面向任务的布尔函数计算：实用码构造

中文摘要 AI 辅助

面向任务的通信仅传递下游任务所需的信息。对于二元决策任务，该范式在信息论上被形式化为通过信道进行布尔函数计算（BFC），其中接收端旨在确定发射端未知的某个函数的值。本文基于里德-所罗门码为BFC问题设计了一种实用的码构造。对于无噪声二进制信道，我们推导了有限块长度的最坏情况误差界。通过定义一个刻画支持消息长度如何随信道使用次数扩展的速率函数，我们刻画了不同布尔函数族的速率-可靠性权衡。在该缩放意义下，所提构造实现了1/2的渐近计算速率。我们进一步将该构造扩展到有噪声信道，方法是将多个BFC任务打包到单个块中，并将其与常规信道码级联。相应的有限块长度保证以每次函数评估的有效信道使用次数表示。当信道码率为R_c时，该构造实现了R_c/2的渐近计算速率，若采用容量可达的信道码，则得到C/2。数值结果展示了所推导的界，并证明了相比传统传输的显著性能增益。例如，所提编码方案在精确权重和秩检验任务中分别实现了约3.4和6.4分贝的信噪比编码增益。

英文摘要

Task-oriented communication conveys information that is necessary for downstream tasks. For binary decision tasks, this paradigm is information-theoretically formalized by Boolean function computation (BFC) via channels, where the receiver aims to determine the value of a function unknown to the transmitter. In this paper, we devise a practical code construction for the BFC problem based on a Reed--Solomon code. For noiseless binary channels, we derive finite-blocklength worst-case error bounds. By defining a rate function that captures how supported message length scales with channel uses, we characterize the rate-reliability tradeoff for different Boolean function families. With respect to this scaling, the proposed construction achieves an asymptotic computation rate of $1/2$. We further extend this construction to noisy channels by packing multiple BFC tasks into a single block and concatenating them with a conventional channel code. The corresponding finite-blocklength guarantees are expressed in terms of effective channel uses per function evaluation. With a channel code rate $R_c$, this construction achieves an asymptotic computation rate of $R_c/2$, yielding $C/2$ when capacity-achieving channel codes are employed. Numerical results illustrate the derived bounds and demonstrate substantial performance gains over conventional transmission. As examples, the proposed coding scheme achieves SNR coding gains of approximately $3.4$ and $6.4$~dB for the exact-weight and rank-test tasks, respectively.

发表机构

• College of Information Science & Electronic Engineering Zhejiang University（浙江大学信息与电子工程学院）
• Department of Electrical and Electronic Engineering University of Melbourne（墨尔本大学电气与电子工程系）

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