Image Processing Reference
In-Depth Information
8.7.2. Possibilistic logic
Possibilistic logic relies on the definition of a possibility measure Π in a Boolean
algebra B with the formulae [DUB 91]:
−→
Π: B
[0 , 1]
such that:
- Π(
)=0;
- Π(
)=1;
-
ϕ, φ , Π( ϕ
ψ ) = max(Π( ϕ ) , Π( ψ ));
)=sup Π ϕ [ a
x ] ,a
D ( x ) (where D ( x ) is the domain of
-
ϕ , Π(
|
the variable x and ϕ [ a
|
x ] is obtained by replacing the occurrences of x in ϕ with a ).
Now let Ω be the set of interpretations and let π be a normalized possibility distri-
bution:
π
−→
[0 , 1]
such that:
ω
Ω ( ω )=1 .
The possibility of a formula is then expressed as:
Π( ϕ ) = sup π ( ω )
= ϕ
|
[8.93]
where ω
|
= ϕ is read “ ω is a model of ϕ ”, meaning that ϕ is satisfied in the world ω .
As we did with sets, a necessity measure is defined for formulae using duality by:
N ( ϕ )=1
Π(
¬
ϕ ) .
[8.94]
We then have the following property:
ψ )=min N ( ϕ ) ,N ( ψ ) .
ϕ, φ, N ( ϕ
[8.95]
This formalism can be used to deal with many situations by modeling them in a
very simple way. For example, a default rule such as “if A then B ”, with possible
exceptions, can be simply expressed as:
Π( A
B )
Π( A
∧¬
B )
[8.96]
Likewise, possibilistic modus ponens reasoning can be modeled by:
- if we have the rule: N ( A
B )= α
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