Alphabeta MathHow 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.
Binary prefix codes and their code trees
Definition
A binary code is a prefix code if it is nonsingular and no codeword is a proper prefix of a different codeword . A word is a proper prefix of when for some nonempty binary word . Nonsingular means whenever .
The code tree is the finite rooted trie whose vertices are all prefixes of codewords, whose root is , and whose edges append one bit, labeled or . Since the code is prefix-free, its codewords are exactly the leaves; unused branches are absent. A binary word path from the root records its successive edge labels.
Depends on
Used by
- Deterministic binary Huffman coding Algorithm
- Exact finite-register arithmetic coding for rational block sources Algorithm
- A uniquely decodable binary code that is not prefix-free Example
- Huffman tree for a dyadic three-symbol source Example
- Every finite binary prefix code is uniquely decodable Lemma
- Binary Kraft criterion for finite positive codeword lengths Theorem
- Ideal-real interval coding is prefix-free and lossless Theorem
- The deterministic Huffman code is prefix-optimal Theorem
- The rational finite-register arithmetic code is prefix-free and lossless Theorem
Dependency tree · two levels
2 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
- Stanford EE376A, Information Theory Course Notes, Chapter 4: Lossless Compression (standard reference, not scraped)
- MIT OpenCourseWare 6.450, Memory-less sources, prefix free codes, and entropy (standard reference, not scraped)