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.
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Acknowledgements
We thank Pavel Pudla'k for inspiration and Amir Yehudayoff for, unwittingly, making the author to write the paper in the first place. Supported by the GACR grant 19-27871X.
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Hrubeš, P. On \(\epsilon\)-sensitive monotone computations. comput. complex. 29, 6 (2020). https://doi.org/10.1007/s00037-020-00196-6
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DOI: https://doi.org/10.1007/s00037-020-00196-6