---
title: Cooperative Binning in Semi-deterministic Channels
url: https://www.emergentmind.com/topics/semi2
type: topic
---

# Cooperative Binning in Semi-deterministic Channels

The problem addressed in "Cooperative Binning for Semi-deterministic Channels with Non-causal State Information" is the capacity characterization of semi-deterministic multiuser channels when the encoder and decoder know the entire state sequence non-causally, while an intermediate terminal does not [1703.08099]. Its central result is that capacity is achieved by **cooperative-bin-forward**, a block-Markov binning scheme in which cooperation is induced through bin indices of deterministic channel outputs rather than by relay-side decoding of a message part. In the non-causal setting, the scheme exploits look-ahead over the state sequence to select cooperation codewords that are jointly typical with future states, thereby implicitly conveying partial state information to a strictly causal relay or cribbing encoder and enlarging the achievable rate region [1703.08099].

## 1. Channel models and assumptions

The principal model is the **semi-deterministic relay channel** (SD-RC). It uses finite alphabets $\mathcal X,\mathcal X_r,\mathcal Y,\mathcal Z,\mathcal S$ and a message $M \in [1:2^{nR}]$. The states $S^n=(S_1,\dots,S_n)$ are i.i.d. with distribution $p_S$, and the channel is memoryless conditioned on $S_i$. The relay observation is deterministic,
$$
Z_i = z(X_i,X_{r,i},S_i),
$$
while the destination observation is stochastic with law
$$
p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).
$$
Equivalently,
$$
p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},
$$
where $1_{Z|X,X_r,S}$ denotes the deterministic conditional PMF induced by $z(\cdot)$. Non-causal CSI means that the encoder and decoder know the full sequence $S^n$ before transmission, whereas the relay has no state information. The relay is strictly causal: at time $i$, $X_{r,i}$ is a function of $Z^{i-1}$ [1703.08099].

A code for this model consists of
$$
x^n : [1:2^{nR}] \times S^n \to \mathcal X^n,\qquad
x_{r,i}: Z^{i-1} \to \mathcal X_r,\qquad
\hat m: Y^n \times S^n \to [1:2^{nR}],
$$
with average error probability $P_e(\mathcal C_n)\le \epsilon$ for sufficiently large $n$. The operational structure is block-Markov: transmission is divided into $B$ blocks of length $n$, and the deterministic relay output in block $b$ determines, through binning, a cooperation index used in block $b+1$ [1703.08099].

The paper also studies the **multiple-access channel with partial cribbing** as a semi-deterministic channel. Encoder 1 knows $S_1^n$ non-causally, Encoder 2 knows $S_2^n$ non-causally, and the decoder knows both. Encoder 2 deterministically cribs
$$
Z_i = z(X_{1,i},S_{1,i}),
$$
either strictly causally or causally. The same semi-deterministic perspective yields capacity results for this model as well. A further specialization is a **point-to-point channel with a state encoder**, where one encoder observes the state and communicates it over a private deterministic link to the main transmitter; this becomes a partial-cribbing MAC with $Z=X_1$ [1703.08099].

## 2. Capacity of the semi-deterministic relay channel with non-causal CSI

For the SD-RC with non-causal CSI at the encoder and decoder, capacity is
$$
C = \max \min \Big\{
I(X,X_r;Y|S),\;
I(X;Y|X_r,Z,S,U) + H(Z|X_r,S,U) - I(U;S)
\Big\},
$$
where the maximization is over distributions of the form
$$
p_{U|S}\, p_{X_r|U}\, p_{X|X_r,U,S},
$$
subject to
$$
I(U;S) \le H(Z|X_r,S,U),
$$
with $Z=z(X,X_r,S)$ deterministic. A cardinality bound is
$$
|\mathcal U| \le \min\{|\mathcal S||\mathcal X||\mathcal X_r|,\; |\mathcal S||\mathcal Y|+1\}.
$$
This is Theorem 1 of the paper [1703.08099].

The first term, $I(X,X_r;Y|S)$, is the cut-set type limitation. The second term combines the packing contribution $I(X;Y|X_r,Z,S,U)$ with a deterministic-link contribution $H(Z|X_r,S,U)$, penalized by the state-coordination cost $I(U;S)$. The feasibility condition $I(U;S)\le H(Z|X_r,S,U)$ states that the deterministic relay observation must support enough binning entropy to coordinate the auxiliary $U$ with the state [1703.08099].

The key structural point is that non-causal CSI changes the nature of cooperation. In causal settings, the cooperation codeword is independent of the state. In the non-causal setting, the encoder can use the states of the next block and choose a cooperation codeword accordingly. Since the relay transmits according to that cooperation codeword, its channel input becomes state-dependent even though the relay never observes the state directly. This implicit state transfer is the mechanism behind the capacity increase [1703.08099].

