Knowledge and Information Systems

, Volume 6, Issue 3, pp 345–365 | Cite as

Fuzzy Conditional Probability Relations and their Applications in Fuzzy Information Systems

  • Rolly IntanEmail author
  • Masao Mukaidono


In our real-world applications, data may be imprecise in which levels or degrees of preciseness of data are intuitively different. In this case, fuzzy set expressions are considered as an alternative to represent the imprecise data. In general, the degree of similarity relationship between two fuzzy (imprecise) data in real-world applications may not necessarily be symmetric or transitive. In order to provide such a degree of similarity between two fuzzy data, we introduced the fuzzy conditional probability relation. The concept of a fuzzy conditional probability relation may be considered as a concrete example of weak similarity relation which in turn is a special type of fuzzy binary relation generalizing similarity relation. Two important applications concerning the application of Knowledge Discovery and Data Mining (KDD) in the presence of a fuzzy data table (usually called fuzzy information system), namely removing redundant objects and recognizing partial or total dependency of (domain) attributes, are considered induced by the fuzzy conditional probability relation. Here, the fuzzy information system contains precise as well as imprecise data (fuzzy values) about objects of interest characterized by some attributes. Related to the dependency of attributes, we introduce the fuzzy functional dependency that satisfies Armstrong’s Axioms. In addition, we also discuss some interesting applications such as approximate data reduction and projection, approximate data querying and approximate joining in order to extend the query system.


Data querying Fuzzy conditional probability relation Fuzzy functional Fuzzy information system dependency Fuzzy proposition Fuzzy sets Knowledge discovery and data mining 


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  1. Armstrong WW (1974) Dependency Structures of Database Relationship. Information Processing, pp 580–583 Google Scholar
  2. Bezdek JC (1981) Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum, New York Google Scholar
  3. Bosc P, Dubois D, Prade H (1998) Fuzzy Functional Dependencies and Redundancy Elimination. J Am Soc Inf Sci 49(3):217–235 CrossRefGoogle Scholar
  4. Buckles BP, Petry FE (1982) A Fuzzy Representation of Data for Relational Database. Fuzzy Sets Syst 5:213–226 CrossRefGoogle Scholar
  5. Codd EF (1970) A Relational Model of Data for Large Shared Data Banks. Commun ACM 13(6):377–387 CrossRefGoogle Scholar
  6. Dubois D, Prade H (1993) Fuzzy Sets and Probability: Misunderstandings, Bridges and Gaps. Proc Second IEEE Int Conf on Fuzzy Systems, San Francisco, pp 1059–1068 Google Scholar
  7. Dubois D, Prade H (1985) A Review of Fuzzy Sets Aggregation Connectives. Inf Sci 36(1–2):85–121 Google Scholar
  8. Intan R, Mukaidono M, Yao YY (2001) Generalization of Rough Sets with α-coverings of the Universe Induced by Conditional Probability Relation. Proc of Int Workshop on Rough Set Theory and Granular Computing, pp 173–176 Google Scholar
  9. Intan R, Mukaidono M (2000a) Conditional Probability Relations in Fuzzy Relational Database. Proc of the Second Inte Conf on RSCTC’00, pp 213–222 Google Scholar
  10. Intan R, Mukaidono M (2000b) Application of Conditional Probability in Constructing Fuzzy Functional Dependency(FFD). Proc of AFSS’00, pp 271–276 Google Scholar
  11. Intan R, Mukaidono M (2000c) Fuzzy Functional Dependency and Its Application to Approximate Querying. Proc of IDEAS’00, pp 47–54 Google Scholar
  12. Klir GJ, Yuan B (1995) Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, NJ Google Scholar
  13. Maier D (1983) The Theory of Relational Database. Computer Science Press Google Scholar
  14. Mukaidono M (1999) Several Extensions of Truth Values in Fuzzy Logic. Proc ISMIS’99, pp 282–291 Google Scholar
  15. Komorowski J, Pawlak Z, Polkowski L, Skowron A (1999) Rough Sets: A Tutorial Google Scholar
  16. Kosko B (1990) Fuzziness vs Probability. Int J Gen Syst 17:211–240 CrossRefGoogle Scholar
  17. Pawlak Z (1991) ROUGH SETS Theoretical Aspects of Reasoning about Data. Kluwer Google Scholar
  18. Raju KVSVN, Majumdar AK (1988) Fuzzy Functional Dependencies and Lossless Join Decomposition of Fuzzy Relational Database Systems. ACM Trans Database Syst 13(2):129–166 CrossRefGoogle Scholar
  19. Richard Jeffrey (1995) Probabilistic Thinking. Princeton University Google Scholar
  20. Shenoi S, Melton A (1989) Proximity Relations in The Fuzzy Relational Database Model. Fuzzy Sets Syst 31:285–296 MathSciNetCrossRefGoogle Scholar
  21. Yager RR (1990) Ordinal Measures of Specificity. Int J Gen Syst 17:57–72 CrossRefGoogle Scholar
  22. Zadeh LA (1965) Fuzzy Sets. Inf Control 8:338–353 MathSciNetCrossRefGoogle Scholar
  23. Zadeh LA (1968) Probability Measures and Fuzzy Events. J Math Anal Appl 23:421–427 MathSciNetCrossRefGoogle Scholar
  24. Zadeh LA (1970) Similarity Relations and Fuzzy Orderings. Inf Sci 3(2):177–200MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of Computer ScienceMeiji UniversityKanagawa-kenJapan
  2. 2.Department of Informatics EngineeringPetra Christian UniversitySurabayaIndonesia

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