Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture practical solving techniques in an elegant way. The most popular of these proof systems is likely DRAT, which is considered the de-facto standard in SAT solving. Moreover, just recently, the proof system DPR has been proposed as a generalization of DRAT that allows for short proofs without the need of new variables. Since every extended-resolution proof can be regarded as a DRAT proof and since every DRAT proof is also a DPR proof, it was clear that both DRAT and DPR generalize extended resolution. In this paper, we show that—from the viewpoint of proof complexity—these two systems are no stronger than extended resolution. We do so by showing that (1) extended resolution polynomially simulates DRAT and (2) DRAT polynomially simulates DPR. We implemented our simulations as proof-transformation tools and evaluated them to observe their behavior in practice. Finally, as a side note, we show how Kullmann’s proof system based on blocked clauses (another generalization of extended resolution) is related to the other systems.

中文翻译：

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模拟具有扩展分辨率的强实用证明系统

命题逻辑的证明系统为确定逻辑公式的可满足性状态的决策程序提供了基础。虽然著名的扩展分辨率证明系统（由 Tseitin 在 60 年代引入）允许证明的紧凑表示，但现代 SAT 求解器（即，用于决定命题逻辑的工具）基于不同的证明系统，这些证明系统在一种优雅的方式。这些证明系统中最流行的可能是 DRAT，它被认为是 SAT 求解中的事实上的标准。此外，就在最近，证明系统 DPR 被提议作为 DRAT 的泛化，它允许在不需要新变量的情况下进行简短的证明。由于每一个扩展分辨率证明都可以看作是一个 DRAT 证明，因为每一个 DRAT 证明也是一个 DPR 证明，很明显，DRAT 和 DPR 都概括了扩展分辨率。在本文中，我们表明——从证明复杂性的角度来看——这两个系统并不比扩展分辨率强。我们通过证明 (1) 扩展分辨率多项式模拟 DRAT 和 (2) DRAT 多项式模拟 DPR 来做到这一点。我们将我们的模拟作为证明转换工具实施并评估它们以观察它们在实践中的行为。最后，作为旁注，我们展示了基于阻塞子句（扩展解析的另一种概括）的库尔曼证明系统如何与其他系统相关。我们通过证明 (1) 扩展分辨率多项式模拟 DRAT 和 (2) DRAT 多项式模拟 DPR 来做到这一点。我们将我们的模拟作为证明转换工具实施并评估它们以观察它们在实践中的行为。最后，作为旁注，我们展示了基于阻塞子句（扩展解析的另一种概括）的库尔曼证明系统如何与其他系统相关。我们通过证明 (1) 扩展分辨率多项式模拟 DRAT 和 (2) DRAT 多项式模拟 DPR 来做到这一点。我们将我们的模拟作为证明转换工具实施并评估它们以观察它们在实践中的行为。最后，作为旁注，我们展示了基于阻塞子句（扩展解析的另一种概括）的库尔曼证明系统如何与其他系统相关。