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Possibility Theory

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Encyclopedia of Complexity and Systems Science

Definition of the Subject and Its Importance

Possibility theory is the simplest uncertainty theory devoted to the modeling of incomplete information. It is characterized by the use of two basic dual set functions that respectively grade the possibility and the necessity of events. Possibility theory lies at the crossroads between fuzzy sets, probability, and nonmonotonic reasoning. Possibility theory is closely related to fuzzy sets if one considers that a possibility distribution is a particular fuzzy set (of mutually exclusive) possible values. However, fuzzy sets and fuzzy logic are primarily motivated by the representation of gradual properties while possibility theory handles the uncertainty of classical (or fuzzy) propositions. Possibility theory can be cast either in an ordinal or in a numerical setting. Qualitative possibility theory is closely related to belief revision theory and commonsense reasoning with exception-tainted knowledge in artificial intelligence. It has been...

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Abbreviations

Guaranteed Possibility:

A guaranteed possibility measure is a set function (decreasing in the wide sense) that returns the minimum of a possibility distribution over a subset representing an event. While possibility measures evaluate the consistency of the information between an event and the available information represented by the underlying possibility distribution, guaranteed possibility measures capture another view of the idea of possibility related to the idea of (guaranteed) feasibility or sufficiency condition.

Necessity Measure:

A necessity measure is a set function, associated by duality to a possibility measure through a relation expressing that an event is all the more necessarily true (all the more certain) as the opposite event is less possible. A necessity measure estimates to what extent the information represented by the underlying possibility distribution entails the occurrence of the event.

Possibilistic Logic:

Standard possibilistic logic is a weighted logic where formulas are pairs made of a classical logical formula and a weight that acts as a lower bound of the necessity of the logical formula. Extended possibilistic logics may include formulas weighted in terms of lower bounds of possibility or guaranteed possibility measures.

Possibility Distribution:

A possibility distribution restricts a set of possible values for a variable of interest in an elastic way. It is represented by a mapping from a universe gathering the potential values of the variable to a scale such as the unit interval of the real line or a finite linearly ordered set, expressing to what extent each value is possible for the variable. Thus, a possibility distribution restricts a set of more or less possible values belonging to a universe that may be also ordered such as a subpart of real line for a numerical variable, or not ordered if, for instance, the variable takes its value in the set of interpretations of a logical language. This may be used for representing uncertainty if the restriction pertains to possible values for an ill-known state of the world, or for representing preferences if the restriction encodes a set of values that are considered as more or less satisfactory for some purpose.

Possibility Measure:

A possibility measure is a set function (increasing in the wide sense) that returns the maximum of a possibility distribution over a subset representing an event.

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  1. IRIT-CNRS, Universite Paul Sabatier, 31062, Toulouse, Cedex 09, France

    Didier Dubois & Henri Prade

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    Robert A. Meyers

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Dubois, D., Prade, H. (2015). Possibility Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_413-2

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