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.

翻译：考虑一个点对点通信系统，其中发送方持有长度为 $m$ 的二进制消息，并传输相应的长度为 $n$ 的码字。接收方的目标是恢复该消息的某个布尔函数值，该函数对发送方未知，但选自一个已知的类 $F$。我们关注 $m$ 与 $n$ 的渐近关系：给定 $n$，$m$ 可以（渐近地）多大，使得布尔函数的值能够被可靠地恢复？该问题推广了由 Ahlswede 和 Dueck 提出的通过信道进行识别的框架。我们提出了计算容量的概念，并针对由函数的汉明权重刻画的特定函数类 $F$，推导了可达性与逆命题结果。本文所得结果对于所考虑的所有 $F$ 情形，在缩放行为的意义上是紧的。