Belief functions on distributive lattices

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 employ Birkhoffʼs representation theorem for finite distributive lattices to 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 use this more general theory to provide a framework for reasoning about belief functions in a deductive approach on non-classical formalisms which assume a setting of distributive lattices. As an illustration of this approach, we investigate the theory of belief functions for a simple epistemic logic the first-degree-entailment fragment of relevance logic R by providing an axiomatization for reasoning about belief functions for this logic and by showing 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.