Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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: decidable CFLs have a decidable intersection-emptiness problem

Statement

Because every context-free language has decidable membership, there is an algorithm that decides from two CFGs whether their intersection is empty.

Facts & Assumptions

Given: Two context-free grammars G and H.

[L1]

Membership in each context-free language is decidable, by Context-free-language membership is decidable.

[L2]

The problem of deciding whether L(G)∩L(H)=∅ is undecidable, by Emptiness of the intersection of two CFGs is undecidable.

Refutation

technique · direct
1.1

Fact [L1] says that for any fixed word w, one can decide separately whether w∈L(G) and whether w∈L(H). That is a pointwise membership statement about the two languages themselves.

L1given
2.1

The claim in the statement is stronger: it asks for one algorithm that, from the grammar descriptions alone, decides whether some word lies in both languages. But [L2] says that this description-level problem is undecidable.

L2step 1.1
3.1

Hence the statement is false. Decidable membership for each individual CFL does not make the intersection-emptiness problem for pairs of CFGs decidable.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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