Function Computation and Identification over Locally Homomorphic Multiple-Access Channels

TL;DR

The paper addresses computing functions over channels beyond exact message recovery by introducing locally homomorphic channels (lhc) and modeling functions via characteristic hypergraphs $H_f$, establishing an approximate equivalence between function codes and edge-bijective lhc. It develops decomposition tools to modularly build encoders and demonstrates how to construct bipartite encoders when two parties hold separate messages, enabling independent encoding. The main contributions include the equivalence between $(f,\phi,\boldsymbol{\lambda})$-codes and edge-bijective lhcs, decomposition results, and practical id/$K$-id constructions with improved rates, illustrated through a deterministic-id example over parallel binary symmetric channels. These methods advance functional communication theory for multi-user settings and have implications for future post-Shannon network designs, while leaving open questions about hypergraph rectangularity and rate-optimality.

Abstract

We develop the notion of a locally homomorphic channel and prove an approximate equivalence between those and codes for computing functions. Further, we derive decomposition properties of locally homomorphic channels which we use to analyze and construct codes where two messages must be encoded independently. This leads to new results for identification and K-identification when all messages are sent over multiple-access channels, which yield surprising rate improvements compared to naive code constructions. In particular, we demonstrate that for the example of identification with deterministic encoders, both encoders can be constructed independently.

Figures (1)

• Figure 1: id and transmission rates achievable over two parallel binary symmetric channels with crossover probability $\gamma = 0.03$, using deterministic encoders. id corresponds to our $f_{\textsf{ID}}$, where the two messages to be compared are sent over the channel, whereas ID-MAC corresponds to $f_{\textsf{ID-MAC}} = f_{\textsf{ID}}^{(1)} \otimes f_{\textsf{ID}}^{(2)}$, where $f_{ID}^{(i)} : \cM_i^2 \to \set{0,1}$, are id functions, i.e., two simultaneous id are performed, each with one message known to the receiver. For deterministic id, the diagonal up to the marked points is achievable. For deterministic id-MAC, the regions below the lines are achievable rosenbergerIbrahimDeppeFerrara2023di_mac_isit. For transmission over a MAC, the red dashed line is the boundary of the capacity region elgamalKim2011network_it. The intersection point id-T marks the rates achievable for $f_{\textsf{ID}}$ if we restrict the codes to transmission codes.

Theorems & Definitions (28)

• definition 1: (discrete memoryless) channel
• definition 2
• definition 3
• definition 4
• definition 5
• definition 6
• definition 7
• remark 1
• definition 8: lhc
• remark 2