A Possibilistic Logic View of Sugeno Integrals

  • Didier Dubois
  • Henri Prade
  • Agnès Rico
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 107)


Sugeno integrals are well-known qualitative aggregation functions in multiple criteria decision making. They return a global evaluation between the minimum and the maximum of the input criteria values. They can model sophisticated aggregation schemes through a system of priorities that applies to any subset of criteria and can take into account some kind of synergy inside subsets of criteria. Although a given Sugeno integral specifies a particular way of implicitly describing a set of entities reaching some global satisfaction level, it is hard to figure out what is the underlying explicit meaning of such an integral in practice (even if the priority level associated to each subset of criteria has a precise meaning). The paper proposes an answer to this problem. Any capacity on a finite set can be represented by a special possibilistic logic base containing positive prioritised clauses, and conversely any possibilistic logic base can represent a set-function. Moreover, Sugeno integral can be represented by a possibilistic logic base expressing how it behaves (thanks to a mapping between the scale and a set of logical atoms reflecting the different values for each criterion). Viewing a Sugeno integral as a set of prioritized logically expressed goals has not only the advantage to make the contents of a Sugeno integral more readable, but it also prompts Sugeno integrals into the realm of logic, and makes it possible to define entailment between them.


Priority Level Propositional Variable Fuzzy Measure Possibility Distribution Possibilistic Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benferhat, S., Dubois, D., Kaci, S., Prade, H.: Bipolar possibility theory in preference modeling: Representation. Fusion and Optimal Solutions, Information Fusion 7, 135–150 (2006)Google Scholar
  2. 2.
    Dubois, D., Fargier, H.: Making discrete Sugeno integrals more discriminant. Inter. J. of Approximate Reasoning 50, 880–898 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dubois, D., Le Berre, D., Prade, H., Sabbadin, R.: Using possibilistic logic for modeling qualitative decision: ATMS-based algorithms. Fundamenta Informaticae 37(1-2), 1–30 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dubois, D., Marichal, J.-L., Prade, H., Roubens, M., Sabbadin, R.: The use of the discrete Sugeno integral in decision-making: A survey. Inter. J. of Uncertainty, Fuzziness and Knowledge-based Systems 9, 539–561 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dubois, D., Prade, H.: Qualitative possibility functions and integrals. In: Pap, E. (ed.) Handbook of Measure Theory, vol. 2, pp. 1469–1521. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  6. 6.
    Dubois, D., Prade, H.: Possibilistic logic: a retrospective and prospective view. Fuzzy Sets and Systems 144, 3–23 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gérard, R., Kaci, S., Prade, H.: Ranking alternatives on the basis of generic constraints and examples - A possibilistic approach. In: Veloso, M.M. (ed.) Proc. 20th Inter. Joint Conf. on Artificial Intelligence (IJCAI 2007), Hyderabad, January 6-12, pp. 393–398 (2007)Google Scholar
  8. 8.
    Grabisch, M.: The Möbius transform on symmetric ordered structures and its application to capacities on finite sets. Discrete Mathematics 287, 17–34 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Annals of Operations Research 175, 247–286 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Greco, S., Matarazzo, B., Slowinski, R.: Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. Europ. J. of Operational Research 158, 271–292 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Labreuche, C.: A general framework for explaining the results of a multi-attribute preference model. Artificial Intelligence 175, 1410–1448 (2011)CrossRefGoogle Scholar
  12. 12.
    Marichal, J.-L.: Aggregation Operations for Multicriteria Decision Aid. Ph.D.Thesis, University of Liège, Belgium (1998)Google Scholar
  13. 13.
    Mesiar, R.: k-order Pan-discrete fuzzy measures. In: Proc. 7th Inter. Fuzzy Systems Assoc. World Congress (IFSA 1997), Prague, June 25-29, vol. 1, pp. 488–490 (1997)Google Scholar
  14. 14.
    Sugeno, M.: Theory of Fuzzy Integrals and its Applications, Ph.D. Thesis, Tokyo Institute of Technology, Tokyo (1974)Google Scholar
  15. 15.
    Sugeno, M.: Fuzzy measures and fuzzy integrals: a survey. In: Gupta, M.M., Saridis, G.N., Gaines, B.R. (eds.) Fuzzy Automata and Decision Processes, pp. 89–102. North-Holland, Amsterdam (1977)Google Scholar
  16. 16.
    Prade, H., Rico, A.: Describing acceptable objects by means of Sugeno integrals. In: Martin, T., et al. (eds.) Proc. 2nd IEEE Inter. Conf. of Soft Computing and Pattern Recognition (SoCPaR 2010), Cergy Pontoise, Paris, December 7-10 (2010)Google Scholar
  17. 17.
    Prade, H., Rico, A., Serrurier, M.: Elicitation of Sugeno integrals: A version space learning perspective. In: Rauch, J., Raś, Z.W., Berka, P., Elomaa, T. (eds.) ISMIS 2009. LNCS, vol. 5722, pp. 392–401. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Prade, H., Rico, A., Serrurier, M., Raufaste, E.: Elicitating Sugeno integrals: Methodology and a case study. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS, vol. 5590, pp. 712–723. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Didier Dubois
    • 1
  • Henri Prade
    • 1
  • Agnès Rico
    • 2
  1. 1.IRITCNRS and Université de ToulouseFrance
  2. 2.ERICUniversité de LyonFrance

Personalised recommendations