We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial $$f\in \mathbb {R}[x_1,\dots , x_n]$$ of degree d has an arithmetic circuit of size s then $$(x_1+\dots +x_n+1)^d+\epsilon f$$ has a monotone arithmetic circuit of size $$O(sd^2+n\log n)$$ , for some $$\epsilon >0$$ . Second, if $$f:\{0,1\}^n\rightarrow \{0,1\}$$ is a Boolean function, we associate with f an explicit exponential-size matrix M(f) such that the Boolean circuit size of f is at least $$\varOmega (\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))- 2n)$$ , where J is the all-ones matrix and $$\mathrm{rk}_{+}$$ denotes the nonnegative rank of a matrix. In fact, the quantity $$\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))$$ characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar $$\epsilon $$ -sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system.

中文翻译：

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在 $$\epsilon$$ 敏感的单调计算上

我们表明，单调算术电路的足够强的下限或矩阵的非负秩意味着算术或布尔电路复杂性的无条件下限。首先，我们证明如果多项式 $$f\in \mathbb {R}[x_1,\dots , x_n]$$ 的阶数为 s，那么 $$(x_1+\dots +x_n+1) ^d+\epsilon f$$ 具有大小为 $$O(sd^2+n\log n)$$ 的单调算术电路，对于某些 $$\epsilon >0$$ 。其次，如果 $$f:\{0,1\}^n\rightarrow \{0,1\}$$ 是一个布尔函数，我们将 f 与一个显式指数大小的矩阵 M(f) 相关联，使得布尔函数f 的电路大小至少为 $$\varOmega (\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))- 2n)$$ ，其中 J是全一矩阵，$$\mathrm{rk}_{+}$$ 表示矩阵的非负秩。事实上，数量 $$\min _{\epsilon > 0}(\mathrm{rk}_{+}(M(f)-\epsilon J))$$ 表征了通过线性规划区分拒绝和接受 f 的输入的难度。最后，我们介绍了一个类似于 Atserias 等人的单调微积分的证明系统。(J Comput Syst Sci 65:626–638, 2002) 并表明单调算术电路上类似的 $$\epsilon $$ 敏感下界意味着系统中证明大小的下界。