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

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.

Computable many-one reductions transfer decidability and recognizability backward

Statement

If $A \leq_{m} B$ , then:

whenever $B$ is decidable, $A$ is decidable, and
whenever $B$ is recognizable, $A$ is recognizable.

Facts & Assumptions

Given: Languages $A \subseteq Σ^{*}$ and $B \subseteq Γ^{*}$ with $A \leq_{m} B$ .

[L1]

A computable many-one reduction is a total computable function $f$ with $x \in A ⟺ f (x) \in B$ for every $x$ , by Computable many-one reductions between languages.

[L2]

A language is decidable when some Turing machine halts on every input and answers membership correctly, and recognizable when some Turing machine accepts exactly its members, by Decidable and recognizable languages.

Proof

technique · direct

1.1

Let $f$ witness $A \leq_{m} B$ . By [L1], $f$ is total and computable and satisfies $x \in A ⟺ f (x) \in B$ for all $x$ .

L1given

2.1

If $B$ is decidable, [L2] supplies a decider $D_{B}$ for $B$ . On input $x$ , first compute $f (x)$ and then run $D_{B}$ on that word. By step 1.1 the output is correct for $A$ , and totality of $f$ and $D_{B}$ makes the composite machine halt on every input. Therefore $A$ is decidable.

L1L2step 1.1construct

3.1

If $B$ is recognizable, [L2] supplies a recognizer $R_{B}$ for $B$ . On input $x$ , compute $f (x)$ and run $R_{B}$ on it. By step 1.1 this machine accepts exactly the words of $A$ , so $A$ is recognizable.

L1L2step 1.1construct∎

Depends on

Used by

Dependency tree · two levels

4 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

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