Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 165

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