Abstract
Resolution proof systems for quantified Boolean formulas (QBFs) provide a formal model for studying the limitations of state-of-the-art search-based QBF solvers that use these systems to generate proofs. We study a combination of two proof systems supported by the solver DepQBF: Q-resolution with generalized universal reduction according to a dependency scheme and long distance Q-resolution. We show that the resulting proof system—which we call long-distance Q(D)-resolution—is sound for the reflexive resolution-path dependency scheme. In fact, we prove that it admits strategy extraction in polynomial time. This comes as an application of a general result, by which we identify a whole class of dependency schemes for which long-distance Q(D)-resolution admits polynomial-time strategy extraction. As a special case, we obtain soundness and polynomial-time strategy extraction for long distance Q(D)-resolution with the standard dependency scheme. We further show that search-based QBF solvers using a dependency scheme D and learning with long-distance Q-resolution generate long-distance Q(D)-resolution proofs. The above soundness results thus translate to partial soundness results for such solvers: they declare an input QBF to be false only if it is indeed false. Finally, we report on experiments with a configuration of DepQBF that uses the standard dependency scheme and learning based on long-distance Q-resolution.
Highlights
Quantified Boolean formulas (QBFs) offer succinct encodings for problems from domains such as formal verification, synthesis, and planning [5,13,16,30,38,43]
Since every polynomial-time algorithm can be implemented by a family of polynomially-sized circuits, and because these circuits can even be computed in polynomial time [1, p. 109], it follows that Long-distance Q-resolution with ordinary ∀/∃-reduction (LDQ)(D) admits polynomial-time strategy extraction when D is normal
We compared the performance of DepQBF in four configurations,8 each using a different proof system for constraint learning: 1. Long-distance Q-resolution with ∀/∃-reduction according to Dstd (LDQD). 2
Summary
Quantified Boolean formulas (QBFs) offer succinct encodings for problems from domains such as formal verification, synthesis, and planning [5,13,16,30,38,43]. Dependency schemes can sometimes identify pairs of variables as independent, allowing the solver to assign them in any order This gives decision heuristics more freedom and results in increased performance [10]. (b) For applications in verification and synthesis, it is not enough for solvers to decide whether an input QBF is true or false—they have to generate a certificate Such certificates can be efficiently constructed from Q-resolution [2] and even long-distance Q-resolution proofs [3]. We define LDQ(D)-resolution as consisting of long-distance Q-resolution with a dependency scheme D, and show that a search-based QBF solver using dependency scheme D and learning based on long-distance Q-resolution generates an LDQ(D)-resolution refutation whenever it declares an input QBF to be false This allows us to partially address (a) by showing that longdistance Q-resolution combined with the reflexive resolution-path dependency scheme [42] is sound. These experiments show that performance with learning based on LDQ(D)-resolution is on par with and—in some cases—even slightly better than the performance of DepQBF with other configurations of constraint learning
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