An Analysis of ATMS-Based Techniques for Computing Dempster-Shafer Belief Functions

This paper analyzes the theoretical under­ pinnings of recent proposals for computing Dempster-Shafer Belief functions from ATMS labels. Such proposals are intended to be a means of integrating symbolic and numeric rep­ resentation methods and of focusing search in the ATMS. This synthesis is formalized us­ ing graph theory, thus showing the relation­ ship between graph theory, the logic-theoretic ATMS description and the set-theoretic Demp­ ster Shafer Theory description. The computa­ tional complexity of calculating Belief functions from ATMS labels using algorithms originally derived to calculate the network reliability of graphs is analyzed. Approximation methods to more efficiently compute Belief functions using this graphical approach are suggested. 1 Introduction To bridge the gap between the claimed lack of a "logical semantics" in uncertainty calculi and the lack of notions of uncertainty (claimed essential to modeling human rea­ soning) in logic, several attempts have been made to integrate formal logic with an uncertainty calculus. In this paper the relationships between an uncertainty cal­ culus, Dempster Shafer Theory, and propositional logic are shown. It has been proposed that Dempster Shafer (DS) The­ ory rivals Probability Theory in expressive power and ef­ fectiveness as a calculus for reasoning under uncertainty. However, because of the computational complexity as­ sociated with computing DS Belief functions, only sub­ sets of the full problem domain expressible in DS The­ ory have been implemented, with the exception of recent Assumption-based TMS (ATMS) implementations. The number of subsets of a set of propositions increases ex­ ponentially with and given that the DS normalizing function can sum over all of these subsets, computing a tute and NSERC grants to A.K. Mackworth. single normalization function can be computationally ex­ pensive. The total space necessary to compute DS belief functions over a set of n propositions is in the worst case. Examples of such restricted implementations include work by Shafer and Logan and by d'Ambrosio. Shafer and Logan [1987] have implemented DS Theory re­ stricted to the case of hierarchical evidence, based on proposals by Barnett [l98l] and Gordon and Short-liffe [1985]. D'Ambrosio [1987] has implemented DS theory for the restricted case defined by the Support Logic Programming of Baldwin [1985]. D'Ambrosio at­ taches a simplification of the Dempster-Shafer uncer­ tainty bounds to ATMS labels. have independently extended the ATMS with the full DS theory in similar manners. Such an extension represents a synthesis of the symbolic (logic-theoretic) ATMS representation and the …