On the Consistency of Circuit Lower Bounds for Non-Deterministic Time
Albert Atserias, Sam Buss, Moritz Müller
备注：
An extended abstract of part of this work appeared in the Proceedings of the 55th ACM Symposium on Theory of Computation (STOC 2023). This is a small revision of the earlier long version that includes some minor corrections, a revised introduction, and some additional observations
We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V$^0_2$ is consistent with the conjecture that NEXP $\not\subseteq$ P/poly, i.e., some problem that is solvable in non-deterministic exponential time does not have polynomial size circuits. We suggest this is the best currently available evidence for the truth of the conjecture. The same techniques establish the same results with NEXP replaced by the class of problems that are decidable in non-deterministic barely superpolynomial time such as NTIME$(n^{O(\log\log\log n)})$. Additionally, we establish a magnification result on the hardness of proving circuit lower bounds.
Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion
在（Kabanets，Impagliazzo，2004）中，给出了如何在确定性次指数时间内确定电路多项式恒等式测试问题（CPIT），假设算术电路的一些显式多线性多项式族的硬度。本文考虑了CPIT的一种特殊情况，即低次非奇异矩阵完备化（NSMC）。对于这个子类的问题，它示出了如何获得相同的确定性的时间范围，使用较弱的假设，在确定性的复杂性。硬度随机性权衡也将显示在相反的方向，在努力取得进展的Valiant的VP与VNP的问题。为了分离VP和VNP，已知证明m × m持久式的行列式复杂度是$m^{ω(\log m)}$就足够了。本文证明了，对于一个适当的显式性概念，具有行列式复杂度m^{ω（\log m）} $ is equivalent to the existence of an efficiently computable generator $ G_n $ for multilinear NSMC with seed length $ O（n^{1/ \sqrt{\log n} }）$. The latter is a combinatorial object that provides an efficient deterministic black-box algorithm for NSMC. ``Multilinear NSMC'' indicates that $ G…
A hitting set construction, with application to arithmetic circuit lower bounds
多项式同一性测试算法必须确定给定的输入多项式是否完全等于0。给出了一元多项式$\sum_{j=0}^t c_j X^{α_j} (a + b X)^{β_j}$的确定性黑盒恒等式检验算法。从我们的算法中，我们推导出一个指数下界表示的多项式，如$\prod_{i=1}^{2^n} (X^i-1)$在这种形式下。已经证明，这些多项式很难用一般的算术电路来计算。我们的结果表明，“从去随机化的硬度”的方法是可行的，为一类受限制的算术电路的下界。证明是基于代数数论的技术，更确切地说，是基于代数数的高度函数的性质。
