# Formal Representations of Uncertainty

*Representing and reasoning under and about uncertainty * is one of the oldest topic of interest in the ADRIA team. Uncertainty is understood as the incapacity to assert the truth or the falsity of a proposition, based on the available information. Three main sources of uncertainty can be highlighted :

- The variability of natural phenomena, which forbids to predict their future behavior * the sheer lack of information
- The excess of information leading to inconsistency.

In other words uncertainty has to do as much with ill-known deterministic situations as with random processes.

The ADRIA team is reputed in part due to the development of a non-probabilistic theory of uncertainty due to incomplete or imprecise information: *possibility theory*. This is the simplest of all uncertainty theories. However it has close links with numerous problems of major concern in Artificial Intelligence and Information processing. In possibility theory, a proposition is qualified by its degree of possibility (expressing the extent to which it is plausible), and its degree of necessity (expressing the extent to which it is certain). The assignment of distinct degrees of plausibility and certainty to propositions is characteristic of uncertainty theories that deal with incomplete information (possibility theory, evidence theory, imprecise probabilities). In possibility theory, though, a proposition is somewhat certain only if it is fully possible.

Due to its elementary formal setting, possibility theory can be developed in a *qualitative* or a *quantitative* form.

The qualitative version of possibility theory has been extensively used for reasoning under and about incomplete or inconsistent information (see the page on reasoning) in the framework of possibilistic logic. It has also been applied to the handling of uncertainty in databases. Quantitative possibility theory has close connections with statistics and has been used in risk analysis information fusion and regression.

### On-going research

- Bridges between multiple -valued logics, modal logics, logic programming and possibility theory.
- Bridges between monotonic qualitative set-functions, uncertainty theories, modal logics, and logics of contradiction.
- Study of a general approach to information fusion that applies to all uncertainty theories
- Relations between statistics and quantitative possibility theory with applications to incomplete information processing.

## Main References about fuzzy sets and possibility theory

- Didier Dubois, Henri Prade.
**Bridging gaps between several forms of granular computing**.*Granular Computing*, Springer, Vol. 1, p. 115-126, 2016. - Didier Dubois, Henri Prade.
**The Emergence of Fuzzy Sets: A Historical Perspective.**Dans / In :*Fuzzy Logic in Its 50th Year 2016. New Developments, Directions and Challenges.*Cengiz Kahraman, Uzay Kaymak, Adnan Yazici (Eds.), Springer, p. 3-19, Vol. 341, Studies in Fuzziness and Soft Computing, 2016. (pdf) - Didier Dubois, Henri Prade.
**Practical Methods for Constructing Possibility Distributions**.*International Journal of Intelligent Systems*, Wiley, Vol. 31 N. 3, p. 215-239, 2016. (pdf) - Didier Dubois, Henri Prade.
**Possibility theory and its applications: where do we stand?**In :*Handbook of Computational Intelligence*. Janusz Kacprzyk, Witold Pedrycz (Eds.), Springer, p. 31-60, 2015. (pdf) - Didier Dubois.
**Possibilistic logic- an overview.**In :*Computational Logic*. Dov M Gabbay, Jörg Siekmann, John Woods (Eds.), Elsevier, p. 283-342, Vol. 9, Handbook of the History of Logic, 2014. (pdf)

## Particular References

- Sébastien Destercke, Didier Dubois.
**Other uncertainty theories based on capacities**. In :*Introduction to Imprecise Probabilities*. Thomas Augustin, Frank Coolen, Gert de Cooman, Matthias Troffaes (Eds.), Wiley, 5, pp. 93-113, 2014. - Didier Dubois, Henri Prade.
**Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets**. In :*Fuzzy Sets and Systems*, Elsevier, Vol. 192, pp. 3-24, 2012. - Didier Dubois, Henri Prade.
**An overview of the asymmetric bipolar representation of positive and negative information in possibility theory**.*Fuzzy Sets and Systems*, Elsevier, Vol. 160 N. 10, pp. 1355-1366, 2009. - Didier Dubois, Henri Prade.
**Formal representations of uncertainty**. In :*Decision-making – Concepts and Methods*. Denis Bouyssou, Didier Dubois, Marc Pirlot, Henri Prade (Eds.), Wiley, 3, pp. 85-156, 2009. - D. Dubois, F. Esteva, L. Godo, H. Prade.
**Fuzzy-set based logics – An history-oriented presentation of their main developments**. In :*Handbook of The history of logic*. Dov M. Gabbay, John Woods (Eds.), V. 8, The many valued and nonmonotonic turn in logic, Elsevier, 2007, p. 325-449. - Didier Dubois, Henri Prade.
**Possibility theory, probability theory and multiple-valued logics: A clarification**. In :*Annals of Mathematics and Artificial Intelligence*, Springer, Vol. 32, pp. 35-66, 2001. - Didier Dubois, Hung T. Nguyen, Henri Prade.
**Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps**. In :*Fundamentals of Fuzzy Sets*. D. Dubois, H. Prade (Eds.), Kluwer, pp. 343-438, The Handbooks of Fuzzy Sets Series, 2000. - Didier Dubois, Henri Prade.
**Possibility theory: qualitative and quantitative**. In :

aspects*Quantified Representation of Uncertainty and Imprecision*. Dov M. Gabbay, Philippe Smets (Eds.), Kluwer Academic Publishers, pp. 169-226, Vol. 1, Handbook of Defeasible Reasoning and Uncertainty Management Systems, 1998. - Didier Dubois, Henri Prade.
**Possibility Theory: An Approach to Computerized Processing of Uncertainty**(traduction revue et augmentée de “Théorie des Possibilités” Masson, 1987), Plenum Press, 1988.