Source
Zhao, Ma, Li, Yuan, Ye and Zhou (2025), LIT-793 (NOTE-638); Chai, Xiao, Shi and Saad (2023), LIT-812 (NOTE-605); Xin, Fan and Letaief (2024), LIT-823 (NOTE-619), for Liu, Zhang and Poor’s framework, which it reports; Tishby, Pereira and Bialek (1999), LIT-338, for the bottleneck.
What was actually shown
- The common form. In the framework the survey reports (Liu et al., Eqs. 21–22 there), in Chai et al. and in Zhao et al., a source emits pairs (S, X) with a joint law known to the designer; the encoder sees X only; fidelity is measured on S (directly, by its marginal law, or by its posterior) and optionally on X. The theorems are coding theorems for that problem: rate I(X; ·) minimised under expected-distortion constraints, proved by the standard random-coding or functional- representation arguments. Nothing in any of them uses a property of S other than its joint law with X and the distortion chosen. Each paper could have shown otherwise, by a limit that changes when S is given logical, compositional or contextual structure with the joint law held fixed; none does.
- The KL identity. Zhao et al. measure semantic distortion as d_p(p(S|x), p(S|y)), with p(S|y) the posterior under the code and S–X–Y Markov. For d_p = KL, E[D(p(S|X) ‖ p(S|Y))] = E log p(S|X)/p(S|Y) = H(S|Y) − H(S|X) = I(S;X) − I(S;Y). Their function with no symbolic constraint is therefore min{I(X;Y) : I(S;Y) ≥ I(S;X) − D_p}, the information bottleneck with the reconstruction as bottleneck variable. This is a two-line derivation from their definitions; the paper names KL as an admissible d_p and does not draw it.
- The TV bound. For d_p = TV and any loss L(a, s) of range at most 1, the receiver acting on p(S|Y) has Bayes risk at most E[d_TV(p(S|X), p(S|Y))] above one acting on p(S|X), since for each pair |min_a E_{p(S|y)}L − min_a E_{p(S|x)}L| ≤ sup_a |E_{p(S|y)}L − E_{p(S|x)}L| ≤ d_TV. A bound D_p therefore limits the loss of informativeness about S uniformly over every bounded decision problem, for the given prior. Also a short derivation; not in the paper.
What this does not say
- Not that semantic communication is “only” classical theory. The frameworks may be the right formalisation; the claim is about what they formalise. The modelling questions the survey lists as open (what S is, where p(S, X) comes from, how knowledge bases enter) are exactly the ones these theorems take as given.
- Not that the survey’s other strands reduce. Logical-probability semantic entropy (Carnap and Bar-Hillel, Bao et al.) and the semantic capacity definitions are not latent-variable rate–distortion and are outside this account. The account covers the rate–distortion strand only, in the three readings above plus the framework the survey reports second-hand.
- Not a Blackwell comparison. The TV bound fixes the prior on S and bounds risk for one prior; the Blackwell order (THEORY-156) compares experiments for every prior and is qualitative. The bound is a prior-dependent, quantitative relative of that order, not the order itself, and D_p = 0 gives Bayes sufficiency of Y for S at this prior, which for a Markov chain S–X–Y is the same as I(S;Y) = I(S;X).
- Not about Chai et al.’s perception constraint. A constraint on the marginal law of the reconstruction (Chai et al.) is not decision-relative: two reconstructions with the same law can differ completely in what they say about S. Only the posterior form carries the TV bound.
- Not about Zhao et al.’s theorems as stated. Their sequence distortion is written as an expected maximum while their proofs use maxima of per-letter expectations (NOTE-638); the identities here are single-letter and unaffected, but the coding theorems hold for the per-letter form.
Note of 2026-10-19: the two identities, re-derived
KL. With S–X–Y Markov, p(s|x,y) = p(s|x). So E_{X,Y}[D(p(S|X) ‖ p(S|Y))] = E_{S,X,Y}[log p(S|X) − log p(S|Y)] = H(S|Y) − H(S|X) = (H(S) − I(S;Y)) − (H(S) − I(S;X)) = I(S;X) − I(S;Y). Minimising I(X;Y) subject to that being at most D_p is min{I(X;Y) : I(S;Y) ≥ I(S;X) − D_p}, the information bottleneck with the reconstruction as the bottleneck variable.
TV. For a loss with values in an interval of length at most 1, |E_P L(a,·) − E_Q L(a,·)| ≤ d_TV(P, Q) for each action, so the minima over actions differ by at most d_TV too. By the tower rule, the receiver acting optimally on p(S|Y) has Bayes risk E_Y[min_a E_{p(S|Y)}L], and the one acting on p(S|X) has E_X[min_a E_{p(S|X)}L]. Their difference is therefore at most E[d_TV(p(S|X), p(S|Y))], for every such loss at once, at the given prior. The bound is on expected risk. Pointwise, the regret of acting on p(S|y) when the posterior is p(S|x) can reach 2 d_TV. Whether frameworks beyond these two share their form is QUESTION-177.