This paper explores algorithms for processing probabilistic and deterministic information when the former is represented as a belief network and the latter as a set of boolean clauses. The motivating tasks are 1. evaluating beliefs networks having a large number of deterministic relationships and2. evaluating probabilities of complex boolean querie over a belief network. We propose a parameterized family of variable elimination algorithms that exploit both types of information, and that allows varying levels of constraint propagation inferences. The complexity of the scheme is controlled by the induced-width of the graph {em augmented} by the dependencies introduced by the boolean constraints. Preliminary empirical evaluation demonstrate the effect of constraint propagation on probabilistic computation.

In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of dependency relationships in their structure. Our method effectively maps such a dense problem to a sparser one which is in some sense "closest". Exact methods can be run on the sparser problem to derive bounds on the original answer, which can be quite sharp. We present empirical results demonstrating that our method works well on the tasks of belief inference and finding the probability of the most probable explanation in belief networks, and finding the cost of the solution that violates the smallest number of constraints in constraint satisfaction problems. On one large CPCS network, for example, we were able to calculate upper and lower bounds on the conditional probability of a variable, given evidence, that were almost identical in the average case.