On $\epsilon$-sensitive monotone computations

Abstract

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.