AI 中文总结
该研究探究随机与分布查询复杂度的直和问题，精确刻画摊销期望随机查询复杂度，证明有界误差经摊销可转化为零误差，改进此前直和界并解决Blais与Brody提出的公开问题，还得到相关分离结果。
AI 中文摘要
本文研究期望随机查询复杂度与分布查询复杂度的直和问题。我们的主要结果给出了摊销期望随机查询复杂度的精确刻画。对于任意全关系$f$和任意误差容限$\varepsilon \in [0,1]$，我们证明：当n趋于无穷时，\\(\frac{\overline{R}_\varepsilon(f^n)}{n}\\)的极限等于\\((1 - \varepsilon) \overline{R}_0(f)\\)。由此，摊销以精确的乘性因子$1-\varepsilon$将有界误差转化为零误差。我们还证明了最坏情况随机查询复杂度与分布查询复杂度对应的下极限/上极限界。这些结果改进了此前仅在常数因子范围内或受限误差区间内成立的直和界，并解决了Blais与Brody（2019）提出的一个公开问题。此外，对于函数$\operatorname{OR}_n \circ f$的单边计算，我们得到了期望代价与最坏情况代价的类似精确摊销恒等式。作为应用，我们得到了摊销代价与单实例代价之间的分离结果，包括分布复杂度与随机关系的无界分离，以及随机全函数的二次屏障。
英文摘要
This paper investigates the direct sum question for expected randomized and distributional query complexity. Our main result gives an exact characterization of the amortized expected randomized query complexity. For any total relation $f$ and any error tolerance $\varepsilon \in [0,1]$, we prove \[ \lim_{n \to \infty} \frac{\overline{R}_\varepsilon(f^n)}{n} = (1 - \varepsilon) \overline{R}_0(f). \] Thus the amortization converts bounded-error into zero error with the exact multiplicative factor $1-\varepsilon$. We also prove corresponding liminf/limsup bounds for worst-case randomized and distributional query complexity. These results improve prior direct-sum bounds that were known only up to constant factors or in restricted error regimes, and they resolve an open question posed by Blais and Brody (2019). Additionally for one-sided computation of the function $\operatorname{OR}_n \circ f$, we obtain analogous exact amortized identities for both expected and worst-case cost. As applications, we obtain separations between amortized and single-instance costs, including unbounded separations for distributional complexity and randomized relations, and a quadratic barrier for randomized total functions.
