The Dempster-Shafer theory of belief functions is an important approach to deal with uncertainty in AI. In the theory, belief functions are defined on Boolean algebras of events. In many applications of belief functions in real world problems, however, the objects that we manipulate is no more a Boolean algebra but a distributive lattice. In this paper, we extend the Dempster-Shafer theory to the setting of distributive lattices, which has a mathematical theory as attractive as in that of Boolean algebras. Moreover, we apply this more general theory to a simple epistemic logic the first-degree-entailment fragment of relevance logic R, provide a sound and complete axiomatization for reasoning about belief functions for this logic and show that the complexity of the satisfiability problem of a belief formula with respect to the class of the corresponding Dempster-Shafer structures is NP-complete.
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