Information Technology Reference
In-Depth Information
In a lot of cases, mainly in the applications, predicates
P
in
X
result indirectly
evaluable in
(
[
,
]
,
)
thanks to a
numerical characteristic C h
P
of the current use
of
P
in
X
. Such characteristic allows to translate '
X
is
P
'into'
Ch
P
(
0
1
x
)
is
Q
', with
an adequate predicate
Q
on the range of the function
Ch
P
:
X
ₒ
R
. For example,
if
X
is a population and
P
=
old
, it could be the case that
Ch
old
=
numerical age
=
Age
, in which case '
x
is
old
' can be translated into '
Age
is
big
', with a modeling
of
big
according with the current use of
old
, and provided this predicate only depends
in the subject's age. In this cases, once the designer is sure that
Ch
P
and
Q
are good
enough for the case, and also that the order in the numerical interval where
Ch
P
ranges is adequate to model the order
(
x
)
P
by the order
Q
, he/she/it should again
be sure that the degrees
.
A
design's process
never can arrive to something “exact”, but approximate and, if
possible, keeping everything under some bounds. For example, by taking the degrees
μ
Q
(
μ
Q
(
Ch
P
(
x
))
agree with the expected degrees
μ
P
(
x
)
Ch
P
(
x
))
in intervals
(
a
(
x
),
b
(
x
))
, where
a
(
x
)
is the minimum of the acceptable
values for
μ
Q
(
Ch
P
(
x
))
, and
b
(
x
)
the maximum of them. This can allow to take
μ
Q
(
Ch
P
(
x
))
as, for instance, an average of
a
(
x
)
and
b
(
x
)
, or as the complex number
a
(
x
)
+
ib
(
x
)
. For example, if
m
x
is the middle point of the interval
(
a
(
x
),
b
(
x
))
,
and the confidence that the value is between
a
and
m
x
can be quantified in a
coefficient
a
1
, and that of being between
m
x
and
b
(
x
)
(
x
)
by
a
2
, it can be taken the
average
a
2
.
A rational design should be carefully made by taking into account all the available
information, or knowledge, in the use of
P
in
X
, as well as of that induced on
Q
in
its numerical universe. If the use of
P
in
X
is not known, it is impossible to design
neither
μ
Q
(
Ch
P
(
x
))
=[
a
1
·
a
(
x
)
+
a
2
·
b
(
x
)
]
/
a
1
+
μ
Q
(
Ch
P
)
μ
P
, nor to accept
μ
P
(
)
=
μ
Q
(
Ch
P
(
)),
∈
, nor
x
x
for all
x
X
.
As it was said, if
[
x
]
is a class in
X
/
=
P
, μ
P
is constant in it. Let us denote by
v
x
the value of
.
Provided
P
is semi-rigid with at most two classes in
X
μ
P
in the class
[
x
]
/
=
P
, then either
X
/
=
P
=
{[
x
]}
,or
X
/
=
P
={[
x
]
,
[
y
]}
. In the first case,
μ
P
only has a single value
v
∈
L
.
μ
P
has at most two values, and, for obvious reasons, we will
only consider the situation where this values
In the second case,
v
x
,v
y
are different. When,
•
X
/
=
P
={[
x
]}
, and either
μ
P
(
x
)
=
ʱ
for all
x
in
X
,or
μ
P
(
x
)
=
ˉ
for all
x
in
X
,itresults
P
∼
= ∅
in the first case, and
P
∼
=
X
in the second. In both cases,
P
is a rigid or binary predicate in
X
.
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