Possibility theory and related representations of uncertainty
In the last 25 years, our group has extensively contributed to the development and to the recognition of a new theory for the representation of uncertainty, the theory of possibility. This theory was initiated by L. A. Zadeh in 1978, independently from a previous, less formalized, proposal by the economist G. L. S. Shackle in the sixties.
Uncertainty in possibility theory is represented by a pair of dual "measures" of possibility and necessity, usually graded on the unit interval (although any pseudo-complemented lattice structure could be used in its place). Possibility measures are max-decomposable for the union of events, in contrast with probability measures which are additive, while necessity measures are min-decomposable for the intersection of events. Possibility and necessity measures can be usually defined from a possibility distribution which assigns a level of plausibility to each element of the referential or universe of discourse. Possibility and necessity degrees should not be confused with degrees of truth in multiple-valued or fuzzy logics. Degrees of truth are generally compositional w.r.t. all connectives, which cannot be the case for degrees of uncertainty.
We can distinguish between qualitative and quantitative possibility theories. Qualitative possibility theory can be defined in purely ordinal settings, while quantitative possibility theory requires the use of a numerical scale. Quantitative possibility measures can be viewed as upper bounds of imprecisely known probability measures. Several operational semantics for possibility degrees have been recently obtained. Qualitative possibility theory is more closely related to nonmonotonic reasoning. Qualitative and quantitative possibility theories differ in the way conditioning is defined (it is respectively based on minimum and product operations).
Apart from possibility and necessity measures, there exist two other noticeable set functions, in possibility theory: the guaranteed possibility and its dual the potential necessity function. Guaranteed possibility function is obtained by taking the minimum of the possibility distribution over subsets of interest, while ordinary possibility is computed by taking the maximum. Guaranteed possibility and potential necessity are decreasing set functions w.r.t. set inclusion, which contrasts with the other set functions used for modelling uncertainty. The joint use of guaranteed and ordinary possibility functions can give birth to bipolar representation frameworks, where we can distinguish between what is possible for sure and what is known as just being not impossible.
A logical counterpart of possibility theory has been developed for almost twenty years, and is known as possibilistic logic. A possibilistic logic formula is a pair of a classical logic formula and a weight understood as a lower bound of a necessity measure. Various extensions of possibilistic logic handle lower bounds of guaranteed or ordinary possibility functions, weights involving variables, fuzzy constants, multiple source information, timed information, ... Graphical representations of possibilistic logic bases, using the two types of conditioning, have been also obtained (see Graphical representations. Independence).
Although possibility theory has been initially developed as a framework for representing imprecise and fuzzy information, it also applies to the modelling of preference. In such a case, a possibility degree should be understood as a satisfaction degree rather than a plausibility degree, and necessity measures express priority rather than certainty levels.
Our group has also contributed to the study of other uncertainty theories including Shafer belief functions (random set interpretation and set operations, unicity of Dempster rule, approximation by possibility measures, ...), imprecise probabilities (study of syllogisms quantified with interval-valued and ill-known conditional probabilities), and rough sets (difference with fuzzy sets, and hybridization with them). The relation between possibility and probability has been explored and possibility-probability transformations have been proposed in connection with Laplace indifference principle, shapley value, and confidence intervals.
Lastly, our group has emphasized the distinction between two types of operations related to the treatment of the arrival of information: revision and focusing. Revision takes place when the input information claims that worlds violating this input information do not exist, while focusing refers to a query of the type "what can be said about items having a given property?" . Frameworks like belief functions, or numerical possibility theory are capable of distinguishing between the two operations. Focusing is closely related to nonmonotonic reasoning where default knowledge has to be applied to an incompletely described contingent situation. (See exception-tolerant reasoning)
contacts: D. Dubois, H. Prade.
some recent publications
- D. Dubois, H. Prade, S. Sandri. Possibilistic logic with fuzzy constants and fuzzily restricted quantifiers. In: Logic Programming and Soft Computing, (Martin,T.P. et Arcelli-Fontana,F., Eds.), Research Studies Press, Baldock, England, 69-90, 1998.
- D. Dubois, W. Ostasiewicz, H. Prade . Fuzzy sets: History and basic notions . In: Fundamentals of Fuzzy Sets . (Dubois,Didier et Prade,Henri, Eds.), Kluwer Academic Publishers , Boston, The Handbooks of Fuzzy Sets Series , 21-124 , 2000.
- D. Dubois, P. Hajek, H. Prade. Knowledge-Driven versus data-driven logics. Journal of Logic, Language, and Information. Eds: Kluwer Academic Publishers, 9, 65-89, 2000.
- D. Dubois, H. Prade. Possibility theory, probability theory and multiple-valued logics: A clarification. Dans: Annals of Mathematics and Artificial Intelligence. Eds: Kluwer, Dordrecht, V. 32, p. 35-66, 2001.
