Information Technology Reference
In-Depth Information
For example, with an
S
-implication, since
T
1
(
,
(
(
),
))
a
S
N
a
b
b
implies
=
0)
T
1
(
,
(
))
=
=
(with
b
a
N
a
0, it should be
T
1
W
˕
for some automorphism
˕
of (
.
With
R
-implications
J
T
, since
[
0
,
1
]
,
)
T
(
a
,
J
T
(
a
,
b
))
=
Min
(
a
,
b
)
b
,
the same
T
in
J
T
allows to have that inequality.
With
Q
-implications, since
T
1
(
a
,
S
(
N
(
a
),
T
(
a
,
b
)))
b
also implies
T
1
(
a
,
N
(
a
))
=
0 for all
a
∈[
0
,
1
]
, it should be also
T
1
=
W
.
˕
Concerning
ML
-implications, since
T
1
(
a
,
T
(
a
,
b
))
Min
(
a
,
Min
(
a
;
b
))
=
Min
(
a
,
b
)
b
,
because both
T
1
Min
, the Modus Ponens inequality is verified for
all t-norms
T
1
and, hence, for
T
1
=
Min
and
T
Min
(the biggest t-norm).
If
T
1
verifies
T
1
(
a
,
J
(
a
,
b
))
b
, because of the well-known result that for
left-continuous t-norms
T
1
,
T
1
(
a
,
t
)
b
is equivalent to
t
J
T
1
(
a
,
b
)
, it results
that the inequality is equivalent to
J
. Hence, among the func-
tions
J
verifying the Modus Ponens inequality with a continuous t-norm
T
1
,the
R
-implication
J
T
1
(
a
,
b
)
J
T
1
(
a
,
b
)
is the biggest one and, consequently,
T
1
(
,
J
T
1
(
,
))
a
a
b
is closer to
b
than
T
1
(
a
,
J
(
a
,
b
))
. In particular, it is
1f
a
b
J
M
(
a
,
b
)
=
Min
(
a
,
b
)
J
Min
(
a
,
b
)
=
b
if
a
>
b
and
1if
a
b
J
L
(
a
,
b
)
=
a
·
b
J
Min
(
a
,
b
)
J
Prod
(
a
,
b
)
=
b
a
if
a
>
b
(since
a
·
b
b
).
Third Step: Zadeh's Compositional Rule of Inference CRI
Once the rule “If
x
is
P
, then
y
is
Q
” is represented by
J
(μ
P
(
x
), μ
Q
(
y
))
, and a
continuous t-norm
T
1
such that
J
J
T
1
is known, the inference:
is obtained by the Zadeh's Compositional Rule of Inference (CRI):
μ
Q
∗
(
y
)
=
Sup
x
T
1
(μ
P
∗
(
x
),
J
(μ
P
(
x
), μ
Q
(
y
))),
for all
y
∈
Y
.
∈
X
Search WWH ::
Custom Search