On Propositional QBF Expansions and Q-Resolution

Over the years, proof systems for propositional satisfiability (SAT) have been extensively studied. Recently, proof systems for quantified Boolean formulas (QBFs) have also been gaining attention. Q-resolution is a calculus enabling producing proofs from DPLL-based QBF solvers. While DPLL has become a dominating technique for SAT, QBF has been tackled by other complementary and competitive approaches. One of these approaches is based on expanding variables until the formula contains only one type of quantifier; upon which a SAT solver is invoked. This approach motivates the theoretical analysis carried out in this paper. We focus on a two phase proof system, which expands the formula in the first phase and applies propositional resolution in the second. Fragments of this proof system are defined and compared to Q-resolution. This paper follows the line of research on proof systems for propositional and quantified Boolean formulas (QBFs). This research is motivated by complexity theory and more recently by the objective to develop and certify QBF solvers [11,18,8,14]. Proof systems for QBF come in different styles and flavors. Krajı́ček and Pudlák propose a Genzen-style calculus KP for QBF [18]. Büning et al. propose a refutation calculus Q-resolution [8], an extension of propositional resolution. Giunchiglia et al. extend the work of Büning et al. into term resolution for proofs of true formulas [14] . Certain separation results were shown between KP and Q-resolution recently by Egly [12]. While many QBF solvers are based on the DPLL procedure [21,9,23,20,13], other solvers tackle the given formula by expanding out quantifiers until a single quantifier type is left. At that point, this formula is handed to a SAT solver [1,4,19,15]. Experimental results show that expansion-based QBF solvers can outperform DPLL-based solvers on a number of families of practical instances. Also, expansion can be used in QBF preprocessing [6,5]. This practical importance of expansion motivates the study carried out in this paper. We define a proof system ∀Exp+Res, which eliminates universal quantification from the given false formula and then applies propositional resolution to refute the remainder. We show that ∀Exp+Res can p-simulate tree Q-resolution refutations. Conversely, we show that Q-resolution can p-simulate ∀Exp+Res refutations under certain restrictions on the propositional resolution part of the proofs.