Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-12
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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.

FALSE: CFG equivalence is decidable by normalization

Statement

False claim: converting context-free grammars to normal form gives a decision procedure for the CFG equivalence problem.

Facts & Assumptions

Given: The CFG equivalence problem.

[A1]

The statement refuted is: converting context-free grammars to normal form gives a decision procedure for the CFG equivalence problem.

[L1]

By The CFG equivalence and ambiguity problems, CFG equivalence asks whether two grammars over the same alphabet generate exactly the same language.

[F1]

Aho's cited Lecture 11 lists the problem “do two given CFG's generate the same language?” among the undecidable problems for context-free grammars.

Refutation

technique · direct
1.1

Normalization preserves the language generated by a grammar, so it may be used as preprocessing for [L1], but it does not produce a canonical normal form whose literal equality settles language equality.

L1given
2.1

More decisively, [F1] states that the general CFG equivalence problem is undecidable. So no normalization-only method can decide it in full generality.

F1step 1.1
3.1

This contradicts [A1]. Therefore CFG equivalence is not decidable merely by normalization.

A1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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