Dempster-Shafer theory with indiscernible states and observations

An internal state s of a system is compatible with an empirical observation value x, if the possibility that s is the actual internal state of the system when x was observed cannot be logically avoided on the grounds of those laws and rules governing the system and its environment which are known to the investigator. The sets of states ana the set of empirical values are divided into equivalence classes by corresponding indiscernibility relations So that two states (two empirical values, resp.) in the same class are indistinguishable from each other and instead of a compatibility relation between particular states and empirical values only a compatibility relation between classes of states and classes of empirical values is given. It will be shown how to define, under such conditions, an approximation of the belief function definable when the original compatibility relation is at hand.