The power of negative reasoning
The power of negative reasoning
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消极推理的力量
DOI：
10.4230/lipics.ccc.2021.40
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Dmitry Sokolov
中科院分区：
文献类型：
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作者：
Susanna F. de Rezende;M. Lauria;Jakob Nordström;Dmitry Sokolov
自20世纪90年代末以来，半代数证明系统在证明复杂性方面得到了广泛的研究，以了解Gröbner基计算、线性和半定规划谱以及其他方法的能力。这样的证明系统只用问题的原始变量交替定义，并使用正负文字的特殊形式变量来定义，但似乎还没有研究这些不同的定义如何影响证明系统的能力。对于Nullstellensatz、多项式演算、Sherali-Adams和平方和，我们证明了添加负文字的形式变量会使证明系统相对于证明中的项数呈指数级增强。CNF公式见证了这些分离，这些公式易于解析，该公式确立了多项式演算、Sherali-Adams和平方和在没有访问负文字变量的情况下无法有效地模拟解析。
Semialgebraic proof systems have been studied extensively in proof complexity since the late 1990s to understand the power of Gröbner basis computations, linear and semidefinite programming hierarchies, and other methods. Such proof systems are defined alternately with only the original variables of the problem and with special formal variables for positive and negative literals, but there seems to have been no study how these different definitions affect the power of the proof systems. We show for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs. These separations are witnessed by CNF formulas that are easy for resolution, which establishes that polynomial calculus, Sherali-Adams, and sums-of-squares cannot efficiently simulate resolution without having access to variables for negative literals.