Algebras of Sets and Coherent Sets of Gambles

In a recent work we have shown how to construct an information algebra of coherent sets of gambles defined on general possibility spaces. Here we analyze the connection of such an algebra with the set algebra of subsets of the possibility space on which gambles are defined and the set algebra of sets of its atoms. Set algebras are particularly important information algebras since they are their prototypical structures. Furthermore, they are the algebraic counterparts of classical propositional logic. As a consequence, this paper also details how propositional logic is naturally embedded into the theory of imprecise probabilities.

Information algebras of coherent sets of gambles in general possibility spaces

In this paper, we show that coherent sets of gambles can be embedded into the algebraic structure of information algebra. This leads firstly, to a new perspective of the algebraic and logical structure of desirability and secondly, it connects desirability, hence imprecise probabilities, to other formalism in computer science sharing the same underlying structure. Both the domain-free and the labeled view of the information algebra of coherent sets of gambles are presented, considering general possibility spaces.

Information algebras of coherent sets of gambles

In this paper, we show that coherent sets of gambles can be embedded into the algebraic structure of information algebra. This leads firstly, to a new perspective of the algebraic and logical structure of desirability and secondly, it connects desirability, hence imprecise probabilities, to other formalism in computer science sharing the same underlying structure. Both the domain free and the labeled view of the information algebra of coherent sets of gambles are presented, considering a special case of possibility space.

Inverses, Conditionals and Compositional Operators in Separative Valuation Algebra

Compositional models were introduce by Jirousek and Shenoy in the general framework of valuation-based systems. They based their theory on an axiomatic system of valuations involving not only the operations of combination and marginalisation, but also of removal. They claimed that this systems covers besides the classical case of discrete probability distributions, also the cases of Gaussian densities and belief functions, and many other systems. Whereas their results on the compositional operator are correct, the axiomatic basis is not sufficient to cover the examples claimed above. We propose here a different axiomatic system of valuation algebras, which permits a rigorous mathematical theory of compositional operators in valuation-based systems and covers all the examples mentioned above. It extends the classical theory of inverses in semigroup theory and places thereby the present theory into its proper mathematical frame. Also this theory sheds light on the different structures of valuation-based systems, like regular algebras (represented by probability potentials), canncellative algebras (Gaussian potentials) and general separative algebras (density functions).

Embedding Default Logic in Propositional Argumentation Systems

In this paper we present a transformation of finite propositional default theories into so-called propositional argumentation systems. This transformation allows to characterize all notions of Reiter's default logic in the framework of argumentation systems. As a consequence, computing extensions, or determining wether a given formula belongs to one extension or all extensions can be answered without leaving the field of classical propositional logic. The transformation proposed is linear in the number of defaults.