ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-02

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.

Reducing halting to language equivalence by a machine that either accepts everything or loops forever

Example

Fix a coded pair $⟨ ⌜ M ⌝, \hat{w} ⟩$ . The standard reduction to machine equivalence builds a machine $E_{M, w}$ that ignores its own input, waits to see whether $M$ halts on $w$ , and then either accepts every input or accepts none.

Facts & Assumptions

Given: A fixed halting instance candidate $⟨ ⌜ M ⌝, \hat{w} ⟩$ .

[L1]

In the reduction proving Language equivalence of Turing machines is undecidable, the machine $E_{M, w}$ recognizes $Σ^{*}$ when $M$ halts on $w$ and recognizes $\emptyset$ when $M$ does not halt on $w$ .

[L2]

The halting language $H A L T_{T M}$ is undecidable, by The halting problem is recognizable and undecidable.

Verification

technique · direct

1.1

If $M$ halts on $w$ , then the simulation inside $E_{M, w}$ eventually terminates on every input, after which $E_{M, w}$ accepts that input. So [L1] gives $L (E_{M, w}) = Σ^{*}$ , the same language recognized by an always-accept machine.

L1given

2.1

If $M$ does not halt on $w$ , then the simulation inside $E_{M, w}$ never terminates, so $E_{M, w}$ accepts no input at all. Again by [L1], $L (E_{M, w}) = \emptyset$ , which is not the language of the always-accept machine. This sharp dichotomy is why deciding equivalence would decide halting, contradicting [L2].

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Stan Nelson, Computability: Undecidable Problems (standard reference, not scraped)