Advertisement

Thick Sets, Multiple-Valued Mappings, and Possibility Theory

  • Didier Dubois
  • Luc JaulinEmail author
  • Henri Prade
Chapter
  • 81 Downloads
Part of the Studies in Computational Intelligence book series (SCI, volume 892)

Abstract

Carrying uncertain information via a multivalued function can be found in different settings, ranging from the computation of the image of a set by an inverse function to the Dempsterian transfer of a probabilistic space by a multivalued function. We then get upper and lower images. In each case one handles so-called thick sets in the sense of Jaulin, i.e., lower and upper bounded ill-known sets. Such ill-known sets can be found under different names in the literature, e.g., interval sets after Y. Y. Yao, twofold fuzzy sets in the sense of Dubois and Prade, or interval-valued fuzzy sets, ... Various operations can then be defined on these sets, then understood in a disjunctive manner (epistemic uncertainty), rather than conjunctively. The intended purpose of this note is to propose a unified view of these formalisms in the setting of possibility theory, which should enable us to provide graded extensions to some of the considered calculi.

Keywords

Thick set Interval analysis Possibility theory Inverse image Uncertainty 

References

  1. 1.
    M. Banerjee, D. Dubois, A simple logic for reasoning about incomplete knowledge. Int. J. Approx. Reason. 55, 639–653 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    I. Couso, D. Dubois, L. Sanchez, Random sets and random fuzzy sets as Ill-perceived random variables, in SpringerBriefs in Computational Intelligence (Springer, Berlin, 2014)Google Scholar
  3. 3.
    A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38, 325–339 (1967)MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Denœux, Z. Younes, F. Abdallah, Representing uncertainty on set-valued variables using belief functions. Artif. Intell. 174(7–8), 479–499 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Desrochers, L. Jaulin, Computing a guaranteed approximation of the zone explored by a robot. IEEE Trans. Automat. Contr. 62(1), 425–430 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Desrochers, L. Jaulin, Thick set inversion. Artif. Intell. 249, 1–18 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Dubois, H. T. Nguyen, H. Prade, M. Sugeno, Introduction: the real contribution of fuzzy systems, in Fuzzy Systems: Modelling and Control, ed. by H.T. Nguyen, M. Sugeno (Kluwer Academic Publishers, Boston, 1998), pp. 1–14Google Scholar
  8. 8.
    D. Dubois, H. T. Nguyen, H. Prade, Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps, in Fundamentals of Fuzzy Sets, ed. by D. Dubois, H. Prade. The Handbooks of Fuzzy Sets Series (Kluwer, 2000), pp. 343–438Google Scholar
  9. 9.
    D. Dubois, H. Prade, Inverse operations for fuzzy numbers, in PRE proceedings of IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, July 19–21 (1983), pp. 391–396. Fuzzy Information, Knowledge Representation and Decision Analysis, ed. by E. Sanchez, M.M. Gupta (Pergamon Press, 1984), pp. 399–404Google Scholar
  10. 10.
    D. Dubois, H. Prade, Fuzzy set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control. Cybern. (Warsaw) 13, 129–146 (1984)MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Dubois, H. Prade, Evidence measures based on fuzzy information. Automatica 21, 547–562 (1985). Preliminary version: Upper and lower possibilities induced by a multivalued mapping, in PRE proceedings of IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, July 19–21 (1983), pp. 174–152. Fuzzy Information, Knowledge Representation and Decision Analysis, ed. by E. Sanchez (Pergamon Press, 1984)Google Scholar
  12. 12.
    D. Dubois, H. Prade, Twofold fuzzy sets and rough sets—some issues in knowledge representation. Fuzzy Sets Syst. 23(1), 3–18 (1987)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Dubois, H. Prade, Possibility Theory (Plenum Press, 1988)Google Scholar
  14. 14.
