Image Processing Reference
In-Depth Information
- and an element of knowledge written as:
N
(
A
)=
β
- then the conclusion can be expressed by: min(
α, β
)
≤
N
(
B
)
≤
α
.
The formalism of possibilistic logic is used in many fields, for example, to repre-
sent preference or utility models in the form of KBS and then to use these systems for
reasoning [DUB 99].
Let us consider a knowledge base of the type:
KB
=
ϕ
i
,α
i
)
,i
=1
...n
where
α
i
is a degree of certainty or priority associated with the formula
ϕ
i
(represent-
ing an element of knowledge).
Satisfying this set of formulae in each world is represented by a possibility distri-
bution defined as follows. In the case where the knowledge base is comprised of only
one formula, we have:
π
(
ϕ,α
)
(
ω
)=
1
if
ω
|
=
ϕ
[8.97]
1
−
α
otherwise
More generally, for a set of elements of knowledge with priorities, we have:
π
KB
(
ω
)= min
i
=1
...n
1
ϕ
i
=min
i
=1
...n
max
1
α
i
,ϕ
i
(
ω
)
.
−
α
i
,ω
|
=
¬
−
[8.98]
This formula is interpreted this way: if a formula is significant (
α
i
close to 1), this
formula's degree of satisfaction in the world
ω
is taken into account. If, on the other
hand, it is not significant (
α
i
close to 0), then it will not come into play in the overall
evaluation of the knowledge base. The min corresponds to the fact that we are trying to
know to what extent the formulae of the knowledge base are simultaneously satisfied
in
ω
.
The inconsistency degree of the base
KB
can be measured using the expression:
1
−
max
ω
π
KB
(
ω
)
.
[8.99]
A base is said to be complete if it can be used to infer whether any formula is
true or whether its opposite is true: either
KB
ϕ
(
KB
allows us to infer
ϕ
), or
KB
¬
ϕ
(
KB
allows us to infer
¬
ϕ
).
When a base is not complete, it leads to an absence of knowledge regarding certain
formulae
ϕ
:
KB
ϕ
and
KB
¬
ϕ
. Using possibilistic logic, this situation can be
represented in a very simple way:
Π(
ϕ
)=Π(
ϕ
)=1
whereas there is no such simple model in probability theory, for example.
¬
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