在尺寸均匀性和深度四种公式上具有较低的个别化程度

N. Kayal, Chandan Saha, Sébastien Tavenas
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引用次数: 27

摘要

设r为整数。我们称多项式f为多重r-ic多项式,如果f对任意变量的阶不超过r(这概括了多重线性多项式的概念)。我们研究的算术电路中,其输出在语法上被迫是一个多r-ic多项式,并将这些称为多r-ic电路。具体来说,首先归纳定义节点a相对于变量x的形式度,如下所示。对于叶节点,如果a被标记为x,则为1,否则为0;对于标记为*(分别为+)的内部节点,它是子节点相对于x的形式度的和(分别为最大值)。如果输出节点相对于任何变量的形式度不超过r,我们称算术电路为多r-ic电路。我们证明了多r-ic电路的各种子类的下界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the size of homogeneous and of depth four formulas with low individual degree
Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.
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