AN ANALYSIS OF ATMS-BASED TECHNIQUES FOR COMPUTING DEMPSTER-SHAFER BELIEF FUNCTIONS Gregory M. Provan* Department of Computer Science University of British Columbia Vancouver, BC Canada V6T 1W5 Abstract 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 *The author completed this research with the support of the University of British Columbia Center for Integrated Computer Systems Research, BC Advanced Systems Insti- 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. Laskey and Lehner [1988], Provan ([1988b], [1989a]) and Pearl [1988] 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 numeric (set- theoretic) DS Theory representation. However, there has been no analysis of the underlying theoretical foun- dations or the complexity of this DS theory implemen- tation. This paper describes the theoretical and computa- tional issues raised by extending an ATMS to compute DS Belief functions without significantly compromising the semantic clarity or computational properties of ei- ther. This extension is motivated by the need to intro- duce a weighting system into the ATMS, and to improve the poor performance of the ATMS as observed in prac- tice and predicted by the average-case results of Provan [1989b]. Existing ATMSs cannot rank hypotheses, but the incorporation of DS Theory allows hypotheses to be ranked and offers efficiency improvements, such as the ability of the Problem Solver to prune the search space by identifying and focusing search only on highly likely partial solutions. Using a graph theoretic formulation of the ATMS we show that simply weighting the edges of the graph corresponds to a representation from which DS Belief functions can be computed. Such a graph theoretic formulation provides clear intuitions into the properties of symbolic ATMS-based DS algorithms and their relationship to the computation of network relia- bility [Agrawal and Barlow, 1984]. The complexity of the problems solved when using the ATMS to compute DS Belief functions is also defined. Provan 1115