On the Representation of Nonmontonic Relations in the Theory of Evidence

A Dempster-Shafer belief structure provides a mechanism for representing uncertain knowledge about a variable. A compatibility relation provides a structure for obtaining information about one variable based upon our knowledge about a second variable. An inference scheme in the theory of evidence involves the use of a belief structure on one variable, called the antecedent, and a compatibility relationship to infer a belief structure on the second variable, called the consequent. The concept of monotonicity in this situation is related to change in the specificity of the consequent belief structure as the antecedent belief structure becomes more specific. We show that the usual compatibility relations, type 1, are always monotonic. We introduce type II compatibility relations and show that a special class of these, which we call irregular, are needed to represent nonmonotonic relations between variables. We discuss a special class of nonmonotonic relations called default relations.