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Computing Uncertainty in Interval Based Sets

  • Luis Mateus Rocha
  • Vladik Kreinovich
  • R. Baker Kearfott
Part of the Applied Optimization book series (APOP, volume 3)

Abstract

The kinds of uncertainty present in different interval based techniques of representing uncertainty in knowledge-based systems will be discussed. Interval-Valued Fuzzy Sets (IVFS) will be shown to describe both fuzziness and nonspecificity in their membership degrees, while a structure called an evidence set further introduces conflict. A more realistic model of uncertainty is described by L-fuzzy sets, interval-valued L-fuzzy sets, and L-evidence sets, where L is a finite set of possible degrees of confidence. Measures of uncertainty of such structures are examined.

Keywords

Membership Function Subjective Probability Membership Degree Fuzzy Measure Focal Element 
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.

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References

  1. [1]
    K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, 1986, Vol. 20, pp. 87–96.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    K. T. Atanassov, “Operators over interval-valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, 1994, Vol. 64, pp. 159–174.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    K. T. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, 1989, Vol. 31, pp. 343–349.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    D. Berleant, “Automatically Verified Arithmetic on Probability Distributions and Intervals”, This Volume.Google Scholar
  5. [5]
    M. B. Gorzalczany, “A method of inference in approximate reasoning based on interval-valued fuzzy sets”, Fuzzy Sets and Systems, 1987, Vol. 21, pp. 1–17.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    D. Harmanec and G. J. Klir, “Measuring total uncertainty in Dempster-Shafer Theory: a novel approach,” International Journal of General Systems, 1994, Vol. 22, pp. 405–419.zbMATHCrossRefGoogle Scholar
  7. [7]
    K. Hirota, “Concepts of probabilistic sets,” Fuzzy Sets and Systems, 1981, Vol. 5, pp. 31–46.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G. J. Klir, “Developments in uncertainty-based information,” In: M. Yovits (ed.), Advances in Computers, 1993, Vol. 36, pp. 255–332.Google Scholar
  9. [9]
    G. J. Klir and A. Ramer, “Uncertainty in the Dempster-Shafer theory: a critical re-examination,” International Journal of General Systems, 1990, Vol. 18, pp. 155–166.zbMATHCrossRefGoogle Scholar
  10. [10]
    G. J. Klir and T. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, 1988.zbMATHGoogle Scholar
  11. [11]
    G. Lakoff, Women, Fire, and Dangerous Things: What Categories Reveal About the Mind, University of Chicago Press, 1987.Google Scholar
  12. [12]
    M. T. Lamata and S. Moral, “Measures of entropy in the theory of evidence,” International Journal of General Systems, 1988, Vol. 14, No. 4, pp. 183–196.MathSciNetCrossRefGoogle Scholar
  13. [13]
    P. R. Medina-Martins, “Metalogues: an abridge of a genetic psychology of non-natural systems”, Communication and Cognition - Artificial Intelligence, 1995, Vol. 12, Nos. 1–2 (Special Issue Self-Reference in Biological and Cognitive Systems), in press.Google Scholar
  14. [14]
    P. R. Medina-Martins and L. M. Rocha, “The in and the out: an evolutionary approach,” In: R. Trappl (ed.), Cybernetics and Systems Research’92, World Scientific Press, 1992, pp. 681–689.Google Scholar
  15. [15]
    P. R. Medina-Martins, L. M. Rocha, et al., CYBORGS: A Fuzzy Conversational System. Final Report for the NATO International Program on Learning Systems, 1993.Google Scholar
  16. [16]
    K. Nakamura and S. Iwai, “A representation of analogical inference by fuzzy sets and its application to information retrieval systems,” In: M. M. Gupta and E. Sanchez (eds.), Fuzzy Information and Decision Processes, 1982, North-Holland, pp. 373–386.Google Scholar
  17. [17]
    A. Ramer, “Concepts of fuzzy information on continuous domains”, International Journal of General Systems, 1990, Vol. 17, pp. 241–248.zbMATHCrossRefGoogle Scholar
