Numerical Considerations in Weighted Model Counting

Abstract

Weighted model counting computes the sum of the rational-valued weights associated with the satisfying assignments for a Boolean formula, where the weight of an assignment is given by the product of the weights assigned to the positive and negated variables comprising the assignment. Weighted model counting finds applications across a variety of domains including probabilistic reasoning and quantitative risk assessment. Most weighted model counting programs operate by (explicitly or implicitly) converting the input formula into a form that enables arithmetic evaluation, using multiplication for conjunctions and addition for disjunctions. Performing this evaluation using floating-point arithmetic can yield inaccurate results, and it cannot quantify the level of precision achieved. Computing with rational arithmetic gives exact results, but it is costly in both time and space. This paper describes how to combine multiple numeric representations to efficiently compute weighted model counts that are guaranteed to achieve a user-specified precision. When all weights are nonnegative, we prove that the precision loss of arithmetic evaluation using floating-point arithmetic can be tightly bounded. We show that supplementing a standard IEEE double-precision representation with a separate 64-bit exponent, a format we call extended-range double (ERD), avoids the underflow and overflow issues commonly encountered in weighted model counting. For problems with mixed negative and positive weights, we show that a combination of interval floating-point arithmetic and rational arithmetic can achieve the twin goals of efficiency and guaranteed precision. For our evaluations, we have devised especially challenging formulas and weight assignments, demonstrating the robustness of our approach.