Information Technology Reference
In-Depth Information
the input space, (
iv
) are designed and activated in groups and not individually, as
explained onwards.
16.2.1
Approximate Reasoning
The approximate (fuzzy) reasoning is achieved when a feasible
uncertain
and
im-
precise
conclusion,
Q
, is deduced from a collection of
imprecise
and
uncertain
premises, represented as vague (fuzzy) sets. A linguistic expression related to an
argument can be represented as IF-THEN rules in the form:
AND
x
j
is
M
j
)
AND
R:IF
x
1
is
M
1
(
x
1
)
AND
...
(
x
j
...
(16.1)
AND
x
n
is
M
n
(
x
n
)
THEN
y
is
N
.
x
j
is
M
j
,for
j
=
=
,...,
=
where
P
j
1
n
,isthe
j
-th input proposition and
Q
y
is
N
is the inferred (deduced) proposition. The
i
-th rule, for
i
=
1
,
2
,...,
m
, compose a
set of fuzzy rules:
AND
x
j
is
M
1
j
(
x
j
)
AND
R
1
:IF
x
1
is
M
11
(
x
1
)
AND
...
...
AND
x
n
is
M
1
n
(
x
n
)
THEN
y
is
N
1
(
y
)
...
AND
x
j
is
M
ij
(
x
j
)
AND
R
i
:IF
x
1
is
M
i
1
(
x
1
)
AND
...
...
AND
x
n
is
M
in
(
x
n
)
THEN
y
is
N
i
(
y
)
...
AND
x
j
is
M
mj
)
AND
(
)
...
(
...
R
m
:IF
x
1
is
M
m
1
x
1
AND
x
j
)
(16.2)
representing, for instance, a multi-input single-output (MISO) linguistic fuzzy logic
system.
The elements
x
j
and
y
refer, respectively, to a
j
-th input and the output objects
within distinct collections named
universe of discourse
,
x
j
∈
(
)
(
AND
x
n
is
M
mn
x
n
THEN
y
is
N
m
y
Y
,alsoas-
signed
linguistic variable
, as well. The amount of
P
n
propositions is related to the
n
-th dimensionality of the argument and so of the human thinking or reasoning.
The input vector
x
X
j
and
y
∈
T
is denominated as premise (antecedent of the rule)
while the output,
y
, is associated to the conclusion (consequent of the rule). The lin-
guistic expressions AND corresponds to the
set operation
, intersection,
=[
x
1
,...,
x
n
]
∩
,the
logic
∧
(
,
)
operation
, conjunction,
.The
terms
M
ij
and
N
j
are assigned
linguistic terms
within the respective universes of
discourse and built as fuzzy sets.
A fuzzy set represents the possibility, similarity, or conformity of an element,
x
, to belong to a set,
M
.Sucha
fuzzy set
,
M
, within a universe of discourse,
X
,
is defined by a
membership function
,
,andthe
Triangular norm
operation (
T-norm
),
t
x
y
)
are, in turn, associate to a degree of truthiness, this is equivalent to the multivalued
logic in which the truth is assigned to continuous values in
μ
M
(
x
)
:
X
→
[
0
,
1
]
.
If the values of
μ
M
(
x
[35]. The null
membership degree denotes that such an element is not in the set, being completely
[
0
,
1
]
