AI 中文总结
研究从含单边噪声的随机约束满足实例中恢复植入赋值的查询复杂度，针对1-CNF和k-CNF问题给出自适应算法查询复杂度界，也给出非自适应算法界，还暗示了噪声排序下界及研究了带否定的模型变体。
AI 中文摘要
我们研究了从具有单边噪声的随机约束满足实例中恢复植入赋值的查询复杂度。考虑以下1-CNF恢复问题：一个未知的二进制字符串，其中有n/2个1和n/2个0，对单个变量进行查询。对1变量的查询以概率p返回“1”，否则返回“0”，而对0变量始终返回“0”。目标是以至少1-δ的概率恢复二进制字符串。我们证明查询复杂度为(1+o(1))c(p)n/2(log₂n+log₂(1/δ))。然后研究了具有单边噪声的植入k-CNF满足问题，证明了查询复杂度为(1+o(1))c(p,k)n/2(log₂n+log₂(1/δ))。这些界适用于自适应算法，也给出了非自适应算法的界，表明对于固定的p，适应性带来指数(Θ(k))的改进。我们的结果还暗示了{0,1}值字符串噪声排序的下界，并研究了带有否定的模型变体。
英文摘要
We study the query complexity of recovering a planted assignment from a random constraint-satisfaction instance with one-sided noise. We consider the following 1-CNF recovery problem: an unknown binary string with $n/2$ ones and $n/2$ zeros is queried at individual variables. A query to a $1$-variable returns "$1$" with probability $p$ and "$0$" otherwise, while a $0$-variable always returns "$0$" (each query is a fresh noisy draw). The goal is to recover the binary string with probability at least $1 - δ$. While the naive counting argument may suggest a query complexity of $\log_2 \binom{n}{n/2}=Θ(n)$, we show that the query complexity is $(1+o(1))c(p) \frac{n}{2} \left( \log_2 n + \log_2(1/δ)\right)$, where $c(p) = \tfrac{1}{-\log_2(1-p)}$. We then study planted $k$-CNF satisfaction with one-sided noise. Each $k$-set containing a $1$-variable is included as a clause independently with probability $p$, and an algorithm may ask whether any given $k$-set is a clause. Unlike the $1$-CNF case, a clause-existence query is one-shot: each $k$-set either is or is not a clause, so repeating yields no new information. The model is one-sided because an observed clause certifies that at least one queried variable is assigned 1, whereas its absence does not certify all are assigned 0. The goal is to recover the planted assignment with probability at least $1 - δ$. The counting baseline is $Θ(n)$, yet we prove a query complexity of $(1+o(1))\,c(p,k)\, \frac{n}{2}\left( \log_2 n + \log_2(1/δ)\right)$, where $c(p,k) = \tfrac{1}{k(-\log_2(1-p))}$. These bounds are for adaptive algorithms. We also prove bounds for nonadaptive algorithms, showing that for fixed $p$, adaptivity gives a factor $\exp(Θ(k))$ improvement. Our results also imply lower bounds for noisy sorting of $\{0,1\}$-valued strings, and we study a variant of the model with negations.
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