Polynomial Calculus Space and Resolution Width

Nicola Galesi, L. Kolodziejczyk, Neil Thapen
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引用次数: 7

Abstract

We show that if a k-CNF requires width w to refute in resolution, then it requires space square root of √ω to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is the first analogue, in polynomial calculus, of Atserias and Dalmau's result lower-bounding clause space in resolution by resolution width. As a by-product of our new approach to space lower bounds we give a simple proof of Bonacina's recent result that total space in resolution (the total number of variable occurrences that must be kept in memory) is lower-bounded by the width squared. As corollaries of the main result we obtain some new lower bounds on the PCR space needed to refute specific formulas, as well as partial answers to some open problems about relations between space, size, and degree for polynomial calculus.
多项式微积分空间和分辨率宽度
我们证明,如果k-CNF在分辨率上需要宽度w来反驳,那么它在多项式微积分中需要√ω的空间来反驳,其中多项式微积分反驳的空间是在进行证明时必须保留在内存中的单项式的数量。这是多项式演算中第一个模拟Atserias和Dalmau在分辨率宽度上的下限子句空间的结果。作为我们新的空间下界方法的副产品,我们对Bonacina最近的结果给出了一个简单的证明,即分辨率的总空间(必须保存在内存中的变量出现的总数)的下界是宽度的平方。作为主要结果的推论,我们得到了一些新的PCR空间下界来反驳特定的公式,以及多项式微积分中一些关于空间、大小和度之间关系的开放性问题的部分答案。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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