## 3. Cooperative-bin-forward achievability

The achievability scheme is based on **cooperative-bin-forward**. The deterministic relay output $Z^n$ is randomly binned:
$$
\mathrm{bin}: \mathcal Z^n \to [1:2^{nR_B}],
$$
with $\mathrm{bin}(z^n)$ chosen i.i.d. uniformly over the bin indices. For each bin index $l$, a cooperation codeword $u^n(l)$ is drawn i.i.d. according to $\prod_{i=1}^n p_U(u_i)$, and for each $u^n(l)$ a relay codeword $x_r^n(u^n(l))$ is drawn i.i.d. according to $\prod_{i=1}^n p_{X_r|U}(x_{r,i}|u_i(l))$ [1703.08099].

A further indirect covering layer is introduced through $z^n$-codewords indexed by a split message part and a covering index. These are drawn according to
$$
\prod_{i=1}^{n} p_{Z|X_r,U,S}(z_i|x_{r,i},u_i,s_i),
$$
consistent with the deterministic relation through
$$
p_{Z|X_r,U,S}\, p_{X|Z,X_r,U,S}
=
p_{X|X_r,U,S}\, 1_{Z|X,X_r,S}.
$$
Finally, transmission codewords are drawn according to
$$
\prod_{i=1}^{n} p_{X|Z,X_r,U,S}(x_i|z_i,x_{r,i},u_i,s_i).
$$

In block $b$, given the previous cooperation index and the current and next state blocks, the encoder searches for a covering index such that the cooperation codeword identified by the bin of the selected $z^n$ sequence is jointly typical with the next block’s state. It then transmits the corresponding $x^n$ codeword. The relay, in block $b$, transmits $x_r^n(u^n(l^{(b-1)}))$ and after observing the actual deterministic output sets
$$
l^{(b)} = \mathrm{bin}(Z^{n(b)}).
$$
Thus the relay follows the encoder’s cooperation pointer **without decoding message bits** [1703.08099].

Decoding is by sliding window. The decoder reconstructs, for each candidate partial message, the encoder’s binning-induced mapping and then declares the unique message pair satisfying joint typicality in one block and coordination in the next. The key intermediate constraints are:

- indirect covering:
$$
\tilde R > I(U;S),\qquad
\tilde R < H(Z|X_r,U,S),\qquad
R_B > \tilde R;
$$

- uniqueness of the pointer:
$$
R' + \tilde R < R_B,\qquad
R' + \tilde R < H(Z|X_r,U,S);
$$

- packing:
$$
R'' \le I(X;Y|X_r,Z,U,S),\qquad
R \le I(X,X_r;Y|S).
$$

Fourier-Motzkin elimination over $(R',R'',\tilde R,R_B)$ yields the single-letter capacity constraints. The auxiliary $U$ therefore serves as the coordination layer among $(S^n,U^n,X_r^n,X^n)$ [1703.08099].

A key tool is the **indirect covering lemma**. If $\{Z^n(k)\}$ are i.i.d. according to $\prod p_{Z|V}$ for a given $v^n$, and the sequences are independently binned uniformly into $2^{nR_B}$ bins, then under
$$
R < H(Z|V)-\delta_1,\qquad
R < R_B-\delta_2,
$$
the number of distinct bins observed is at least $2^{n(R-\delta_n)}$ with high probability. This furnishes enough bin diversity to associate bin indices with $U^n$ codewords and choose one jointly typical with the target state sequence [1703.08099].

## 4. Converse and the role of implicit state transfer

The converse begins with Fano’s inequality and the cut-set bound:
$$
nR \le I(M;Y^n|S^n) + n\epsilon_n
\le \sum_{i=1}^n I(X_i,X_{r,i};Y_i|S_i) + n\epsilon_n
= n[I(X,X_r;Y|S)+\epsilon_n].
$$
This recovers the first branch of the capacity expression [1703.08099].

The second branch uses determinism more directly. Define
$$
U_i = (Z^{i-1},S^{i-1}).
$$
Then
$$
H(Z^n,X_r^n|S^n)
=
\sum_{i=1}^n \big[ H(Z_i|X_{r,i},S_i,U_i) - I(U_i;S_i)\big]
=
n\big[ H(Z|X_r,S,U) - I(U;S)\big].
$$
Also,
$$
H(M|Z^n,X_r^n,S^n)
\le I(M;Y^n|Z^n,X_r^n,S^n) + n\epsilon_n
\le \sum_{i=1}^n I(X_i;Y_i|Z_i,X_{r,i},S_i,U_i) + n\epsilon_n,
$$
so that
$$
R \le I(X;Y|Z,X_r,S,U) + H(Z|X_r,S,U) - I(U;S) + \epsilon_n.
$$
The same argument yields the consistency condition
$$
I(U;S) \le H(Z|X_r,S,U).
$$
Thus the converse matches achievability [1703.08099].

Conceptually, the converse formalizes why the non-causal problem differs from the causal one. The relay never sees $S^n$, yet its transmission can still depend on the state through the bin-selected auxiliary $U^n$. The encoder chooses the bin index so that $U^n$ is typical with the **upcoming** state block, and the relay then transmits according to that $U^n$. A plausible implication is that the deterministic observation $Z$ functions as a constrained coordination resource: it is not merely a relay observation, but also the carrier of a state-adaptive cooperation pointer [1703.08099].

