超越识别：通过信道计算布尔函数
Beyond Identification: Computing Boolean Functions via Channels
AI总结：
研究通过信道可靠恢复未知布尔函数的问题，提出计算容量概念，分析不同函数类的渐进行为。
摘要：
考虑一个点对点通信系统，其中发送端持有长度为$m$的二进制消息，并发送对应的长度为$n$的码字。接收端的目标是恢复该消息的布尔函数，其中该函数未知于发送端，但选择自已知的函数类$F$。我们关注$m$和$n$的渐近关系：给定$n$，如何大$m$（渐近而言），使得布尔函数的值可以可靠恢复？该问题推广了Ahlswede和Dueck引入的通过信道识别框架。我们提出了计算容量的概念，并为选定的函数类$F$推导了可实现性和反向结果，这些函数类由函数的汉明重量特征化。我们获得的结果在所有考虑的$F$情况中都是紧的，即在缩放行为方面是紧的。
英文摘要：
Consider a point-to-point communication system in which the transmitter holds a binary message of length $m$ and transmits a corresponding codeword of length $n$. The receiver's goal is to recover a Boolean function of that message, where the function is unknown to the transmitter, but chosen from a known class $F$. We are interested in the asymptotic relationship of $m$ and $n$: given $n$, how large can $m$ be (asymptotically), such that the value of the Boolean function can be recovered reliably? This problem generalizes the identification-via-channels framework introduced by Ahlswede and Dueck. We formulate the notion of computation capacity, and derive achievability and converse results for selected classes of functions $F$, characterized by the Hamming weight of functions. Our obtained results are tight in the sense of the scaling behavior for all cases of $F$ considered in the paper.
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