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.
For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function
\Pi
2\Omega
Axiom 1:
\Pi(\varnothing)=0
Axiom 2:
\Pi(\Omega)=1
Axiom 3:
\Pi(U\cupV)=max\left(\Pi(U),\Pi(V)\right)
U
V
It follows that, like probability on finite probability spaces, the possibility measure is determined by its behavior on singletons:
\Pi(U)=max\omega\Pi(\{\omega\}).
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
\Pi
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:
\Pi(U\cupV)=max\left(\Pi(U),\Pi(V)\right)
U
V
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:
\Pi(U\capV)\leqmin\left(\Pi(U),\Pi(V)\right)\leqmax\left(\Pi(U),\Pi(V)\right).
When Ω is not finite, Axiom 3 can be replaced by:
For all index sets
I
Ui,
\Pi\left(cupiUi\right)=\supi\Pi(Ui).
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
U
N(U)=1-\Pi(\overlineU)
In the above formula,
\overlineU
U
\Omega
U
N(U)\leq\Pi(U)
U
and that:
N(U\capV)=min(N(U),N(V))
Note that contrary to probability theory, possibility is not self-dual. That is, for any event
U
\Pi(U)+\Pi(\overlineU)\geq1
However, the following duality rule holds:
For any event
U
\Pi(U)=1
N(U)=0
Accordingly, beliefs about an event can be represented by a number and a bit.
There are four cases that can be interpreted as follows:
N(U)=1
U
U
\Pi(U)=1
\Pi(U)=0
U
U
N(U)=0
\Pi(U)=1
U
U
N(U)
N(U)=0
U
U
\Pi(U)
The intersection of the last two cases is
N(U)=0
\Pi(U)=1
U
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 classic example.
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 the 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
K=\{P\mid\forallS P(S)\leq\Pi(S)\}.
This allows one to study possibility theory using the tools of imprecise probabilities.
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 to a very simple and interesting fuzzy logic called 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. 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).