Cooperative Binning in Semi-deterministic Channels

• The paper demonstrates that capacity is achieved through cooperative-bin-forward, using non-causal CSI to coordinate relay transmissions without explicit message decoding.
• It introduces an indirect covering method that bins deterministic outputs to select cooperation codewords, aligning future state information with relay actions.
• The study extends the approach to multiple-access and state-encoder models, proving the benefits of implicit state transfer for improved rate regions.

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 (Gattegno et al., 2017). 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 (Gattegno et al., 2017).

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 $M \in [1:2^{nR}]$0 before transmission, whereas the relay has no state information. The relay is strictly causal: at time $M \in [1:2^{nR}]$1, $M \in [1:2^{nR}]$2 is a function of $M \in [1:2^{nR}]$3 (Gattegno et al., 2017).

A code for this model consists of

$M \in [1:2^{nR}]$4

with average error probability $M \in [1:2^{nR}]$5 for sufficiently large $M \in [1:2^{nR}]$6. The operational structure is block-Markov: transmission is divided into $M \in [1:2^{nR}]$7 blocks of length $M \in [1:2^{nR}]$8, and the deterministic relay output in block $M \in [1:2^{nR}]$9 determines, through binning, a cooperation index used in block $S^n=(S_1,\dots,S_n)$0 (Gattegno et al., 2017).

The paper also studies the multiple-access channel with partial cribbing as a semi-deterministic channel. Encoder 1 knows $S^n=(S_1,\dots,S_n)$1 non-causally, Encoder 2 knows $S^n=(S_1,\dots,S_n)$2 non-causally, and the decoder knows both. Encoder 2 deterministically cribs

$S^n=(S_1,\dots,S_n)$3

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 $S^n=(S_1,\dots,S_n)$4 (Gattegno et al., 2017).

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

$S^n=(S_1,\dots,S_n)$5

where the maximization is over distributions of the form

$S^n=(S_1,\dots,S_n)$6

subject to

$S^n=(S_1,\dots,S_n)$7

with $S^n=(S_1,\dots,S_n)$8 deterministic. A cardinality bound is

$S^n=(S_1,\dots,S_n)$9

This is Theorem 1 of the paper (Gattegno et al., 2017).

The first term, $p_S$0, is the cut-set type limitation. The second term combines the packing contribution $p_S$1 with a deterministic-link contribution $p_S$2, penalized by the state-coordination cost $p_S$3. The feasibility condition $p_S$4 states that the deterministic relay observation must support enough binning entropy to coordinate the auxiliary $p_S$5 with the state (Gattegno et al., 2017).

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 (Gattegno et al., 2017).

3. Cooperative-bin-forward achievability

The achievability scheme is based on cooperative-bin-forward. The deterministic relay output $p_S$6 is randomly binned:

$p_S$7

with $p_S$8 chosen i.i.d. uniformly over the bin indices. For each bin index $p_S$9, a cooperation codeword $S_i$0 is drawn i.i.d. according to $S_i$1, and for each $S_i$2 a relay codeword $S_i$3 is drawn i.i.d. according to $S_i$4 (Gattegno et al., 2017).

A further indirect covering layer is introduced through $S_i$5-codewords indexed by a split message part and a covering index. These are drawn according to

$S_i$6

consistent with the deterministic relation through

$S_i$7

Finally, transmission codewords are drawn according to

$S_i$8

In block $S_i$9, 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_i = z(X_i,X_{r,i},S_i),$0 sequence is jointly typical with the next block’s state. It then transmits the corresponding $Z_i = z(X_i,X_{r,i},S_i),$1 codeword. The relay, in block $Z_i = z(X_i,X_{r,i},S_i),$2, transmits $Z_i = z(X_i,X_{r,i},S_i),$3 and after observing the actual deterministic output sets

$Z_i = z(X_i,X_{r,i},S_i),$4

Thus the relay follows the encoder’s cooperation pointer without decoding message bits (Gattegno et al., 2017).

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:

$Z_i = z(X_i,X_{r,i},S_i),$5

• uniqueness of the pointer:

$Z_i = z(X_i,X_{r,i},S_i),$6

• packing:

$Z_i = z(X_i,X_{r,i},S_i),$7

Fourier-Motzkin elimination over $Z_i = z(X_i,X_{r,i},S_i),$8 yields the single-letter capacity constraints. The auxiliary $Z_i = z(X_i,X_{r,i},S_i),$9 therefore serves as the coordination layer among $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$0 (Gattegno et al., 2017).

