THE DEVELOPMENT OF AFS THEORY UNDER PROBABILITY THEORY
Xiaodong Liu · 2007
As moving further into the age of machine intelligence and auto- mated decision-making, we have to deal with both the subjective imprecision of human perception-based information described in natural language and the ob- jective uncertainty of randomness universally existing in the real world. A basic limitation of standard probability theory which cannot deal with information described in natural language becomes a serious problem. With its abilities to represent natural language, the notion of AFS (Axiomatic Fuzzy Set) theory has proven useful in the clustering, classiflcations, concept representations and decision trees. In this paper, we apply AFS theory and probability theory to propose a new interpretation of the membership functions taking both fuzziness (subjective imprecision) and randomness (objective uncertainty) into account. So that uncertainty of randomness and of imprecision can be treated in a uni- fled and coherent manner under the AFS and probability framework. It opens a door to explore the deep mathematical analysis properties of fuzzy set the- ory and to a major enlargement of the role of natural languages in probability theory.