We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in the case of the Pigeonhole principle are tight. The technique known so far to prove depth lower bounds for Stabbing Planes is a generalization of that used for the Cutting Planes proof system. In this work we introduce two new approaches to prove length/depth lower bounds in Stabbing Planes: one relying on Sperner's Theorem which works for the Pigeonhole principle and Tseitin contradictions over the complete graph; a second proving the lower bound for Tseitin contradictions over a grid graph, which uses a result on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz.

中文翻译：

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组合原则中刺伤平面的深度下限

我们证明了在刺平面中对数原理的对数深度下界，即称为鸽孔原理和Tseitin矛盾的组合原理的类别。深度下界是新的，它是通过给出几乎线性的长度下界而获得的，该下界不依赖于不等式的位大小，并且在Pigeonhole原理的情况下是紧密的。迄今为止，证明刺刀深度下界的技术是对“切割飞机”证明系统的推广。在这项工作中，我们引入了两种新方法来证明刺伤平面中的长度/深度下界：一种是基于Sperner定理（适用于Pigeonhole原理），另一种是Tseitin矛盾。一秒钟证明网格图上Tseitin矛盾的下界，