Problem
Research questions and friction points this paper is trying to address.
Boolean function computation
computation capacity
identification-via-channels
asymptotic rate
Hamming weight
Capacity regimes for Boolean function computation via channels
🤖 AI Summary
This study investigates the asymptotic performance of computing Boolean functions over point-to-point communication channels, where the receiver must reliably recover the value of an unknown Boolean function—known to belong to a specified class—from a length-$n$ codeword. The goal is to characterize the rate function describing how the message length $m$ scales with $n$, and the associated computation capacity. By extending classical channel coding to the setting of function computation and leveraging information-theoretic tools together with Hamming-weight-based characterizations of Boolean function classes, the paper fully determines the rate function for a broad class of Boolean functions and establishes tight upper and lower bounds on the computation capacity, whose gap is at most a factor of two.
📝 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. 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 $2$.
Innovation
Methods, ideas, or system contributions that make the work stand out.
computation capacity
Boolean function computation
identification via channels
rate function
Hamming weight
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