Summary
Probability theory (P-theory) and Zadeh fuzzy system (Z-system) have both been used as foundations for mining structures or relations among data. The goal of this paper is to study the links and differences between the two systems. We start by considering the Z −-system, derived by discarding the complement set definition in the Z-system. A theorem is proved to state that P-theory and the Z −-system perform equivalently when one set is a subset of the other (i.e., A ⊆ B or B ≨ A) either in a sense of almost surely or in a sense of the Zadeh fuzzy set. Furthermore by jointly considering the B-system that modifies the Z-system with the “MIN-MAX” operations replaced by so-called Bold operations, another theorem is proved to state that the Z −-system and the B-system attempt to approximate the P-theory in two opposite ways, with success in some cases and failure in others. The failures come from either or both of an incapability of capturing additive structures and an inadequate handling of the dependence relation across two sets. Finally, we examine the controversial definition of a complement set in the Z-system and clarify that it arises from confusion about the “complement” concept and the “conjugate” concept. The confusion and thus the controversy can be avoided by honoring the additive principle, as in the B-system, or by renaming the complement set as the conjugate set.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Dubois, D. and Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications, Academic, New York
Dubois, D. and Prade, H. (1994) Partial Truth is not Uncertainty, IEEE Expert, pp 15–19
Elkan, C. (1994) The Paradoxical Success of Fuzzy Logic, IEEE Expert, pp 3–8. Also obtained a honorable mention in a best-written paper competition on Proceedings of AAAI’93, July 1993, pp 698–703
Gao, Q.S. (2005) The errors and shortcomings of Zadeh’s fuzzy set theory and its overcoming – C-fuzzy set theory, Journal of Dalian University of Technology, 45(5)
Giles, R. (1976) Lukasiewicz logic and fuzzy theory, International Journal of Man–Machine Studies, 8, pp 313–327
Jensen, F.V. (1996) An Introduction to Bayesian Networks, University of College London Press, London, UK
Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufman, San Fransisco, CA
Special Issue, (1994) Responses on the Paradoxical Success of Fuzzy Logic, IEEE Expert, pp 9–46
Xu, L. and Yan, P.F. (1985) Some Investigations on Subjective Probability Distribution, Proceedings of 5th National Conference on Pattern Recognition and Machine Intelligence, May 27–30, 1986. Xian, China
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gao, Q.S., Gao, X.Y., Xu, L. (2008). A Probability Theory Perspective on the Zadeh Fuzzy System. In: Lin, T.Y., Xie, Y., Wasilewska, A., Liau, CJ. (eds) Data Mining: Foundations and Practice. Studies in Computational Intelligence, vol 118. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78488-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-78488-3_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78487-6
Online ISBN: 978-3-540-78488-3
eBook Packages: EngineeringEngineering (R0)