A key tool is the indirect covering lemma. If $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$1 are i.i.d. according to $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$2 for a given $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$3, and the sequences are independently binned uniformly into $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$4 bins, then under

$p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$5

the number of distinct bins observed is at least $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$6 with high probability. This furnishes enough bin diversity to associate bin indices with $p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$7 codewords and choose one jointly typical with the target state sequence (Gattegno et al., 2017).

4. Converse and the role of implicit state transfer

The converse begins with Fano’s inequality and the cut-set bound:

$p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$8

This recovers the first branch of the capacity expression (Gattegno et al., 2017).

The second branch uses determinism more directly. Define

$p_{Y|X,X_r,Z,S}(y_i|x_i,x_{r,i},z_i,s_i).$9

Then

$p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$0

Also,

$p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$1

so that

$p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$2

The same argument yields the consistency condition

$p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$3

Thus the converse matches achievability (Gattegno et al., 2017).

Conceptually, the converse formalizes why the non-causal problem differs from the causal one. The relay never sees $p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$4, yet its transmission can still depend on the state through the bin-selected auxiliary $p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$5. The encoder chooses the bin index so that $p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$6 is typical with the upcoming state block, and the relay then transmits according to that $p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$7. A plausible implication is that the deterministic observation $p_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$8 functions as a constrained coordination resource: it is not merely a relay observation, but also the carrier of a state-adaptive cooperation pointer (Gattegno et al., 2017).

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_{Y,Z|X,X_r,S} = 1_{Z|X,X_r,S}\, p_{Y|X,X_r,Z,S},$9

with $1_{Z|X,X_r,S}$0 and

$1_{Z|X,X_r,S}$1

by the inequalities

$1_{Z|X,X_r,S}$2

$1_{Z|X,X_r,S}$3

$1_{Z|X,X_r,S}$4

$1_{Z|X,X_r,S}$5

For causal cribbing, the region is the same except that the distribution becomes

$1_{Z|X,X_r,S}$6

so Encoder 2 may condition on the instantaneous cribbed symbol (Gattegno et al., 2017).

The same cooperative-bin-forward logic is retained. Encoder 1 chooses the bin index to match $1_{Z|X,X_r,S}$7 to the next block’s $1_{Z|X,X_r,S}$8, and Encoder 2 uses $1_{Z|X,X_r,S}$9—and possibly the current $z(\cdot)$0 in the causal case—to adapt $z(\cdot)$1. In this model, $z(\cdot)$2 serves both as a coordination layer with the state and as a common layer in a superposition structure (Gattegno et al., 2017).

The point-to-point channel with a state encoder is a particularly revealing specialization. The state encoder observes $z(\cdot)$3 and transmits $z(\cdot)$4 over a private deterministic link to the main transmitter, which then sends $z(\cdot)$5 over a state-dependent channel $z(\cdot)$6. The decoder knows $z(\cdot)$7. If the main transmitter has strictly causal access to the state encoder’s outputs, the capacity is

$z(\cdot)$8

If access is causal, then

$z(\cdot)$9

The paper gives an explicit example with $M \in [1:2^{nR}]$00, $M \in [1:2^{nR}]$01, and $M \in [1:2^{nR}]$02 showing

$M \in [1:2^{nR}]$03

at $M \in [1:2^{nR}]$04. 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 $M \in [1:2^{nR}]$05, non-causal CSI at the state encoder can still be strictly better than causal CSI (Gattegno et al., 2017).

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

$M \in [1:2^{nR}]$06

If cooperation is independent of state, or equivalently $M \in [1:2^{nR}]$07, the non-causal formula reduces to the causal-bin-forward capacity

$M \in [1:2^{nR}]$08

over $M \in [1:2^{nR}]$09. For the MAC, setting $M \in [1:2^{nR}]$10 constant recovers the no-cribbing case with non-causal state at Encoder 1 (Gattegno et al., 2017).

The work sits in direct continuity with the earlier paper "Cooperative Binning for Semideterministic Channels" (Kolte et al., 2015). 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 (Kolte et al., 2015).

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 $M \in [1:2^{nR}]$11 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 (Gattegno et al., 2017).

A broader implication is that the deterministic entropy term $M \in [1:2^{nR}]$12 acts as a coordination budget. The auxiliary $M \in [1:2^{nR}]$13 consumes this budget through $M \in [1:2^{nR}]$14, 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 $M \in [1:2^{nR}]$15 enables bin-forward cooperation, and non-causal CSI enables look-ahead selection of cooperation codewords that compress state into a shared $M \in [1:2^{nR}]$16-layer, thereby inducing state-dependent relay or cribbing behavior and achieving capacity (Gattegno et al., 2017).