Consider a random sequence of n bits that has entropy at least n−k, where $${k\ll n}$$k≪n . A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query $${\approx\frac{n}{k}}$$≈nk other coordinates of the sequence, even if the adversary is non-deterministic. This implies corresponding results for decision trees and certificates for Boolean functions.As an application of this result, we prove a new result on depth-3 circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions due to Boppana (Circuits Inf Process Lett 63(5):257–261, 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIAM J Discrete Math 3(2):255–265, 1990), and, in particular, it is a “top-down” proof (Håstad et al. in Computat Complex 5(2):99–112, 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest.

中文翻译：

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部分信息和事后预测，并应用于电路下界

考虑具有至少 n−k 熵的 n 位随机序列，其中 $${k\ll n}$$k≪n 。一个常用的观察结果是，这个随机序列的平均坐标接近于均匀分布，即坐标“看起来是随机的”。在这项工作中，我们证明了一个更强的结果，粗略地说，对于允许查询 $${\approx\frac{n}{k}}$$≈nk 其他坐标的对手来说，平均坐标看起来是随机的。顺序，即使对手是不确定的。这意味着决策树的相应结果和布尔函数的证书。作为该结果的应用，我们证明了深度 3 电路的新结果，该结果作为直接推论恢复了奇偶函数和多数函数的已知下界，以及由于 Boppana 导致的敏感函数的下限（Circuits Inf Process Lett 63(5):257–261, 1997）。这个证明的一个有趣的特点是它在 Karchmer 和 Wigderson 的框架中工作（SIAM J Discrete Math 3(2):255–265, 1990），特别是它是一个“自上而下”的证明（Håstad等人在 Computat Complex 5(2):99–112, 1995 中。最后，它为非产品分布产生了一种新的随机限制引理，这可能是独立的。