On Logical Foundations of Evidential Reasoning Based on the Dempster-Shafer Probability Model

Abstract

First this paper discusses logical meanings of frames of discernments (FOD, in short) in the Dempster-Shafer probability model. In order to clarify formal semantics of FOD, their characteristic logical formulae are defined in terms of the first order predicate calculus, where assumptions used to define FOD are explicitly specified. For example, let A⟶P_1+P_2+…+P_n be a characteristic formula, where A is an assumption, P_i's are ground logical formulae representing meanings of elements in an FOD (= {P_1, P_2, …, P_n}).⟶ and + denote implication and exclusive OR respectively. Then, we augment an FOD by introducing a new element representing its basic assumption (i. e. A). With augmented FOD, we can eliminate the heuristic normalization involved in Dempster's combination rule. Moreover, we can define algorithms to consistently integrate probabilities defined on different FOD. Logical foundations of the integration are described by using characteristic logical formulae of FOD. Several illustrative examples are given to show the process of the integration. In the last part of the paper, we discuss applications of the algorithms to image understanding.