Backjumping for Quantified Boolean Logic satisfiability

Abstract

The implementation of effective reasoning tools for deciding the satisfiability of Quantified\ud Boolean Formulas (QBFs) is an important research issue in Artificial Intelligence. Many decision\ud procedures have been proposed in the last few years, most of them based on the Davis, Logemann,\ud Loveland procedure (DLL) for propositional satisfiability (SAT). In this paper we show how it is\ud possible to extend the conflict-directed backjumping schema for SAT to the satisfiability of QBFs:\ud When applicable, conflict-directed backjumping allows search to skip over existentially quantified\ud literals while backtracking. We introduce solution-directed backjumping, which allows the same\ud behavior for universally quantified literals. We show how it is possible to incorporate both conflict-\ud directed and solution-directed backjumping in a DLL-based decision procedure for satisfiability of\ud QBFs. We also implement and test the procedure: The experimental analysis shows that, because of\ud backjumping, significant speed-ups can be obtained.\ud Summing up: We present the first algorithm that applies conflict and solution directed backjumping\ud to QBF, and demonstrate the performance of this algorithm via an empirical study

QBF Reasoning on Real-World Instances

“…Q n x n Φ, where Φ = Ψ ∨ Θ, Ψ is a conjunction of clauses, and Θ is a disjunction of terms. Even assuming that Θ is initially ⊥, formulas of this kind arise during the learning process as shown in [10]. SOLVE returns TRUE if the input QBF is satisfiable and FALSE otherwise.…”

“…Running a profiler on a DPLL-based QBF solver like QUBE++ confirms this result: on all the instances that we have tried, lookahead always amounted to more than 70% of the total runtime. The need for a fast lookahead procedure is accentuated by the use of smart lookback techniques such as learning [10], where the solver augments the initial set of constraints with other ones discovered during the search. With learning, possibly large amounts of lengthy constraints have to be processed quickly, otherwise the overhead will dwarf the benefits of learning itself.…”

“…This is indeed the case when the conflicts/solutions are caused by some choice done way up in the search tree. To solve this problem [13] introduced conflict backjumping and solution backjumping (CBJ, SBJ) for QBFs. Using CBJ (resp.…”

“…For details regarding this process, see [10]. With CBJ/SBJ reasons are discarded while backtracking over the nodes that caused the conflict or the solution, and this may lead to a subsequent redundant exploration.…”

“…The techniques are: (i) efficient detection of unit and monotone literals using lazy data structures as in [9]; (ii) learning, as introduced in [10], with improvements that generalize the techniques first used in SAT by GRASP [11]; (iii) a branching strategy that exploits information gleaned from the input formula initially, and leverages the information extracted in the learning phase. Using formal verification and planning benchmarks from the QBF evaluation [8], we conducted an experimental campaign to highlight the relative efficiency of the above techniques, and of their combination: the results show that each single technique contributes to the effectiveness of QUBE++ and that the resulting system is a clear improvement over current state-of-the-art QBF solvers.…”