Possibility theory

Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise.
Formalization of possibility.
For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function formula_1 from formula_2 to [0, 1] such that:
It follows that, like probability, the possibility measure is determined by its behavior on singletons:
provided that "U" is finite or countably infinite.
Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω.
Axiom 2 could be interpreted as the assumption that the evidence from which formula_1 was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1.
Axiom 3 corresponds to the additivity axiom in probabilities. However there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that:
Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is "compositional" with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally:
When Ω is not finite, Axiom 3 can be replaced by:
Necessity.
Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the "possibility" and the "necessity "of the event. For any set formula_6, the necessity measure is defined by
In the above formula, formula_19 denotes the complement of formula_6, that is the elements of formula_21 that do not belong to formula_6. It is straightforward to show that:
and that:
Note that contrary to probability theory, possibility is not self-dual. That is, for any event formula_6, we only have the inequality:
However, the following duality rule holds:
Accordingly, beliefs about an event can be represented by a number and a bit.
Interpretation.
There are four cases that can be interpreted as follows:
formula_31 means that formula_6 is necessary. formula_6 is certainly true. It implies that formula_29.
formula_35 means that formula_6 is impossible. formula_6 is certainly false. It implies that formula_30.
formula_29 means that formula_6 is possible. I would not be surprised at all if formula_6 occurs. It leaves formula_42 unconstrained.
formula_30 means that formula_6 is unnecessary. I would not be surprised at all if formula_6 does not occur. It leaves formula_46 unconstrained.
The intersection of the last two cases is formula_30 and formula_29 meaning that I believe nothing at all about formula_6. Because it allows for indeterminacy like this, possibility theory relates to the graduation of a many-valued logic, such as intuitionistic logic, rather than the classical two-valued logic.
Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classical example.
Possibility theory as an imprecise probability theory.
There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator.
A possibility measure can be seen as a consonant plausibility measure in Dempster–Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence.
Possibility can be seen as an upper probability: any possibility distribution defines a unique credal set set of admissible probability distributions by
This allows one to study possibility theory using the tools of imprecise probabilities.
Necessity logic.
We call "generalized possibility" every function satisfying Axiom 1 and Axiom 3. We call "generalized necessity" the dual of a generalized possibility. The generalized necessities are related with a very simple and interesting fuzzy logic we call "necessity logic". In the deduction apparatus of necessity logic the logical axioms are the usual classical tautologies. Also, there is only a fuzzy inference rule extending the usual Modus Ponens. Such a rule says that if α and α → β are proved at degree λ and μ, respectively, then we can assert β at degree min{λ,μ}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).