TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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The Turing-machine acceptance problem is computably enumerable complete

Statement

The language $A_{T M}$ of The Turing-machine acceptance problem $A_{T M}$ is c.e.-complete under computable many-one reductions.

Facts & Assumptions

Given: The acceptance language $A_{T M}$ .

[L1]

A language is c.e.-complete exactly when it is recognizable and every recognizable language many-one reduces to it, by Computably enumerable hardness and completeness under computable many-one reductions.

[L2]

The language $A_{T M}$ is recognizable, by The Turing-machine acceptance problem is recognizable.

[L3]

A coded pair $⟨ ⌜ M ⌝, \hat{w} ⟩$ belongs to $A_{T M}$ exactly when the coded machine $M$ accepts the word $w$ , by The Turing-machine acceptance problem $A_{T M}$ .

Proof

technique · direct

1.1

By [L2], $A_{T M}$ is recognizable. To use [L1], it remains to prove c.e.-hardness.

L1L2given

1.2

Let $L$ be any recognizable language, and choose a recognizer $R$ for $L$ . Define $f (x) : = ⟨ ⌜ R ⌝, \hat{x} ⟩$ . Because $R$ is fixed, encoding its chosen code and pairing it with the canonical word-code of $x$ is a total computable procedure. By [L3], $x \in L$ exactly when $R$ accepts $x$ , and that happens exactly when $f (x) \in A_{T M}$ . Thus $L \leq_{m} A_{T M}$ .

L3givenconstruct

2.1

Step 1.2 proves that every recognizable language many-one reduces to $A_{T M}$ , and step 1.1 already recorded recognizability. Therefore [L1] gives that $A_{T M}$ is c.e.-complete.

L1step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

13 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

Kevin Kelly, Many-one Reduction (standard reference, not scraped)
MIT 6.045J / 18.400J, Lecture 9: Mapping Reducibility and Rice's Theorem (standard reference, not scraped)