    D. Dubois, H. Prade, On incomplete conjunctive information. Comput. Math. Appl. 15(10), 797–810 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Dubois, H. Prade, Gradualness, uncertainty and bipolarity: making sense of fuzzy sets. Fuzzy Sets Syst. 192, 3–24 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    H. Farreny, H. Prade, Uncertainty handling and fuzzy logic control in navigation problems, in Proceedings of International Conference on Intelligent Autonomous Systems, ed. by L.O. Hertzberger, F.C.A. Groen (Amsterdam, 1986), pp. 218–225, (North-Holland, 1987)Google Scholar
  17. 17.
    H. Farreny, H. Prade, Tackling uncertainty and imprecision in robotics, in Proceedings of 3rd International Symposium of Robotics Research, Gouvieux (Chantilly), ed. by O. Faugeras, G. Giralt (M.I.T. Press, 1985), pp. 85–91Google Scholar
  18. 18.
    Y. Gentilhomme, Les ensembles flous en linguistique. Cahiers de Linguistique Théorique and Appliquée (Bucarest) 5, 47–63 (1968)Google Scholar
  19. 19.
    L. Jaulin, Solving set-valued constraint satisfaction problems. Computing 94(2–4), 297–311 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    L. Jaulin, M. Kieffer, O. Didrit, E. Walter, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics (Springer, London, 2001)Google Scholar
  21. 21.
    R. Moore, Interval Analysis (Prentice-Hall, 1966)Google Scholar
  22. 22.
    H.T. Nguyen, A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. 64(2), 369–380 (1978)MathSciNetCrossRefGoogle Scholar
  23. 23.
    H.T. Nguyen. On random sets and belief functions. J. Math. Anal. Appl. 65(3), 531–542 (1978). Reprinted in Classic Works of the Dempster-Shafer Theory of Belief Functions, ed. by R. Yager, L. Liu (A.P. Dempster, G. Shafer, advisory eds.). Studies in Fuzziness and Soft Computing, 219, Chap. 5 (2008), pp. 105–116Google Scholar
  24. 24.
    H.T. Nguyen, V. Kreinovich, From numerical intervals to set intervals (Interval-related results presented at the first international workshop on applications and theory of random sets). Reliab. Comput. 3(1), 95–102 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    H.T. Nguyen, V. Kreinovich, B. Wu, G. Xiang, Computing statistics under interval and fuzzy uncertainty, in Applications to Computer Science and Engineering. Studies in Computational Intelligence 393 (Springer, 2012)Google Scholar
  26. 26.
    H.T. Nguyen, V. Kreinovich, O. Kosheleva, Membership functions representing a number vs. representing a set: proof of unique reconstruction, in Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’16), Vancouver, July 24–29 (2016), pp. 657–662Google Scholar
  27. 27.
    Z. Pawlak, Rough Sets. Theoretical Aspects of Reasoning about Data (Kluwer Academic Publishers, Dordrecht, 1991)Google Scholar
  28. 28.
    R. Sambuc, Fonctions \(\Phi \)-floues. Application à l’aide au diagnostic en pathologie thyroidienne. Thèse Université de Marseille (1975)Google Scholar
  29. 29.
    E. Sanchez, Solution of fuzzy equations with extended operations. Fuzzy Sets Syst. 12, 237–248 (1984)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Y.Y. Yao, Interval sets and interval-set algebras, in Proceedings of the 8th IEEE International Conference on Cognitive Informatics (ICCI’09), June 15–17, ed. by G. Baciu, Y. Wang, Y. Yao, W. Kinsner, K. Chan, L.A. Zadeh (Hong Kong, 2009), pp. 307–314Google Scholar
  31. 31.
    J.T. Yao, Y. Yao, V. Kreinovich, P. Pinheiro da Silva, S.A. Starks, G. Xiang, H.T. Nguyen, Towards more adequate representation of uncertainty: from intervals to set intervals, with possible addition of probabilities and certainty degrees, in Proceedings of International Conference on Fuzzy Systems (FUZZ-IEEE408), June 1–6 (Hong Kong, 2008), pp. 983–990Google Scholar
  32. 32.
    L.A. Zadeh, Quantitative fuzzy semantics. Inf. Sci. 3, 159–176 (1971)MathSciNetCrossRefGoogle Scholar
  33. 33.
    L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

Authors and Affiliations

  1. 1.IRIT - CNRSToulouse Cedex 09France
  2. 2.Lab-STICC, ENSTA-BretagneBrestFrance

Personalised recommendations