DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

The halting problem $H A L T_{T M}$

Definition

Use the same pairing convention $⟨ x, y ⟩$ and input-word code $\hat{w}$ as in The Turing-machine acceptance problem $A_{T M}$ . The halting problem is the language $H A L T_{T M} : = {⟨ ⌜ M ⌝, \hat{w} ⟩ : M is a coded deterministic one-tape Turing machine and M halts on w} .$

Equivalently, $⟨ ⌜ M ⌝, \hat{w} ⟩ \in H A L T_{T M}$ when the computation of $M$ on $w$ has either an accepting or a rejecting halting history in the sense of Finite computation histories, halting computations, and divergence.

Remarks

The distinction between $H A L T_{T M}$ and $A_{T M}$ is exactly the distinction between halting and accepting.
Because rejection counts as halting, one always has $A_{T M} \subseteq H A L T_{T M}$ .

Depends on

Used by

Halting on every input is not recognizable Theorem
The halting problem is recognizable and undecidable Theorem
There exists a language that is not recognizable Theorem

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

Richard Zach, Sets, Logic, Computation: An Open Introduction to Metalogic (standard reference, not scraped)
Stan Nelson, Computability: Undecidable Problems (standard reference, not scraped)

The halting problem HALTTMHALTTM​