Beyond Identification: Computing Boolean Functions via Channels

2026-01-19

AI Summary

This work addresses the problem of reliably computing, at the receiver, the value of a Boolean function of the transmitted message in point-to-point communication, where the function is unknown but belongs to a known class, without requiring full recovery of the original message. For a given channel code length \(n\), the goal is to maximize the supported message length \(m\). Inspired by the identification framework of Ahlswede and Dueck, the paper introduces the notion of “computation capacity” and focuses on classes of Boolean functions categorized by Hamming weight. By integrating information-theoretic techniques with structural properties of Boolean functions, the authors establish tight achievability and converse theorems that fully characterize the asymptotic relationship between \(m\) and \(n\), thereby providing the first precise characterization of the optimal scaling law between message length and code length in this functional computation setting.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Evolutionary Computation

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• Boolean functions
• computation capacity
• identification via channels
• point-to-point communication
• asymptotic analysis

Innovation

• computation capacity
• Boolean functions
• identification via channels
• Hamming weight
• asymptotic analysis