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Many knowledge-based expert systems employ numerical schemes to represent evidence, rate competing hypoth eses, and guide search through the domains problem space. This paper has two objectives: first, to introduce one such scheme developed by Arthur Dempster and Glen Shafer, to a wider audience; second, to present results that can reduce the compu tationtime complexity from exponential to linear allowing this scheme to be implemented in many more systems. In order to enjoy this reduction, some assumptions about the structure of the type of evidence represented and combined must be made The assumption made here is that each piece of the evidence either confirms or denies a single proposition rather than a disjunction For any domain in which the assumption is justified the savings are available.