In this paper we create a bridge between desirability and information algebras: we show how coherent sets of gambles, as well as coherent lower previsions, induce such structures. This allows us to enforce the view of such imprecise-probability objects as algebraic and logical structures; moreover, it enforces the interpretation of probability as information, and gives tools to manipulate them as such.

In recent works, we have shown how to construct an information algebra of coherent sets of gambles, considering firstly a particular model to represent questions, called the multivariate model, and then generalizing it. Here we further extend the construction made to the highest level of generality, setting up an associated information algebra of coherent lower previsions, analyzing the connection of both the information algebras constructed with an instance of set algebras and, finally, establishing and inspecting a version of the marginal problem in this framework. Set algebras are particularly important information algebras since they are their prototypical structures. They also represent the algebraic counterparts of classical propositional logic. As a consequence, this paper details as well how propositional logic is naturally embedded into the theory of imprecise probabilities.

In this paper, we study the relationship between ordering structures on semirings and semiring-induced valuation algebras. We show that a semiring-induced valuation algebra is a complete (resp. continuous) lattice if and only if the semiring is complete (resp. continuous) lattice with respect to the reverse order relation on semirings. Furthermore, a semiring-induced information algebra is compact, if the dual of the semiring is an algebraic lattice.

Generally, technical systems are not guaranteed to work correctly. They are more or less reliable. One main problem for technical systems is the computation of the reliability of a system as studied in reliability theory. A second main problem for technical systems is the problem of diagnostic, i.e., explanations why something does not work. In fact, these problems are, in some sense, dual to each other. In this article, we will use the concept of probabilistic argumentation systems (PASs) for modeling the system description as well as observation and specifications of behavior in one common framework. We show that PASs provide a framework that allows us to formulate and solve reliability and diagnostic problems, and all concepts for these two problems can be defined clearly therein. Using PASs, reliability and diagnostics can be considered as dual problems. PASs offer one common strategy for computing answers to the questions in these different situations. © 2003 Wiley Periodicals, Inc.

Resolution is an often used method for deduction in propositional logic. Here a proper organization of deduction is proposed which avoids redundant computations. It is based on a generic framework of decompositions and local computations as introduced by Shenoy and Shafer. The system contains the two basic operations with information, namely marginalization (or projection) and combination; the latter being an idempotent operation in the present case. The theory permits the conception of an architecture of distributed computing. As an important application assumption-based reasoning is discussed.