Beyond Identification: Computing Boolean Functions via Channels

TL;DR

This paper explores the limits of computing Boolean functions over communication channels, generalizing identification frameworks and establishing tight bounds on message length relative to codeword length.

Contribution

It introduces the concept of computation capacity and derives tight asymptotic bounds for classes of Boolean functions based on Hamming weight.

Findings

Derived achievability and converse bounds for computation capacity.

Results are tight in the asymptotic scaling for all function classes considered.

Generalized the identification-via-channels framework to Boolean function computation.

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.