Towards More Adequate Representation of Uncertainty: From Intervals to Set Intervals, with the Possible Addition of Probabilities and Certainty Degrees

摘要

In the ideal case of complete knowledge, for each property P{sub}i (such as "high fever", "headache", etc.), we know the exact set S{sub}i of all the objects that satisfy this property. In practice, we usually only have partial knowledge. In this case, we only know the set S{sub}i of all the objects about which we know that P{sub}i holds and the set S{sub}i about which we know that P{sub}i may hold (i.e., equivalently, that we have not yet excluded the possibility of P{sub}i). This pair of sets is called a set interval. Based on the knowledge of the original properties, we would like to describe the set S of all the values that satisfy some combination of the original properties: e.g., high fever and headache and not rash. In the ideal case when we know the exact set S{sub}i of all the objects satisfying each property, it is sufficient to apply the corresponding set operation (composition of union, intersection, and complement) to the known sets S{sub}i. In this paper, we describe how to compute the class S of all possible sets S.