The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients SP* can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.

中文翻译：

---

论切分法的权力与局限

引入了Stabbing Planes证明系统来对在实际的混合整数规划求解器中进行的推理建模。作为证明系统，它足够强大，可以模拟“切割平面”并驳斥Tseitin公式（某些线性方程组mod 2无法满足的系统），这是许多代数证明系统的典型示例。在最近（且令人惊讶）的结果中，Dadush和Tiwari表明，对Tseitin公式的这些简短反驳可以转化为准多项式大小和深度Cuting Planes证明，从而驳斥了长期以来的推测。此翻译提出了几个有趣的问题。首先，是否可以通过“切割平面”有效地模拟所有“刺刀平面”证明。这将允许在“切割平面”系统上进行的大量分析被应用到实际的混合整数编程求解器中。其次，这些证明的准多项式深度是否是“切割平面”所固有的。在本文中，我们在回答这两个问题上均取得了进展。首先，我们证明可以将任何具有有限系数SP *的刺刀平面证明转换为切割平面。由于切割平面的已知下限，这将在SP *上建立第一个指数下限。使用此转换，我们扩展了Dadush和Tiwari的结果，以显示“切割平面”对有限域上任何不满足要求的线性方程组都有简短的反驳。像达杜什（Dadush）和蒂瓦里（Tiwari）的切割飞机样机一样，我们的驳斥也引起了深度上的准多项式爆炸，我们猜想这是内在的。作为对此猜想的一个步骤，我们开发了一种新的几何技术来证明“切割平面”证明的深度的下界。这使我们能够在Tseitin公式的语义切割平面证明的深度上建立第一个下界。