A Probability Theory Perspective on the Zadeh Fuzzy System

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.