近似度的一般复合定理
A General Composition Theorem for Approximate Degree
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中文总结 AI 辅助
本文证明了所有全布尔函数的近似度在块复合下满足乘法复合定理，即近似度乘积的阶数匹配，解决了该领域的长期开放问题。
中文摘要 AI 辅助
布尔函数复杂度中一个长期悬而未决的问题询问近似度在块复合下是否乘法性地复合。虽然已知一个一般的乘法上界，但匹配的下界此前仅对受限函数类建立。我们通过证明匹配下界解决了所有全布尔函数的这一问题。结合Sherstov的上界，我们的结果表明，对于每一对全布尔函数$f:\{0,1\}^n\to\{0,1\}$和$g:\{0,1\}^m\to\{0,1\}$，有\\[ \widetilde{deg}(f\circ g) = \Theta\\!\left( \widetilde{deg}(f)\\,\widetilde{deg}(g) \right), \\] 其中$\widetilde{deg}$表示常数误差近似度。
英文摘要
A longstanding open question in Boolean function complexity asks whether approximate degree composes multiplicatively under block composition. Although a general multiplicative upper bound is known, matching lower bounds have previously been established only for restricted classes of functions. We resolve this question for all total Boolean functions by proving the matching lower bound. Together with Sherstov's upper bound, our result shows that, for every pair of total Boolean functions $f:\{0,1\}^n\to\{0,1\}$ and $g:\{0,1\}^m\to\{0,1\}$, \[ \widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right), \] where $\widetilde{deg}$ denotes constant-error approximate degree.
发表机构
- Stony Brook University（石溪大学）
机构由 AI 辅助整理，请以论文原文为准。