  18. [18]
    A. Ramer and V. Kreinovich, “Maximum entropy approach to fuzzy control”, Information Sciences, 1994, Vol. 81, No. 3–4, pp. 235–260.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    A. Ramer and V. Kreinovich, “Information complexity and fuzzy control”, Chapter 4 in: A. Kandel and G. Langholtz (Eds.), Fuzzy Control Systems, CRC Press, Boca Raton, FL, 1994, pp. 75–97.Google Scholar
  20. [20]
    L. M. Rocha, “Fuzzification of Conversation Theory,” In: Principia Cybernetica Conference, Free University of Brussels, 1991.Google Scholar
  21. [21]
    L. M. Rocha, “Cognitive Categorization revisited: extending interval valued fuzzy sets as simulation tools for concept combination,” In: Proceedings of the 1994 International Conference of NAFIPS/IFIS/NASA, 1994, IEEE, pp 400–404.Google Scholar
  22. [22]
    L. M. Rocha, “Von Foerster’s cognitive tiles: semantically closed building blocks for AI and Alife.” In: R. Trappl (ed.), Cybernetics and Systems’94, World Scientific Press, 1994, Vol. 1, pp. 621–628.Google Scholar
  23. [23]
    L. M. Rocha, “Interval Based Evidence Sets”, In: Proceedings of the 1995 NAFIPS/IFIS Joint Conference, IEEE Press, 1995 (in press).Google Scholar
  24. [24]
    L. M. Rocha, “Artificial semantically closed objects,” In: L. Rocha (ed.), Special issue in Self-Reference in Biological and Cognitive Systems, Communication and Cognition - AI, 1995, Vol. 12, No. 1–2.Google Scholar
  25. [25]
    L. M. Rocha, “Evidence (interval based) Sets: Modelling Subjective Categories,” International Journal of General Systems, 1995 (in press).Google Scholar
  26. [26]
    D. A. Schum, Evidential Foundations of Probabilistic Reasoning, John Wiley and Sons, 1994.Google Scholar
  27. [27]
    G. Shafer and J. Pearl (editors), Readings in Uncertain Reasoning, Morgan Kauffman, San Mateo, CA, 1990.zbMATHGoogle Scholar
  28. [28]
    G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, 1976.zbMATHGoogle Scholar
  29. [29]
    I. B. Turksen, “Interval valued fuzzy sets based on normal forms,” Fuzzy Sets and Systems, 1986, Vol. 20, pp. 191–210.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    J. Vejnarova and G. J. Klir, “Measure of strife in Dempster-Shafer theory”, International Journal of General Systems, 1993, Vol. 22, pp. 25–42.zbMATHCrossRefGoogle Scholar
  31. [31]
    Z. Wang and G. Klir, Fuzzy Measure Theory, Plenum Press, N.Y., 1992.zbMATHGoogle Scholar
  32. [32]
    Webster’s Dictionary, Random House, 1991.Google Scholar
  33. [33]
    R. R. Yager, “On the measure of fuzziness and negation, part I: membership in the unit interval,” International Journal of Man-Machine Studies, 1979, Vol. 11, pp. 189–200.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    R. R. Yager, “On the measure of fuzziness and negation. Part II: Lattices,” Information and Control, 1980, Vol. 44, pp. 236–260.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    Bo Yuan, W. Wu, and Y. Pan, “On normal form based interval valued fuzzy sets and its application to approximate reasoning,” International Journal of General Systems, 1995 (in press).Google Scholar
  36. [36]
    L. A. Zadeh, “Fuzzy Sets.” Information and Control, 1965, Vol. 8, pp. 338–353.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning,” Information Science, 1975, Vol. 8, pp. 199–249.MathSciNetCrossRefGoogle Scholar
  38. [38]
    Q. Zhu and E. S. Lee, “Evidence theory in multivalued logic systems”, International Journal of Intelligent Systems, 1995, Vol. 10, pp. 185–199.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Luis Mateus Rocha
    • 1
  • Vladik Kreinovich
    • 2
  • R. Baker Kearfott
    • 3
  1. 1.Department of Systems Science and Industrial Engineering, T.J. Watson SchoolState University of New York at BinghamtonBinghamtonUSA
  2. 2.Department of Computer ScienceUniversity of Texas at El PasoEl PasoUSA
  3. 3.Department of MathematicsUniversity of Southwestern Louisiana, U.S.LLafayetteUSA

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