传递函数的组合测度之间的分离
Separations between Combinatorial Measures for Transitive Functions
AI总结：
本文针对传递布尔函数，研究多对组合复杂性测度的最大分离，并通过改造非传递的指针函数与备忘单函数，构造出可展示类似分离的传递函数。
摘要：
对称性在布尔函数 $f:\{0,1\}^n \to \{0,1\}$ 中的作用已在复杂性理论中得到广泛研究。例如，对称函数，即在 $S_n$ 的作用下保持不变的函数，是布尔函数研究中的一类重要函数。函数 $f:\{0,1\}^n \to \{0,1\}$ 被称为传递函数（或弱对称函数），如果存在 $S_n$ 的一个传递群 $G$，使得 $f$ 在 $G$ 的作用下保持不变——也就是说，即使 $f$ 的输入比特按照某个置换 $\sigma\in G$ 被重新排列，函数值仍保持不变。在过去几十年中，理解传递函数的各种复杂性测度一直是一个内容丰富的研究领域。在本工作中，我们从若干组合测度的角度研究传递函数。我们考察传递函数的各对测度之间的最大分离。多年来，针对一般布尔函数的此类研究一直在进行。Aaronson 等人（STOC，2021）对一般布尔函数的最著名结果进行了很好的汇编。一对组合测度之间的分离是通过构造能够展示这种分离的有趣函数来证明的。但许多著名的分离结果都是通过构造非传递函数得到的，例如 Ambainis 等人（JACM，2017）的“指针函数”和 Aaronson 等人（STOC，2016）的“备忘单函数”。因此，对于传递函数，我们尚不清楚各对测度之间是否存在这样的分离。在本文中，我们展示了如何修改其中一些函数，以构造出能够在各对组合测度之间展示类似分离的传递函数。
英文摘要：
The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class of functions in the study of Boolean functions. A function $f:\{0,1\}^n \to \{0,1\}$ is called transitive (or weakly-symmetric) if there exists a transitive group $G$ of $S_n$ such that $f$ is invariant under the action of $G$ - that is the function value remains unchanged even after the bits of the input of $f$ are moved around according to some permutation $σ\in G$. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. In this work, we study transitive functions in light of several combinatorial measures. We look at the maximum separation between various pairs of measures for transitive functions. Such study for general Boolean functions has been going on for past many years. The best-known results for general Boolean functions have been nicely compiled by Aaronson et. al (STOC, 2021). The separation between a pair of combinatorial measures is shown by constructing interesting functions that demonstrate the separation. But many of the celebrated separation results are via the construction of functions (like "pointer functions" from Ambainis et al. (JACM, 2017) and "cheat-sheet functions" Aaronson et al. (STOC, 2016)) that are not transitive. Hence, we don't have such separation between the pairs of measures for transitive functions. In this paper we show how to modify some of these functions to construct transitive functions that demonstrate similar separations between pairs of combinatorial measures.
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