Expansion-based QBF solving versus Q-resolution

Mikoláš Janota, Joao Marques-Silva
2015 Theoretical Computer Science  
This article introduces and studies a proof system ∀Exp+Res that enables us to refute quantified Boolean formulas (QBFs). The system ∀Exp+Res operates in two stages: it expands all universal variables through conjunctions and refutes the result by propositional resolution. This approach contrasts with the Q-resolution calculus, which enables refuting QBFs by rules similar to propositional resolution. In practice, Q-resolution enables producing proofs from conflict-driven DPLL-based QBF solvers.
more » ... The system ∀Exp+Res can on the other hand certify certain expansion-based solvers. So a natural question is to ask which of the systems, Q-resolution and ∀Exp+Res, is more powerful? The article gives several partial responses to this question. On the positive side, we show that ∀Exp+Res can p-simulate tree Q-resolution. On the negative side, we show that ∀Exp+Res does not p-simulate unrestricted Q-resolution. In the favor of ∀Exp+Res we show that ∀Exp+Res is more powerful than a certain fragment of Q-resolution, which is important for DPLL-based QBF solving. formula. Consequently, QBFs and QBF solving are fundamentally different from propositional formulas and satisfiability solving (SAT). While SAT is NP-complete, deciding whether a given QBF is true or not is PSPACEcomplete. Further complexity differences exist. For instance, QBF remains PSPACE-complete even for bounded tree-width [1] whereas SAT becomes tractable in such case [2]. This article 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 [3, 4, 5, 6]. Proof systems for QBF come in different styles and flavors. Krajíček and Pudlák propose a Gentzen-style calculus KP for QBF [4]. Büning et al. propose a refutation calculus Q-resolution [5], an extension of propositional resolution. Giunchiglia et al. extend the work of Büning et al. into term resolution for proofs of true formulas [6] . Certain separation results were shown between KP and Q-resolution recently by Egly [7]. Van Gelder introduces a generalization of Q-resolution, called QU-resolution [8]. While many QBF solvers are based on the DPLL procedure and conflictdriven learning [9, 10, 11, 12, 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 [14, 15, 16, 17] . 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 [18, 19] . This practical importance of expansion motivates the theoretical study carried out in this article. We define a proof system ∀Exp+Res, which eliminates universal quantification from the given false QBF and then applies propositional resolution to refute the remainder. We show several results on how ∀Exp+Res compares to Q-resolution. The article is organized as follows. Section 2 introduces concepts and notation used throughout the paper. Section 3 introduces the proof system ∀Exp+Res. Section 4 shows that tree Q-resolution is polynomially simulated by ∀Exp+Res. Section 5 investigates the relation in the opposite direction and shows that Q-resolution polynomially simulates a certain fragment of ∀Exp+Res. Section 6 shows that the result shown in Section 4 "cannot be improved", i.e. that unrestricted Q-resolution is not polynomially simulated by ∀Exp+Res. Section 7 shows a somewhat weaker negative result in the opposite direction; it shows that a certain restriction of Q-resolution does not
doi:10.1016/j.tcs.2015.01.048 fatcat:ejpfeirhvfb3fghyodkibwxuvu