ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Capacity of the binary erasure channel

Example

The binary erasure channel outputs the input bit with probability $1 - ϵ$ and the erasure symbol $e$ with probability $ϵ$ , independently at each use. For $ϵ \in [0, 1]$ , its capacity is $(1 - ϵ) \ln 2$ nats per use, attained by a uniform input.

Facts & Assumptions

Given: A binary erasure channel with erasure probability $ϵ \in [0, 1]$ and input law $P (X = 1) = r$ .

[F1]

Finite discrete memoryless channels: conditional on $X = x$ , the output is $x$ with probability $1 - ϵ$ and $e$ with probability $ϵ$ .

[F2]

Operational capacity formula for a finite DMC: capacity is the maximum of finite mutual information over input laws.

[F3]

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

[F4]

Finite conditional entropy and Finite distributions, entropy, divergence, and stochastic matrices: conditional entropy averages the entropy of each positive-mass conditional law; zero coordinates contribute $0 \ln 0 = 0$ .

[F5]

Finite KL divergence is nonnegative with equality exactly at equal laws: for a binary law $Bern (r)$ , $D (Bern (r) ∥ Bern (1 / 2)) = \ln 2 - h (r) \geq 0$ , with equality at $r = 1 / 2$ .

Verification

Given: $X \sim Bern (r)$ and the independent erasure mechanism.

1.1F1F3F4givencases

On the events $Y = 0$ and $Y = 1$ , $X$ is known exactly. The erasure indicator is independent of $X$ because [F1] gives the same conditional erasure probability $ϵ$ for both input symbols. Hence on $Y = e$ the conditional law of $X$ remains $Bern (r)$ . By [F4], $H (X ∣ Y) = ϵ h (r)$ , and [F3] gives $I (X; Y) = (1 - ϵ) h (r)$ .

2.1F2F5step 1.1algebracases∎

By [F5], $h (r) \leq \ln 2$ , with equality at $r = 1 / 2$ . Therefore the maximum mutual information is $(1 - ϵ) \ln 2$ by [F2]. At $ϵ = 0$ the channel is noiseless and has capacity $\ln 2$ ; at $ϵ = 1$ its output is constant and the capacity is $0$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

MIT 6.441 Information Theory, Lecture 8, Binary Erasure Channel (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 19 §19.3 (standard reference, not scraped)