TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Some Turing reductions are not truth-table reductions

Statement

There are sets $Y, X \subseteq N$ with $Y \leq_{T} X$ but $Y ̸ \leq_{t t} X$ .

Facts & Assumptions

Given: the non-computable domination of $0^{'}$ and the characterization of truth-table reducibility.

Proof

technique · contradiction

1.1given

Let $X = 0^{'}$ . It is not computably dominated by The halting oracle is not computably dominated.

2.1step 1.1assume-contra

If every $Y \leq_{T} X$ were truth-table reducible to $X$ , the reverse direction of Computable domination characterizes truth-table reducibility would make $X$ computably dominated.

3.1step 2.1discharge-contradiction∎

This contradiction supplies a set $Y \leq_{T} X$ with $Y ̸ \leq_{t t} X$ .

Depends on

Used by

A Turing reduction without a truth table Example

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

Ludovic Patey, Computability Theory, Proposition 5.13 (standard reference, not scraped)