QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies.

中文翻译：

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了解 QBF CDCL 求解器和 QBF 分辨率的相对强度

实现 QCDCL 范式的 QBF 求解器是强大的算法，可成功解决许多计算复杂的应用程序。然而，我们对这些 QCDCL 求解器的强度和局限性的理论理解非常有限。在本文中，我们建议将 QCDCL 求解器正式建模为证明系统。我们定义了可用于决策启发式和单元传播的不同策略，并产生了许多健全且完整的 QBF 证明系统（以及新的 QCDCL 算法）。关于实际 QCDCL 求解中使用的标准策略，我们表明相应的 QCDCL 证明系统与文献中使用的经典 QBF 分辨率系统 Q-分辨率是不可比的（通过指数分离）。这与已知 CDCL 和分辨率为 p 等价的命题设置形成鲜明对比。这提出了标准 QCDCL 难以使用哪些公式的问题，因为 Q 分辨率下限不一定适用于我们在此处显示的 QCDCL。为了回答这个问题，我们证明了 QCDCL 的几个下界，包括一大类随机 QBF 的指数下界。我们还引入了经典 QCDCL 中使用的决策启发式的强化，它不一定按照前缀的顺序来决定变量，但仍然允许学习断言子句。我们表明，使用此决策策略，QCDCL 在某些公式上的速度可以呈指数级增长。我们进一步展示了 p 等效于 Q 分辨率的 QCDCL 证明系统。与经典的 QCDCL 相比，