A Set-Theoretic Framework for the Processing of Uncertain Knowledge

Abstract

In this paper, a knowledge base is represented by an input space, an output space, and a set of mappings that associate subsets of the two spaces. Under this representation, knowledge propocessing has three major parts: (1) The user enters observations of evidence in the input space and assigns a degree of certainty to each observation. (2) A piece of evidence that receives a non-zero certainty activates a mapping. This certainty is multiplied by the certainty associated with the mapping, and is thus propagated to a proposition in the output space. (3) The consensus among all the propositions that have non-zero certainties is computed, and a final set of conclusions is drawn. A degree of support is associated with each conclusion. The underlying model of certainty in this processing scheme is based on the Dempster-Shafer mathematical theory of evidence. The computation of the consensus among the propositions uses Dempster's rule of combination. The inverse of the rule of combination, which we call the rule of decomposition, is derived in this paper. Given an expected consensus, the inverse rule can generate the certainty required for each proposition. Thus, the certainties in the mappings can be inferred iteratively through alternating use of the rule of combination and the rule of decomposition.