Databases Reference
In-Depth Information
The definition
µ
p∨q
(
a
)=max
{
µ
p
(
a
)
,µ
q
(
a
)
}
can not automatically lead to
that the value of
p
(
a
)
.
Also, missing information can not be a real excuse too. It would be one
if the conflicts and confusion could not be avoided by any other operation
based on the same information. Actually, the bold operations in (11) refute
this argument. The success of (11) on resolving the above discussed conflicts
and confusion actually comes from honoring the additive principle behind
both the set theory and the P-theory. Still, one may further argue that (11)
brings other problems, e.g., making the idempotent law (i.e.
A
∨
q
(
a
) is given by max
{
µ
p
(
a
)
,µ
q
(
a
)
}
∪
A
=
A
and
A
A
=
A
) invalid. Actually, the reason behind this problem is just what has
been discussed at the end of Sect. 3.3, and can be solved if the dependence
type of
P
B|A
(
a
)
,P
A|B
(
a
) in (16) are also considered [4]. Further ahead along
this line, we are finally lead to the P-theory.
If one wants all the axioms of Boolean algebra to remain hold, the P-theory
is a best choice for the development of models of uncertainty or partial truth.
If one does not want to use the P-theory due to higher computation costs and
requiring extra information for handling
P
B|A
(
a
)
,P
A|B
(
a
), one has to abandon
some axioms of Boolean algebra. Of course, depending on applications, one
may choose to abandon the idempotent law or the exclude-middle law or
other. However, one can not introduce a conceptual confusion by duplicately
using a terminology with a well known meaning to name a new and different
concept. If one abandons the complement definition 1
∩
−
µ
A
(
a
) or just simply
renaming 1
µ
A
(
a
) by (21), everything is fine with the min-max operations,
which may be worth some further study, especially on the joint role of
µ
A
(
a
)
and
µ
ΞA
(
a
) in the Z-system.
−
5 Conclusions
We have elucidated the links and differences between probability theory and
Zadeh's fuzzy system. When one set is included in the other (i.e.,
A
⊆
B
or
B
A
either in a almost surely sense or in a Zadeh fuzzy set sense), we have
proved that the P-theory and the
Z
−
-system perform equivalently in com-
puter reasoning that does not involve complement operation. Moreover, we
have proved that both the
Z
−
-system and the B-system attempt to approx-
imate the P-theory. In some cases, these approximations are acceptable, and
they have the advantage of easy implementation. In other cases, however we
get bad approximations that are incapable of producing reasonable results.
The failures arise from either its incapability of capturing additive structures
or inadequate handling of the dependence relation across two sets, or both.
It seems that many efforts are needed to investigate some error bounds of
approximation and to control the bounds. Furthermore, we show that the
controversy about the definition of the complement set arises from confusion
over the “complement” concept.
⊆
