Counting QBF solutions at level two

Abstract

We lift the problem of enumerative solution counting to quantified Boolean formulas (QBFs) at the second quantifier block. In contrast to the well-explored model counting problem for SAT (#SAT), where models are simply assignments to the Boolean variables of a formula, we are now dealing with tree (counter-)models reflecting the dependencies between the variables of the first and the second quantifier block. It turns out that enumerative counting on the second level does not give the complete solution count and more fine-grained view is necessary. We present a level-2 solution counting approach that works for true and false formulas. We implemented the presented approach in a counting tool exploiting state-of-the-art QBF solving technology. We present several kinds of benchmarks for testing our implementation and show that even with this very basic approach of solution enumeration the solution counts of challenging benchmarks can be found.