fuzzy interval. This sit-
uation can be represented by the trapezoidal fuzzy set
shown in figure 4 which incorporates the semantics of
our problem. As can be seen in such figure, the right
side of the interval (VET) is set to 1 indefinitely while
the tuple remains valid. On the contrary, the left side
of the interval is the part that reflects the imprecision
about the starting time point of the interval or VST.
In this figure, the fuzzy value represents an interval
where the VST corresponds to middle february and
the VET corresponds to now or indefinitely.
9877
9877
BOSS
JUNIOR
TRAINEE
EXPERTISE
~ 31-05-1998~15-06-199715001245
GRANT
~undefined~01-06-199815001245
GRANT
VETVSTSALARYEMPIDEMPNAM
9877
9877
BOSS
JUNIOR
TRAINEE
EXPERTISE
~ 31-05-1998~15-06-199715001245
GRANT
~undefined~01-06-199815001245
GRANT
VETVSTSALARYEMPIDEMPNAM
(01-06-1998,01-06-1998,2,2)
(31-12-2050,31-12-2050,0,0)
Figure 3: Internal representation of a fuzzy date.
01/06/199830/05/1998
1
Days
31/12/2050
Figure 4: Fuzzy Period of Time for a Valid Tuple.
This representation has the advantage that, not
only periods of time, but fuzzy dates can also be rep-
resented in a unified way. Think that a parametrical
representation as (m,m,a,b) represents a central time
point with some imprecision at both sides, what is in-
terpreted as a fuzzy date.
As it was explained in section 1.2, it is
quite easy to represent a fuzzy interval with this
characteristics since only four parameters need to
be stored in order to specify it. In our ex-
ample, the parametrical representation should be
(17/02/1996,31/12/2050,5,0)
1
. This representation is
more flexible, since admits to represent both precise
1
We have assumed that 31/12/2050 is the ending point
of the time domain
and imprecise time intervals. For example, a precise
interval like [18/03/1997,25/06/1998] is represented
as (18/03/1997,25/06/1998,0,0).
In (Medina et al., 1994), (Medina et al., 1995) is
presented a generalized model of fuzzy DB that sup-
ports this representation for fuzzy data and the corre-
sponding implementation in a classical relational DB
system (Oracle).
3 INSERTING AND UPDATING IN
THE FUZZY TDB
As we explained in the introduction, in a TDB in-
formation is never deleted when an update operation
is being carried out. The process now is to leave
the old version of data in the DB and to add a new
version with the suitable modifications achieved, but
closing the old one by adding a valid end time value
which is the immediately previous granule to the valid
start time of the tuple inserted. Note that closing the
old version of a tuple is a deletion operation in the
TDB environment whereas adding a new version cor-
responds to an insert operation.
One the most important problems that arise when
the time period considered is fuzzy is that we can not
say which is the time point immediately previous to
a given one for a concrete granularity, since many
values with different possibility degree are possible.
To compute this value is very important for the update
operation since, as explained below, we need to close
the old version for the new one is valid. So now, the
given solution is not precise but also imprecise.
Definition 2. Let us note by µ
O
(x) the membership
function associated to the fuzzy interval of the old
version of the tuple to be updated and µ
N
(x) the mem-
bership function associated to the new fuzzy interval.
Then, the membership function of the fuzzy interval
(µ
0
O
(x)) that serves to close the validity time of the old
one is:
µ
0
O
(x) =
½
µ
O
(x) ∀x | µ
N
(x) = 0
1 − µ
N
(x) ∀x | µ
N
(x) > 0
This result can be graphically seen in figure 5.
It is obvious, that the non-overlapping condition
required for the crisp TDB is not valid now, but the
overlapping degree will never reach value 1 for the
sake of consistency.
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