Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators
Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators
复制标题
无取消电路：证明线性布尔运算符超线性下界的方法
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Magnus Find
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文献类型：
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作者：
J. Boyar;Magnus Find
我们继续研究无取消线性电路的概念。我们表明，每个矩阵都可以通过取消电路来计算，几乎所有矩阵最多都是恒定因子大于计算矩阵的最佳线性电路。与一般的线性电路相比，证明有关取消线性电路结构的说明似乎更容易。我们证明了两个非平凡的超线性下限。我们表明，无取消的线性电路计算$ n imes n $ sierpinski垫片矩阵必须使用至少1/2 N logn大门，这很紧。这支持了亚伦森的猜想。此外，我们表明，证明在单调电路上证明下限的证明策略几乎可以直接转换为证明无取消线性电路的下限。我们将其与Andreev的极端图理论的结果一起使用，以证明{omega}（n^（2- epsilon））的下限，对于每$ epsilon> 0 $ for for for Inverivallibly $ n imes n $矩阵。这些混凝土矩阵的下限几乎是最佳的，因为所有矩阵都可以使用$ O（n^2/log n）$门计算。
We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellation- free circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit that computes the matrix. It appears to be easier to prove statements about the structure of cancellation-free linear circuits than for linear circuits in general. We prove two nontrivial superlinear lower bounds. We show that a cancellation-free linear circuit computing the $n imes n$ Sierpinski gasket matrix must use at least 1/2 n logn gates, and that this is tight. This supports a conjecture by Aaronson. Furthermore we show that a proof strategy for proving lower bounds on monotone circuits can be almost directly converted to prove lower bounds on cancellation-free linear circuits. We use this together with a result from extremal graph theory due to Andreev to prove a lower bound of {Omega}(n^(2- epsilon)) for infinitely many $n imes n$ matrices for every $epsilon > 0$ for. These lower bounds for concrete matrices are almost optimal since all matrices can be computed with $O(n^2/log n)$ gates.