ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cost-constrained capacity of a binary identity channel with feedback

Example

Let $X = Y = {0, 1}$ , let $W (y ∣ x) = 1_{{y = x}}$ , and let $c (0) = 0, c (1) = 1$ . For every budget $Γ \geq 0$ , both ordinary and feedback capacities are $C (Γ) = C_{f b} (Γ) = h_{2} (\min {Γ, 1 / 2}),$ where $h_{2} (p) = - p \ln p - (1 - p) \ln (1 - p)$ and $0 \ln 0 = 0$ . In particular the values at $Γ = 0, 1 / 2, 1$ are $0, \ln 2, \ln 2$ .

Facts & Assumptions

Given: The binary identity transition law, letter costs $0, 1$ , and a nonnegative budget $Γ$ .

[F1]

Finite DMC with a nonnegative input cost and feasible budget: an input law with $p = P_{X} (1)$ has expected cost $p$ .

[F2]

Finite entropy and mutual-information chain rules: for the identity channel, $I (X; Y) = H (X) - H (X ∣ Y) = H (X)$ .

[F3]

Finite-simplex entropy is continuous and concave: binary entropy is continuous, concave, and symmetric under $p \mapsto 1 - p$ by its formula.

[F4]

Average-cost fixed-length codes and operational DMC capacity, Average-cost capacity formula for a finite DMC, and Causal feedback does not increase finite DMC capacity under average cost: ordinary and feedback capacities use the stated average-cost conventions and equal the maximum feasible mutual information: ordinary and feedback capacities equal the maximum feasible mutual information.

[F5]

The natural logarithm as the inverse of the exponential function and Finite distributions, entropy, divergence, and stochastic matrices: entropy is in natural-log nats, with $0 \ln 0 = 0$ .

Verification

1.1F1F2F3F5givenalgebra

An input law is feasible exactly when $0 \leq p \leq \min {Γ, 1}$ , and [F2] makes its mutual information $h_{2} (p)$ . If $0 \leq p \leq q \leq 1 / 2$ and $p < 1 / 2$ , put $θ = (1 - p - q) / (1 - 2 p) \in [1 / 2, 1]$ . Then $q = θ p + (1 - θ) (1 - p)$ ; concavity and symmetry in [F3] imply $h_{2} (q) \geq θ h_{2} (p) + (1 - θ) h_{2} (1 - p) = h_{2} (p)$ . Thus $h_{2}$ is nondecreasing on $[0, 1 / 2]$ .

2.1F1F3F5step 1.1givenalgebra

For any $p \in [0, 1]$ , concavity and symmetry give $h_{2} (1 / 2) \geq \frac{1}{2} h_{2} (p) + \frac{1}{2} h_{2} (1 - p) = h_{2} (p)$ . Therefore the maximum over the feasible interval occurs at $p = \min {Γ, 1 / 2}$ and equals the displayed formula. Substitution yields $C (0) = 0$ , $C (1 / 2) = \ln 2$ , and $C (1) = \ln 2$ .

3.1F4step 2.1given∎

The feedback theorem in [F4] makes the feedback capacity equal to the same ordinary constrained capacity. This identifies the values for all three requested budgets.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 20 §20.1 (standard reference, not scraped)
Yury Polyanskiy, MIT 6.441 Information Theory, Chapter 21 §21.1 (standard reference, not scraped)