通过信道实现布尔函数计算的容量区间
Capacity regimes for Boolean function computation via channels
AI总结：
该研究针对发送方未知的布尔函数计算问题，定义计算容量，推导其上下界及速率函数，推广了Ahlswede与Dueck的信道识别框架。
AI中文摘要：
考虑一个点对点通信系统，其中发送方持有长度为$m$的二元消息，并发送对应长度为$n$的码字。接收方的目标是恢复该消息的布尔函数值，该函数对发送方未知，但选自已知类$F$。我们关注$m$与$n$的渐近关系：给定$n$时，$m$可渐近多大，才能可靠恢复布尔函数值？该问题是Ahlswede与Dueck提出的“通过信道识别”框架的推广。本文中，我们定义了计算容量的概念，并针对以函数汉明重量为特征的大类函数$F$，推导了可达性与 converse 结果。与经典传输问题不同，函数计算问题的性能由计算容量与速率函数共同刻画，速率函数即$m$随$n$的渐近缩放关系。我们的结果完整刻画了计算问题的速率函数，并给出了计算容量的上下界，二者差异最多为2倍。
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. In this paper, we formulate the notion of computation capacity, and derive achievability and converse results for a large class of functions $F$, characterized by the Hamming weight of functions. Different from the classical transmission problem, the performance of the function computation problem is jointly characterized by the computation capacity and the rate function, namely how $m$ scales with $n$ asymptotically. Our results give a complete characterization of the rate function of the computation problem, and provide upper and lower bounds on the computation capacity, where they differ by a factor of at most $2$.
