Generalization of the Dempster-Shafer Theory

The Dempster-Shafer theory gives a solid basis for reasoning appl icat ions characterized by uncertainty. A key feature of the theory is that proposi t ions are represented as subsets of a set which represents a hypothesis space. Th is power set along w i t h the set operations is a Boolean algebra. Can we generalize the theory to cover a rb i t ra ry Boolean algebras ? We show that the answer is yes. The theory then covers, for example, in f in i te sets. The pract ical advantages of general ization are tha t increased f lex ib i l i ty of representation is a l lowed and that the performance of evidence accumula t ion can be enhanced. In a previous paper we generalized the Dempster-Shafer or thogonal sum operat ion to support pract ical evidence pool ing. In the present paper we provide the theoret ical underp inn ing of tha t procedure, by systematical ly considering fami l iar evident ia l funct ions in t u rn . For each we present a "weaker f o r m " and we look at the relat ionships between these var iat ions of the funct ions. The relat ionships are not so st rong as for the conventional funct ions. However, when we specialize to the fami l ia r case of subsets, we do indeed get the wel lknown relat ionships.