Algorithmic Perspective on Toda's Theorem

Abstract: Toda's Theorem is a fundamental result in computational complexity theory, whose proof relies on a reduction from a QBF problem with a constant number of quantifiers to a model counting problem. While this reduction, henceforth called Toda's reduction, is of a purely theoretical nature, the recent progress of model counting tools raises the question of whether the reduction can be utilized to an efficient algorithm for solving QBF. In this work, we address this question by looking at Toda's reduction from an algorithmic perspective. We first convert the reduction into a concrete algorithm that given a QBF formula and a probability measure, produces the correct result with a confidence level corresponding to the given measure. Beyond obtaining a naive prototype, our algorithm and the analysis that follows shed light on the fine details of the reduction that are so far left elusive. Then, we improve this prototype through various theoretical and algorithmic refinements. While our results show a significant progress over the naive prototype, they also provide a clearer understanding of the remaining challenges in turning Toda's reduction into a competitive solver.