Uncertain information: Random variables in graded semilattices

Random sets can be considered as random variables with values in a Boolean algebra, in particular in a field of sets. Many properties of random sets, in particular those relative to belief functions, can also be obtained by relaxing the algebraic structure of the domain. In fact, functions monotone to order infinity, like Choquet capacities, can be defined on semilattices. In this paper random variables with values in some kind of graded semilattices are studied. It is shown that this algebraic structure models important operations regarding information. It turns out that random variables in this algebra form themselves an algebra of the same kind. Their probability distribution corresponds to functions monotone of order infinity or to belief functions in the sense of Dempster-Shafer theory of evidence. This paper proposes therefore a natural generalization of evidence theory to a general structure related to probabilistic argumentation systems. Those systems have many interesting models like probabilistic assumption-based reasoning with different kind of logics, systems of linear equations and inequalities with stochastic disturbances, etc. The theory presented leads thus to an interesting and novel way of combining logic and probability. (C) 2007 Elsevier Inc. All rights reserved.