A logic-based analysis of Dempster-Shafer theory

Dempster-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{variant}), where Σ is a set of propositional clauses and ρ{variant} is an assignment of mass to each clause Σ_i ε{lunate} Σ. It is shown that the disjunction of minimal support clauses for a clause Σ_i with respect to a set Σ of propositional clauses, ξ(Σ_i, Σ), when represented in terms of symbols for the ρ{variant}_i 's, corresponds to a symbolic representation of the Dempster-Shafer belief function for δ_i. 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.