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discrete-memoryless-two-way-channel

Discrete memoryless two-way channel

Two terminals exchange messages while adapting each input to their own past observations.

Two-way Finite alphabet Discrete memoryless Feedback Capacity region Bounds only

Channel and question

Input
Terminal i chooses \(X_{i,t}\) from its message and past observations.
Output
Terminals receive \(Y_{1,t}\) and \(Y_{2,t}\).
Law
A memoryless law \(p(y_1,y_2|x_1,x_2)\).
Quantity
Capacity region \(\mathcal C_{\mathrm{TWC}}\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average error in both directions.

  • Encoding is interactive and causal.
  • Messages are independent.

Current status

\[\mathcal R_{\mathrm{Shannon,in}}\subseteq\mathcal C_{\mathrm{TWC}}\subseteq\mathcal R_{\mathrm{Shannon,out}}\]

The bounds coincide for several symmetric or decomposable channels.

ResultRelationMethodYear
Inner Region\(\mathcal R_{\mathrm{in}}\subseteq\mathcal C\)Independent stationary inputs without adaptation.1961
Outer Region\(\mathcal C\subseteq\mathcal R_{\mathrm{out}}\)Allow arbitrary correlated inputs in a cut-style single-letter outer bound.1961

Research frontier

Determine when interaction enlarges the two-way capacity region and characterize the general region.

Why it remains open. The terminals create input dependence dynamically through noisy observations, which is difficult to summarize by a single-letter distribution.

What would count as progress

  • Find a sharper dependence-aware converse.
  • Close an explicit channel where Shannon's bounds differ.
  • Formalize interactive two-way codes.

Lean formalization

Canonical statementNone

Version 1 · Lean. Interactive causal encoders are not yet in the shared Lean layer.

Substantial proofs0 linked

No external Lean proof is registered. Proofs longer than roughly 50 lines or requiring problem-specific infrastructure should live in a dedicated repository and link back to this statement version.

References

  1. Claude E. Shannon (1961). Two-Way Communication Channels. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability.
  2. Andries P. Hekstra and Frans M. J. Willems (1989). Dependence Balance Bounds for Single-Output Two-Way Channels. IEEE Transactions on Information Theory. DOI 10.1109/18.42175.
  3. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

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