Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove: (1) An exponential separation between sum of ROFs and read-$k$ formulas for some constant $k$. (2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs. Our results are based on analysis of the partial derivative matrix under different distributions. These results highlight richness of bounded read restrictions in arithmetic formulas and ABPs. Finally, we consider a generalization of multilinear ROABPs known as strict-interval ABPs defined in [Ramya-Rao, MFCS2019]. We show that strict-interval ABPs are equivalent to ROABPs upto a polynomial size blow up. In contrast, we show that interval formulas are different from ROFs and also admit depth reduction which is not known in the case of strict-interval ABPs.

中文翻译：

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有界读取公式之和的限制

证明计算显式多项式的各类算术电路的超多项式大小下界是代数复杂性理论中一项非常重要且具有挑战性的任务。我们将多项式表示为较弱模型的总和，例如读取一次公式 (ROF) 和读取一次不经意代数分支程序 (ROABP)。我们证明：（1）ROF 总和与某些常数 $k$ 的 read-$k$ 公式之间的指数分离。(2) ROABP 之和和句法多线性 ABP 之间的次指数分离。我们的结果基于对不同分布下的偏导矩阵的分析。这些结果突出了算术公式和 ABP 中有限读取限制的丰富性。最后，我们考虑多线性 ROABP 的泛化，称为 [Ramya-Rao, MFCS2019] 中定义的严格间隔 ABP。我们表明严格间隔 ABP 等价于 ROABP，达到多项式大小的膨胀。相比之下，我们表明区间公式与 ROF 不同，并且还允许深度减小，这在严格区间 ABP 的情况下是未知的。