Alphabeta Math
TheoremStatement: Literature-sourcedProof: 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.

CFG equivalence is undecidable

Statement

The CFG equivalence problem is undecidable.

Facts & Assumptions

Given: A context-free grammar G over a terminal alphabet Σ.

[L1]

The CFG equivalence problem asks whether two CFGs generate the same language, by The CFG equivalence and ambiguity problems.

[L2]

CFG universality is undecidable, by CFG universality is undecidable.

Proof

technique · direct
1.1

Let UΣ be a fixed grammar generating Σ∗. Such a grammar is easy to write, for example with productions S→aS∣ε(a∈Σ). Then L(G)=Σ∗  ⟺  L(G)=L(UΣ).

L1givenconstruct
2.1

Therefore a decision procedure for CFG equivalence would decide whether L(G)=Σ∗ by comparing G with the fixed universal grammar UΣ. That would decide CFG universality, contradicting [L2].

L2step 1.1contradiction
3.1

Hence CFG equivalence is undecidable.

step 2.1discharge-contradiction: equivalence with a fixed universal grammar would decide universality∎

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