Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

翻译：穿透平面（也称为分支切割）是一种近期引入的证明系统，其非正式地说，通过分支于整数线性不等式而非单个变量来扩展DPLL方法。目前用于证明穿透平面大小和深度下界的技术是切割平面证明系统所用方法的推广。大小下界通过单调电路论证建立，而深度下界则通过通信复杂性与保护技术获得。这些下界适用于组合陈述的提升版本。切割平面的秩下界也通过称为保护引理的几何论证得到。本文引入两种新的几何方法来证明任意公式在穿透平面中的大小/深度下界：（1）基于Sperner定理的反链方法，以及（2）基于布尔立方体被线性多项式本质覆盖的覆盖方法，后者又依赖于Alon的组合零点定理。我们展示了这些方法在鸽巢原理、Tseitin矛盾式和线性序原理等组合原理类中的应用。通过第一种方法，我们证明了鸽巢原理的接近线性大小下界和最优对数深度下界，以及完全图上Tseitin矛盾式和线性序原理的类似下界。通过覆盖方法，我们获得了网格图上Tseitin矛盾式的穿透平面证明的超线性大小下界和对数深度下界。