关于块敏感度、证书复杂度和AND（和OR）决策树复杂度的凝结
On Condensation of Block Sensitivity, Certificate Complexity and the $\mathsf{AND}$ (and $\mathsf{OR}$) Decision Tree Complexity
AI总结：
研究了块敏感度、证书复杂度和AND/OR查询复杂度的凝结性，证明块敏感度在限制下不凝结，且AND/OR决策树也不存在凝结性。
摘要：
给定一个n位布尔函数，其复杂度度量（如块敏感度、查询复杂度等）为M(f)=k，硬性凝结问题询问是否能将f限制到O(k)变量，使得复杂度度量为Ω(k)。本文研究了块敏感度、证书复杂度、AND（和OR）查询复杂度和傅里叶稀疏性凝结性。我们证明块敏感度在限制下不凝结，不同于敏感度：存在一个布尔函数f，其查询复杂度为k，但任何对f限制到O(k)变量后，其块敏感度为O(k^{2/3})。这回答了Göös等人(2024)的一个开放问题的否定。同一函数也得出证书复杂度的不凝结结果。我们进一步证明AND（和OR）决策树也不存在凝结性。
英文摘要：
Given an $n$-bit Boolean function with a complexity measure (such as block sensitivity, query complexity, etc.) $M(f) = k$, the hardness condensation question asks whether $f$ can be restricted to $O(k)$ variables such that the complexity measure is $Ω(k)$? In this work, we study the condensability of block sensitivity, certificate complexity, AND (and OR) query complexity and Fourier sparsity. We show that block sensitivity does not condense under restrictions, unlike sensitivity: there exists a Boolean function $f$ with query complexity $k$ such that any restriction of $f$ to $O(k)$ variables has block sensitivity $O(k^{\frac{2}{3}})$. This answers an open question in Göös, Newman, Riazanov, and Sokolov (2024) in the negative. The same function yields an analogous incondensable result for certificate complexity. We further show that $\mathsf{AND}$(and $\mathsf{OR}$) decision trees are also incondensable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