## 5. Extensions: partial cribbing and the state-encoder point-to-point model

For the MAC with partial cribbing and strictly causal cribbing, the capacity region is given over distributions
$$
p_{X_1,U|S_1}\, p_{X_2|U,S_2},
$$
with $Z=z(X_1,S_1)$ and
$$
I(U;S_1|S_2) \le H(Z|S_1,U),
$$
by the inequalities
$$
R_1 \le I(X_1;Y|X_2,Z,S_1,S_2,U) + H(Z|S_1,U) - I(U;S_1|S_2),
$$
$$
R_2 \le I(X_2;Y|X_1,S_1,S_2,U),
$$
$$
R_1 + R_2 \le I(X_1,X_2;Y|Z,S_1,S_2,U) + H(Z|S_1,U) - I(U;S_1|S_2),
$$
$$
R_1 + R_2 \le I(X_1,X_2;Y|S_1,S_2).
$$
For causal cribbing, the region is the same except that the distribution becomes
$$
p_{X_1,U|S_1}\, p_{X_2|Z,U,S_2},
$$
so Encoder 2 may condition on the instantaneous cribbed symbol [1703.08099].

The same cooperative-bin-forward logic is retained. Encoder 1 chooses the bin index to match $U^n$ to the next block’s $S_1^n$, and Encoder 2 uses $U^n$—and possibly the current $Z_i$ in the causal case—to adapt $X_2^n$. In this model, $U$ serves both as a coordination layer with the state and as a common layer in a superposition structure [1703.08099].

The point-to-point channel with a state encoder is a particularly revealing specialization. The state encoder observes $S^n$ and transmits $X_1^n$ over a private deterministic link to the main transmitter, which then sends $X_2^n$ over a state-dependent channel $p_{Y|X_2,S}$. The decoder knows $S^n$. If the main transmitter has strictly causal access to the state encoder’s outputs, the capacity is
$$
C_{\mathrm{nc}}
=
\max_{p_{U|S},\, p_{X_2|U}: \, I(U;S)\le \log_2 |\mathcal X_1|}
I(X_2;Y|U,S).
$$
If access is causal, then
$$
C_c
=
\max_{1_{X_1|S},\, p_{X_2|X_1}}
I(X_2;Y|S,X_1).
$$
The paper gives an explicit example with $S\in\{0,1,2\}$, $|\mathcal X_1|=2$, and $p=0.2$ showing
$$
C_{\mathrm{nc}} \approx 0.8644 > C_c \approx 0.8633 > \text{no-CSI} \approx 0.8623
$$
at $\alpha=0.5$. This directly refutes the common intuition that receiver-side state knowledge eliminates any benefit of non-causal state availability upstream: even when the decoder knows $S^n$, non-causal CSI at the state encoder can still be strictly better than causal CSI [1703.08099].

## 6. Reductions, comparisons, and conceptual significance

Several classical models are recovered as special cases. If the state is degenerate, the SD-RC formula reduces to the classic semi-deterministic relay capacity
$$
C = \max_{p_{X,X_r}}
\min\{ I(X,X_r;Y),\; H(Z|X_r) + I(X;Y|X_r,Z)\}.
$$
If cooperation is independent of state, or equivalently $U \perp S$, the non-causal formula reduces to the causal-bin-forward capacity
$$
C
=
\max
\min\{
I(X,X_r;Y|S),\;
H(Z|X_r,S) + I(X;Y|X_r,Z,S)
\}
$$
over $p_{X_r}p_{X|X_r,S}$. For the MAC, setting $Z$ constant recovers the no-cribbing case with non-causal state at Encoder 1 [1703.08099].

The work sits in direct continuity with the earlier paper "Cooperative Binning for Semideterministic Channels" [1508.05149]. That paper introduced cooperative-bin-forward as a generalization of partial-decode-forward for semideterministic multiuser channels, emphasizing that explicit recovery of a message part at the intermediate node is not necessary to induce cooperation. The 2017 non-causal extension shows why this distinction matters: partial-decode-forward becomes restrictive when side information is asymmetric, whereas cooperative-bin-forward continues to achieve capacity [1508.05149].

This comparison also clarifies the relation to other relay strategies. **Decode-forward** requires relay-side decoding and is suboptimal when the relay lacks the state known to the encoder. **Compress-forward** is designed for noisy relay observations and explicit compression of received signals, whereas in the present setting $Z$ is deterministic and therefore already predictable at the encoder. Cooperative-bin-forward instead uses exactly the deterministic structure available: the encoder dictates the relay’s observation, predicts its bin index, and uses that index as a pointer to a future cooperation codeword [1703.08099].

A broader implication is that the deterministic entropy term $H(Z|\cdot)$ acts as a coordination budget. The auxiliary $U$ consumes this budget through $I(U;S)$, and the non-causal encoder spends it to make a strictly causal terminal effectively state-adaptive. Within the paper’s scope, this is the semi-deterministic core: deterministic $Z$ enables bin-forward cooperation, and non-causal CSI enables look-ahead selection of cooperation codewords that compress state into a shared $U$-layer, thereby inducing state-dependent relay or cribbing behavior and achieving capacity [1703.08099].

Source: https://www.emergentmind.com/topics/semi2