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arXiv 2101.08600cs.CC
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关于布尔函数的次数与块灵敏度的分离

On Separation between the Degree of a Boolean Function and the Block Sensitivity

Nikolay V. Proskurin

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16 pages in total (12 for article + 4 for references and appendix)
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摘要：

在本文中，我们研究了两个复杂性度量之间的分离：布尔函数作为实数上多项式的次数和它的块敏感性。我们表明，这两种措施之间的分离可以提高从$ d^2(f) \geq bs(f) $，建立塔尔，$ d^2(f) \geq (\sqrt{10} - 2)bs(f) $。作为推论，我们证明了其他一些复杂性度量之间的分离也不是紧密的，例如，我们可以将Huang最近的敏感性猜想结果改进为$s^4(f) \geq (\sqrt{10} - 2)bs(f)$。我们的技术是基于Nisan和Szegedy的论文，包括更详细的对称化多项式分析。在我们的下一个结果中，我们在布尔函数的近似度和它的块敏感性之间的分离中显示了相同类型的改进：我们显示了$deg_{1/3}^2(f) \geq \sqrt{6/101} bs(f)$，并改进了Nisan和Szegedy的先前结果$ deg_{1/3}(f) \geq \sqrt{bs(f)/6} $。此外，我们构造了一个例子，表明在下限和已知的上限之间的差距是小于$0.2$的常数。在我们的最后一个结果中，我们研究了10个变量的4次完全敏感函数的性质，它的存在将导致这两个措施之间的最大已知差距的改善。我们证明，有唯一的一元多项式，可以通过使用插值和线性规划技术相结合的对称化这个功能。

英文摘要：

In this paper we study the separation between two complexity measures: the degree of a Boolean function as a polynomial over the reals and its block sensitivity. We show that separation between these two measures can be improved from $ d^2(f) \geq bs(f) $, established by Tal, to $ d^2(f) \geq (\sqrt{10} - 2)bs(f) $. As a corollary, we show that separations between some other complexity measures are not tight as well, for instance, we can improve recent sensitivity conjecture result by Huang to $s^4(f) \geq (\sqrt{10} - 2)bs(f)$. Our techniques are based on paper by Nisan and Szegedy and include more detailed analysis of a symmetrization polynomial. In our next result we show the same type of improvement in the separation between the approximate degree of a Boolean function and its block sensitivity: we show that $deg_{1/3}^2(f) \geq \sqrt{6/101} bs(f)$ and improve the previous result by Nisan and Szegedy $ deg_{1/3}(f) \geq \sqrt{bs(f)/6} $. In addition, we construct an example which shows that gap between constants in the lower bound and in the known upper bound is less than $0.2$. In our last result we study the properties of conjectured fully sensitive function on 10 variables of degree 4, existence of which would lead to improvement of the biggest known gap between these two measures. We prove that there is the only univariate polynomial that can be achieved by symmetrization of this function by using the combination of interpolation and linear programming techniques.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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