We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C . Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC 0 [ p ] circuits. In contrast, any non-trivial lower bound for propositional AC 0 [ p ]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.

中文翻译：

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量化布尔逻辑的弗雷格系统

我们为量化布尔公式 (QBF) 定义和研究 Frege 系统。对于这些新的证明系统，我们开发了一种下限技术，可以直接提升电路类的电路下限C到 QBF Frege 系统，其线路从C. 这种从电路到证明复杂性下限的直接转移通常被假设用于命题系统，但在这项工作之前，尚未正式为任何证明系统建立如此普遍性。这导致 QBF Frege 的受限版本有很强的下限，特别是使用 AC 运行的 QBF Frege 系统的指数下限0[p] 电路。相反，命题 AC 的任何非平凡下界0[p]-弗雷格构成了一个主要的开放性问题。将这些下界改进为不受限制的 QBF Frege 与这电路复杂性和命题证明复杂性的主要问题。特别是，证明使用任意 P/poly 电路运行的 QBF Frege 系统的下限等效于显示 P/poly 或命题扩展 Frege（使用 P/poly 电路运行）的下限。我们还将我们的新 QBF Frege 系统与 QBF 的标准连续演算进行比较，并建立与直觉有界算术的对应关系。