多数函数的自顶向下深度四电路下界
A Top-Down Depth-Four Circuit Lower Bound for Majority
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中文总结 AI 辅助
该研究扩展了FOCS 2023的工作，针对多数函数构造镜像集以解决翻转多位输入的挑战，得到其自顶向下深度四电路下界。
中文摘要 AI 辅助
我们通过扩展Göös、Riazanov、Sofronova和Sokolov（FOCS 2023）的最新工作，给出了多数函数的自顶向下深度四电路下界。该团队曾给出奇偶函数的深度四电路下界的自顶向下证明，其依赖于鲁棒向日葵构造镜像集，并利用块不可预测性寻找局部极限。多数函数情形的主要挑战在于构造对应的镜像集，二者的区别在于：翻转奇偶函数的一位输入即可改变其值，而翻转多数函数的值可能需要改变布尔串的多位输入；我们通过考虑布尔立方体的切片（即汉明重量约为n/2的固定布尔串）来避免这种多位翻转问题。
英文摘要
We present a top-down depth-four circuit lower bound for Majority function by extending recent work of Göös, Riazanov, Sofronova, and Sokolov (FOCS 2023), who gave a top-down proof of a depth-four circuit lower bound for Parity which relies on the robust sunflower to construct a mirror set and the block unpredictability to find the local limits. The main challenge for the case of Majority is to construct a corresponding mirror set, the difference is that to flip the value of Majority function, one may have to flip many bits of the input Boolean string, while for Parity, flipping one bit suffices. We avoid this flipping by considering slices of the Boolean cube, that is, Boolean strings of fixed Hamming weight approximately $n/2$.
