Computation in Valuation Algebras

Many different formalisms for treating uncertainty or, more generally, information and knowledge, have a common underlying algebraic structure. The essential algebraic operations are combination, which corresponds to aggregation of knowledge, and marginalization, which corresponds to focusing of knowledge. This structure is called a valuation algebra. Besides managing uncertainty in expert systems, valuation algebras can also be used to to represent constraint satisfaction problems, propositional logic, and discrete optimization problems. This chapter presents an axiomatic approach to valuation algebras. Based on this algebraic structure, different inference mechanisms that use local computations are described. These include the fusion algorithm and, derived from it, the Shenoy-Shafer architecture. As a particular case, computation in idempotent valuation algebras, also called information algebras, is discussed. The additional notion of continuers is introduced and, based on it, two more computational architectures, the Lauritzen-Spiegelhalter and the HUGIN architecture, are presented. Finally, different models of valuation algebras are considered. These include probability functions, Dempster-Shafer belief functions, Spohnian disbelief functions, and possibility functions. As further examples, linear manifolds and systems of linear equations, convex polyhedra and linear inequalities, propositional logic and information systems, and discrete optimization are mentioned. ∗Research supported by grant No. 2100–042927.95 of the Swiss National Foundation for Research. 1 Appeared in: J. Kohlas and S. Moral (eds.), Algorithms for Uncertainty and Defeasible Reasoning, Handbook of Defeasible Reasoning and Uncertainty Managment Systems, Vol. 5, 2000, pp. 5--39, Kluwer Academic Publishers, Dordrecht