Using propositional logic to compute probabilities in multistate systems

Abstract

In this paper we start with presentation of a general language for representing subsets of a cartesian product of finite sets. This language is used to represent the set of diagnoses in the general theory of model-based diagnosis presented by Reiter (R. Reiter, Artif. Intell. 32 (1987) 57–95) when the components have more than two possible operating modes. After having established some general results about Boolean algebras, which turn out to be the appropriate mathematical structure to define the language precisely, they are applied in the special case of propositional logic and product spaces, thereby defining a language for the description of events in product spaces. Then we present three different symbloic methods for computing the probability of a formula in the language without explicitly constructing the corresponding system states. The first two methods are based on the algorithm of Abraham (J.A. Abraham, IEEE Transactions on Reliablity 28 (1979) 58–61) whereas the last method is based on the algorithm of Bertschy-Monney (R. Bertschy, P.A. Monney, J. Comput. Appl. Math. 76 (1996) 55–76). All these methods transform the original formula into an equivalent formula for which it is very simple to compute the probability. The problem of computing the probability of a logical formula 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.