Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

FALSE: CYK needs no normal-form preprocessing

Statement

False claim: the CYK algorithm can be applied directly to any context-free grammar without first converting it to a suitable normal form.

Facts & Assumptions

Given: The grammar S→aSb∣ab.

[A1]

The statement refuted is: the CYK algorithm can be applied directly to any context-free grammar without first converting it to a suitable normal form.

[L1]

By The CYK parsing table and span variables, the CYK table recurrence uses only productions of the forms A→a and A→BC.

[L2]

By The CYK algorithm decides membership for CNF grammars, the CYK algorithm is proved correct exactly for grammars in the required CNF shape.

[L3]

By Every context-free language has an equivalent grammar in Chomsky normal form, an arbitrary context-free grammar can first be converted to an equivalent CNF grammar.

Refutation

technique · direct
1.1

In the displayed grammar, the production S→aSb has length three and mixes terminals with a variable, while S→ab has two terminals on the right-hand side. Neither production has one of the two forms listed in [L1].

L1given
2.1

Therefore the literal CYK recurrence does not know how to place S into any table cell directly from those productions, so the proof of correctness in [L2] does not apply. One must first normalize the grammar as in [L3], or else modify the algorithm itself.

L2L3step 1.1
3.1

This contradicts [A1]. So CYK does require the normal-form preprocessing built into its proof.

A1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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