Databases Reference
In-Depth Information
The following t-norm examples are typically used as interpretations of fuzzy conjunctions:
T m(x, y)
=
min
(x, y)
(minimum t-norm)
Tp(x, y)
y
(product t-norm)
T l(x, y)
=
max
(x
+
y
−
1
,
0
)
(Lukasiewicz t-norm)
It is worth noting that
Tm
is the only idempotent t-norm. That is,
T m(x, x)
=
x
·
x
.
1
This
becomes handy when comparing t-norms with fuzzy aggregate operators (Section
2.2.2
). It can be
easily proven (see [
Hajek
,
1998
]) that
=
T l(x, y)
≤
Tp(x, y)
≤
T m(x, y)
for all
x,y
.
All t-norms over the unit interval can be represented as a combination of the triplet
(T m, Tp, T l)
(see [
Hajek
,
1998
] for a formal presentation of this statement). For example, the
Dubois-Prade family of t-norms
T
dp
, often used in fuzzy set theory and fuzzy logic, is defined using
Tm
,
Tp
and
Tl
as:
∈[
0
,
1
]
λ
·
Tp(
λ
,
λ
)
2
(x,y)
∈[
0
,λ
]
T
dp
(x, y)
=
T m(x, y)
otherwise
2.2.2 FUZZY AGGREGATE OPERATORS
The
average
operator belongs to another large family of operators termed
fuzzy aggregate operators
[
Klir and Yuan
,
1995
]. A fuzzy aggregate operator
H
n
:[
0
,
1
]
→[
0
,
1
]
satisfies the following axioms
for every
x
1
,...,x
n
∈[
0
,
1
]
:
H(x
1
,x
1
,...,x
1
)
=
x
1
(idempotency)
(2.1)
for every
y
1
,y
2
,...,y
n
∈[
y
i
,
H(x
1
,x
2
,...,x
n
)
≤
H(y
1
,y
2
,...,y
n
)
(increasing monotonicity)
0
,
1
]
such that
x
i
≤
(2.2)
H
is a continuous function
(2.3)
¯
=
(
1
, ...,
n
)
be a weight vector that sums to unity. Examples of fuzzy aggregate operators include the
average
operator
Ha(x)
=
x
=
(x
1
,...,x
n
)
be a vector such that for all 1
≤
i
≤
n
,
x
i
∈[
0
,
1
]
Let
, and let
n
1
x
i
and the
weighted average
operator
Hwa(x,
1
¯
)
=
x
·¯
.
1
Clearly,
average
is a special case of the
weighted average
operator, where
1
= ··· =
n
=
n
.It
is worth noting that
Tm
(the min t-norm) is also a fuzzy aggregate operator, due to its idempotency
(its associative property provides a way of defining it over any number of arguments). However,
Tp
and
Tl
are not fuzzy aggregate operators.
T-norms and fuzzy aggregate operators are comparable, using the following inequality:
min
(x
1
,...,x
n
)
≤
H(x
1
,...,x
n
)
for all
x
1
,...,x
n
∈[
0
,
1
]
and function
H
satisfying axioms
2.1
-
2.3
.
1
For a binary operator
f
, idempotency is defined to be
f(x,x)
=
x
(similar to
Klir and Yuan
[
1995
], p. 36).
