Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-12
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.

Context-free-language membership is decidable

Statement

Given a context-free grammar G and a word w, one can decide whether w∈L(G).

Facts & Assumptions

Given: A context-free grammar G and a word w.

[L1]

By Every context-free language has an equivalent grammar in Chomsky normal form, G has an equivalent grammar in Chomsky normal form.

[L2]

By The CYK algorithm decides membership for CNF grammars, the CYK procedure decides membership for CNF grammars.

Proof

technique · direct
1.1

Convert G to an equivalent CNF grammar G′ using [L1].

L1givenconstruct
2.1

Run the CYK membership procedure from [L2] on G′ and w. Because G′ is equivalent to G, the answer is "yes" exactly when w∈L(G).

L2step 1.1
3.1

Therefore context-free-language membership is decidable.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

8 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