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Beyond Identification: Computing Boolean Functions via Channels

Jingge Zhu, Matthias Frey

TL;DR

This work studies computing Boolean functions over noisy channels, formalizing the Boolean function computation (BFC) problem and introducing a computation capacity $C_{\text{BFC}}$ that depends on the function class weight. By constructing BFC codes via a maximal-code approach grounded in Ahlswede's identification framework, it derives explicit scaling laws for how large the message length $m$ can be relative to channel uses $n$ across six weight regimes, ranging from constant to exponential in $m$ relative to $n$. The converse arguments reduce BFC codes to identification codes to transfer known ID-capacity bounds, yielding tight (in scaling) upper bounds that match the achievability in most cases. The results reveal rich phase transitions in scaling—from exponential to linear—governed by the Hamming weight profile of the target function class, with connections to propositional logic semantics and Ahlswede’s models. These findings have implications for semantic communication and function-specific transmission strategies in noisy channels.

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.

Beyond Identification: Computing Boolean Functions via Channels

TL;DR

This work studies computing Boolean functions over noisy channels, formalizing the Boolean function computation (BFC) problem and introducing a computation capacity that depends on the function class weight. By constructing BFC codes via a maximal-code approach grounded in Ahlswede's identification framework, it derives explicit scaling laws for how large the message length can be relative to channel uses across six weight regimes, ranging from constant to exponential in relative to . The converse arguments reduce BFC codes to identification codes to transfer known ID-capacity bounds, yielding tight (in scaling) upper bounds that match the achievability in most cases. The results reveal rich phase transitions in scaling—from exponential to linear—governed by the Hamming weight profile of the target function class, with connections to propositional logic semantics and Ahlswede’s models. These findings have implications for semantic communication and function-specific transmission strategies in noisy channels.

Abstract

Consider a point-to-point communication system in which the transmitter holds a binary message of length and transmits a corresponding codeword of length . 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 . We are interested in the asymptotic relationship of and : given , how large can 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 , characterized by the Hamming weight of functions. Our obtained results are tight in the sense of the scaling behavior for all cases of considered in the paper.
Paper Structure (12 sections, 10 theorems, 116 equations, 1 table)

This paper contains 12 sections, 10 theorems, 116 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Examples: $\mathcal{F}_m^{\text{W}}(S)$ and $\mathcal{F}_m^{\text{W}}(\leq S)$ for different $S$
  4. Formulae in propositional logic
  5. Connection to identification via channels and other Ahlswede models in ahlswede_general_2008
  6. Main result
  7. Proof of achievability
  8. Proof of the converse
  9. Proof of Proposition \ref{['prop:maximal_code']}
  10. Proof of Lemma \ref{['lemma:error_probability']}
  11. A proof of Case 5 in Theorem \ref{['thm:achievability']}
  12. Lemma \ref{['lemma:gilbert_cw']} and its proof

Key Result

Theorem 1

Assume the channel $W(\cdot | x^n)$ has a Shannon capacity $C$, and let $c>0$ be a constant independent from $m$ and $n$. Then,

Theorems & Definitions (25)

  • Definition 1: Boolean function computation (BFC) code
  • Definition 2: Achievable computation rate
  • Definition 3: Computation capacity
  • Example 1: $\mathcal{F}_m^{\text{W}}(c)$
  • Example 2: $\mathcal{F}_m^{\text{W}}(\leq cm^{\beta})$
  • Example 3: $\mathcal{F}_m^{\text{W}}(c2^{\gamma m})$
  • Example 4: ranking
  • Example 5: Hamming weight of DNF
  • Theorem 1: Achievability
  • Theorem 2: Converse
  • ...and 15 more