Efficiently Representing Existential Dependency Sets for Expansion-based QBF Solvers

Given a quantified boolean formula (QBF) in prenex conjunctive normal form (PCNF), we consider the problem of identifying variable dependencies. In related work, a formal definition of dependencies has been suggested based on quantifier prefix reordering: two variables are independent if swapping them in the prefix does not change satisfiability of the formula. Instead of the general case, we focus on the sets of depending existential variables for all universal variables. This is relevant particularly for expansion-based QBF solvers. We present an approach for efficiently computing existential dependency sets by means of a directed connection relation over variables and demonstrate how this relation can be compactly represented as a tree using a union-find data structure. Experimental results show the effectiveness of our approach.