The function π represents the knowledge of an agent and in particular encodes to what ex- tent the agent finds ω plausible to be the real world. By convention, π(ω) = 0 means that ω is impossible and π(ω) = 1 means that no available information prevents ω from being the actual world. A possibility distribution π is said to be normalized if ∃ω ∈ Ω · π(ω) = 1, i.e. at least one interpretation is entirely plausible. Conversely, when ∀ω ∈ Ω · π(ω) = 0 we say that the possibility distribution π is vacuous. A normalized possibility distribution expresses consistent belief and is thus preferred, as a possibility distribution that is not normalized indicates the presence of conflicting information. Possibility degrees are mainly interpreted qualitatively: when π(ω) > π(ω0), ω is considered more plausible than ω0.
For two possibility distributions π1 and π2 with the same domain Ω we write π1 ≥ π2
when ∀ω ∈ Ω · π1(ω) ≥ π2(ω) and π1 > π2 when π1 ≥ π2 as well as π1 6= π2. When
we impose constraints on possibility distributions, we are usually only interested in the
least specific possibility distributions, i.e. the greatest possibility distribution w.r.t. the
ordering >, that satisfies the constraints.
A possibility distribution π induces two uncertainty measures. The possibility measure Π is defined by [Dubois and Prade 1988]:
Π(A) = max {π(ω) | ω ∈ A} with A ⊆ Ω
and evaluates the extent to which a world ω in A is consistent with the beliefs expressed by π. The dual necessity measure N is defined by:
N (A) = 1 − Π(A)with A ⊆ Ω
and evaluates the extent to which all possible worlds belong to A. We always have that
N (Ω) = 1since Π({}) = 0. However, we only have Π(Ω) = 1 and, conversely, N({}) = 0
when the possibility distribution is normalized. To identify the possibility/necessity measure associated with a specific possibility distribution πX, we will use a subscript notation,
i.e. ΠX and NX are the corresponding possibility and necessity measure, respectively.
We omit the subscript when the possibility distribution is clear from the context.
2.4
Possibilistic Logic
Now we introduce the concepts of possibilistic logic [Dubois et al. 1994], which is a logic capable of dealing with uncertainty based on possibility theory. Thus far, we have seen that interpretations in ASP can be partial. As such, they are different from interpretations
in classical logic. The semantics of possibilistic logic, on the other hand, are defined w.r.t. classical interpretations. We represent such an interpretation as a set of atoms ω, where ω |= a if a ∈ ω and otherwise ω |= ¬a, with |= the satisfaction relation from classical logic. The set of all interpretations is defined as Ω = 2A, with A a finite set of
atoms. At the semantic level, possibility distributions over Ω are considered. Example 12
Consider the possibility distribution π12 defined as:
π({a, b, c}) = 0 π({b, c}) = 0 π({a, b}) = 1 π({b}) = 0.4 π({a, c}) = 0 π({c}) = 0
π({a}) = 0.7 π({}) = 0.2
The possibility distribution π12is normalized since the world {a, b} is entirely possible.
We can see that a world in which ‘a’ and ‘b’ are true simultaneously is preferred over worlds in which either ‘a’ or either ‘b’ is true since π12({a, b}) > π12({a}) and
π12({a, b}) > π12({b}). We can also see that, given this possibility distribution, we
do not entertain the possibility that ‘c’ is true since for every world ω ∈ Ω with ω |= c we have π12(ω) = 0.
The two uncertainty measures from possibility theory can then be used to rank proposi- tions. Indeed, the possibility measure Π can now be written as
Π(p) = max {π(ω) | ω |= p}
which evaluates the extent to which a proposition p is consistent with the beliefs expressed by π [Dubois et al. 1994]. The dual necessity measure N defined as
N (p) = 1 − Π(¬p)
evaluates the extent to which a proposition p is entailed by the available beliefs expressed by π [Dubois et al. 1994]. In particular, the notations Π(p) and N(p) with p a proposi- tion are thus defined as shorthands for Π({w | w |= p}) and N({w | w |= p}). Note that we now always have N(>) = 1 for any possibility distribution, while Π(>) = 1 (and
N (⊥) = 0) only holds when the possibility distribution is normalized, i.e. only normalized
2.4. POSSIBILISTIC LOGIC Example 13
Consider the possibility distribution π12 from Example 12. Since π({a, b}) = 1 we
find that Π(a) = Π(b) = 1, i.e. it is consistent to believe that ‘a’ and ‘b’ will be true in the actual world. Since {a, b} |= ¬c we furthermore have that N(c) = 1−Π(¬c) = 0. Indeed, given this possibility distribution, we have no reason to conclude that ‘c’ is true. We furthermore find that N(a) = 0.6 and N(b) = 0.3, i.e. we are more certain that ‘a’ is true.
An important property of necessity measures is their min-decomposability w.r.t. conjunc- tion: N(p ∧ q) = min(N(p), N(q)) for all propositions p and q. However, for disjunction only the inequality N(p ∨ q) ≥ max(N(p), N(q)) holds. As possibility measures are the dual measures of necessity measures, they have the property of max-decomposability w.r.t. disjunction, while for the conjunction we have that only the inequality Π(p ∧ q) ≤ min (Π(p), Π(q)) holds.4
At the syntactic level, a possibilistic knowledge base consists of pairs (p, c) where p is a propositional formula and c ∈ ]0, 1] expresses the certainty that p is the case. Formulas of the form (p, 0) are not explicitly represented in the knowledge base since they encode trivial information. A formula (p, c) is interpreted as the constraint N(p) ≥ c, i.e. a possibilistic knowledge base Σ corresponds to a set of constraints on possibility distributions. Typically, there can be many possibility distributions that satisfy these constraints. In practice, we are usually only interested in the minimally specific possibility distributions, which are the possibility distributions that make minimal commitments, i.e. the maximal possibility distributions w.r.t. the ordering >. For the constraints induced by a possibilistic logic base, there is a unique minimally specific distribution, which is called the least specific distribution [Dubois et al. 1994].
Example 14
Consider the possibilistic knowledge base consisting of the pairs:
(¬c, 1) (a ∨ b ∨ c, 0.8)
(¬a ∨ b, 0.3) (¬b ∨ a, 0.6).
4This is a notable difference with fuzzy logic, which is truth-functional. Furthermore, both logics handle
different sources of information. Indeed, in fuzzy logic we are able to express that a bottle is half empty, i.e. we deal with multi-valuedness. On the other hand, in possibilistic logic we are able to express that N(full) = 0.5, i.e. we are somewhat certain that the bottle is full. Still, in the actual world the bottle will
This knowledge base imposes the constraints N (¬c) = 1 N (a ∨ b ∨ c) ≥ 0.8 N (¬a ∨ b) ≥ 0.3 N (¬b ∨ a) ≥ 0.6 or, equivalently Π(c) = 0 Π(¬a ∧ ¬b ∧ ¬c) ≤ 0.2 Π(a ∧ ¬b) ≤ 0.7 Π(b ∧ ¬a) ≤ 0.4
The least specific possibility distribution that satisfies these constraints is π12 from
Example 12.