Image Processing Reference
InDepth 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
));
xϕ
)=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
:Ω
−→
[0
,
1]
such that:
∃
ω
∈
Ω
,π
(
ω
)=1
.
The possibility of a formula is then expressed as:
Π(
ϕ
) = sup
π
(
ω
)
,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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