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 Turing-machine acceptance problem $A_{T M}$

Definition

Fix the chosen machine code $⌜ M ⌝$ from A fixed effective binary encoding of deterministic one-tape Turing machines. For a binary word $z$ , write $c (z) : = 1^{∣ z ∣} 0 z .$ For binary words $x, y$ , write $⟨ x, y ⟩ : = 1^{∣ x ∣} 0 x y .$ If $w = a_{1} \dots a_{ℓ} \in Σ_{s}^{*}$ , where each $a_{i} \in {1, \dots, s}$ , put $\hat{w} : = c (bin (ℓ)) c (bin (a_{1})) \dots c (bin (a_{ℓ})) .$

The Turing-machine acceptance problem is the language $A_{T M} : = {⟨ ⌜ M ⌝, \hat{w} ⟩ : M is a coded deterministic one-tape Turing machine and M accepts w} .$

Remarks

The member of $A_{T M}$ is the binary code of a machine/input pair, not the machine or word themselves.
The empty input word is allowed: if $w = ε$ , then $\hat{w} = c (bin (0))$ .

Depends on

Used by

An immediate-accept machine gives an easy yes-instance inside an undecidable language Counterexample
The halting problem HALT_TM Definition
FALSE: undecidable means no individual instance can be answered False statement
A hypothetical decider for A_TM yields a diagonal self-application machine Lemma
The Turing-machine acceptance problem is recognizable Theorem
The Turing-machine acceptance problem is undecidable Theorem

Dependency tree · two levels

8 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)
EECS 376 Course Notes, Part 6: Computability (standard reference, not scraped)

The Turing-machine acceptance problem ATMATM​