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Possibility Theory

  • Reference work entry
Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

Historical Background

Basic Notions of Possibility Theory

Qualitative Possibility Theory

Quantitative Possibility Theory

Probability-Possibility Transformations

Applications and Future Directions

Bibliography

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Abbreviations

Possibility distribution :

A possibility distribution restricts a set of possible values fora variable of interest in an elastic way. It is represented by a mapping from a universe gathering the potential values of the variable toa 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 thevariable. Thus, a possibility distribution restricts a set of more or less possible values belonging to a universe that may be also ordered suchas 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 ofa logical language. This may be used for representing uncertainty if the restriction pertains to possible values for an ill-known state of the world, orfor representing preferences if the restriction encodes a set of values that are considered as more or less satisfactory for somepurpose.

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.

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.

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.

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.

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

    Didier Dubois & Henri Prade

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Dubois, D., Prade, H. (2012). Possibility Theory. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_139

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