Computing the Probability of Formulas Representing Events in Product Spaces

Abstract

In this paper we present a general language for representing subsets of a product of finite sets. For example this type of language is used for representing the set of diagnoses in the general theory of model-based diagnosis presented by Reiter [8]. In this paper it is shown that the mathematical structure of a Boolean algebra is the appropriate concept to define and describe the language. After having established some general results about Boolean algebras, these results are applied in the special case of product spaces, thereby defining a language for the description of events in product spaces. Then we give two algorithms for computing the probability of a formula in the language. This problem appears for example in model-based diagnostics when we need to compute the conditional probability of a diagnosis given the observations made on the system.