A logic-based analysis of Dempster-Shafer theory

AbstractDempster-Shafer (DS) theory is formulated in terms of propositional logic, using the implicit notion of provability underlying DS theory. Dempster-Shafer theory can be modeled in terms of propositional logic by the tuple (Σ, ϱ), where Σ is a set of propositional clauses and ϱ is an assignment of mass to each clause Σi ϵ Σ. It is shown that the disjunction of minimal support clauses for a clause Σi with respect to a set Σ of propositional clauses, ξ(Σi, Σ), when represented in terms of symbols for the ϱi 's, corresponds to a symbolic representation of the Dempster-Shafer belief function for δi. The combination of Belief functions using Dempster's rule of combination corresponds to a combination of the corresponding support clauses. The disjointness of the Boolean formulas representing DS Belief functions is shown to be necessary. Methods of computing disjoint formulas using network reliability techniques are discussed.In addition, the computational complexity of deriving DS Belief functions, including that of the logic-based methods which are the focus of this paper, is explored. Because of intractability even for moderately sized problem instances, efficient approximation methods are proposed for such computations. Finally, implementations of DS theory based on domain restrictions of DS theory, hypertree embeddings, and the ATMS, are examined

Rejoinder to comments on “reasoning with belief functions: An analysis of compatibility”

AbstractAn earlier position paper has examined the applicability of belief-functions methodology in three reasoning tasks: (1) representation of incomplete knowledge, (2) belief-updating, and (3) evidence pooling. My conclusions were that the use of belief functions encounters basic difficulties along all three tasks, and that extensive experimental and theoretical studies should be undertaken before belief functions could be applied safely. This article responds to the discussion, in this issue, of my conclusions and the degree to which they affect the applicability of belief functions in automated reasoning tasks