Soundness of Q-resolution with dependency schemes

Q-resolution and Q-term resolution are proof systems for quantified Boolean formulas (QBFs). We introduce generalizations of these proof systems named Q(D)Q(D)-resolution and Q(D)Q(D)-term resolution. Q(D)Q(D)-resolution and Q(D)Q(D)-term resolution are parameterized by a dependency scheme D   and use more powerful ∀-reduction and ∃-reduction rules, respectively. We show soundness of these systems for particular dependency schemes: we prove (1) soundness of Q(D)Q(D)-resolution parameterized by the reflexive resolution-path dependency scheme, and (2) soundness of Q(D)Q(D)-term resolution parameterized by the resolution-path dependency scheme. These results entail soundness of the proof systems used for certificate generation in the state-of-the-art solver DepQBF.