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On the probabilistic degree of OR over the reals
Random Structures and Algorithms ( IF 0.9 ) Pub Date : 2021-01-19 , DOI: 10.1002/rsa.20991
Siddharth Bhandari 1 , Prahladh Harsha 1 , Tulasimohan Molli 1 , Srikanth Srinivasan 2
Affiliation  

We study the probabilistic degree over urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0001 of the OR function on n variables. For urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0002, the urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0003-error probabilistic degree of any Boolean function f : {0, 1}n → {0, 1} over urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0004 is the smallest nonnegative integer d such that the following holds: there exists a distribution P of polynomials urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0005 of degree at most d such that for all urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0006, we have urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0007. It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0008-error probabilistic degree of the OR function is at most urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0009. Our first observation is that this can be improved to urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0010 which is better for small values of urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0011. In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution P have the following special structure: urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0012 where each Li(x1, … , xn) is a linear form in the variables x1, … , xn, that is, the polynomial urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0013 is a product of affine forms. We show that the urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0014-error probabilistic degree of OR when restricted to polynomials of the above form is urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0015, thus matching the above upper bound (up to poly-logarithmic factors).

中文翻译:

关于 OR 在实数上的概率度

我们研究了urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0001OR 函数对n 个变量的概率度。对于urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0002urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0003任何布尔函数f的-error 概率度 :{0, 1} n  → {0, 1} overurn:x-wiley:rsa:media:rsa20991:rsa20991-math-0004是最小的非负整数d,使得以下成立:存在度数最多为d的多项式的分布P这样对于所有,我们有。从 Tarui ( Theoret. Comput. Sci. 1993) 和 Beigel, Reingold, and Spielman ( Proc. 6th CCC 1991) 的作品可知,OR 函数的-error 概率度至多为urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0005urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0006urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0007urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0008urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0009. 我们的第一个观察结果是,urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0010对于较小的 值,这可以改进为更好urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0011。在所有已知的对于该或函数的概率多项式(包括上述改善)的结构中,多项式P在支撑分布P具有如下特殊结构:urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0012其中每个大号X 1,...,  X Ñ)为线性形式在变量x 1 , … ,  x n 中,即多项式urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0013是仿射形式的乘积。我们表明,urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0014当限制为上述形式的多项式时,OR的-error 概率度为urn:x-wiley:rsa:media:rsa20991:rsa20991-math-0015,从而匹配上述上限(最多为多对数因子)。
更新日期:2021-01-19
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