A Logic-Based Analysis of Dempster Shafer Theory

We formulate Dempster Shafer Theory in terms of Propositional Logic, using the implicit notion of probability underlying Dempster Shafer Theory. Dempster Shafer theory can be modeled in terms of propositional logic by the tuple $(\Sigma , \varrho )$, where $\Sigma $ is a set of propositional clauses and $\varrho $ is an assignment of measure to each clause $\Sigma_i \in \Sigma $. We show that the disjunction of minimal support clauses for a clause $i$ with respect to a set $\Sigma $ of propositional clauses, $ \xi ( \Sigma_{i}, \Sigma )$, is a symbolic representation of the Dempster Shafer Belief function for $\Sigma_{i}$. 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 formulae representing DS Belief functions is shown to be necessary. Methods of computing disjoint formulae using Network Reliability techniques are discussed.

In addition, we explore the computational complexity of deriving Dempster Shafer Belief functions, including that of the logic-based methods which are the focus of this paper. Because of Intractability even for moderately-sized problem instances, we propose the use of effluent approximation methods for such computations. Finally, we examine implementations of Dempster Shafer theory, based on domain restrictions of DS theory, hypertree embeddings, and the ATMS.