CounterexampleConstruction: AI-adaptedVerification: 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.

A reduction to the complement is not a reduction to the original target

Statement refuted

A computable many-one reduction may reverse yes-instances and no-instances.

Facts & Assumptions

Given: $A = {0}$ , $B = {1}$ , and the computable map $r$ on ${0, 1}^{*}$ defined by $r (0) = 0$ and $r (x) = 1$ for $x \neq 0$ .

[L1]

A many-one reduction to $B$ must satisfy $x \in A ⟺ f (x) \in B$ , by Computable many-one reductions between languages.

[L2]

The false slogan being refuted is recorded in FALSE: a valid many-one reduction may reverse yes-instances and no-instances.

Counterexample

technique · direct

1.1

The map $r$ reverses membership relative to $B$ : the yes-instance $0 \in A$ is sent to $0 \notin B$ , while every no-instance $x \notin A$ is sent to $1 \in B$ . In particular, already at $x = 0$ the biconditional required by [L1] fails, so $r$ is not a many-one reduction from $A$ to $B$ .

L1given

2.1

The same map does satisfy $x \in A ⟺ r (x) \in \overline{B}$ , so it is a reduction from $A$ to $\overline{B}$ . Thus an answer-reversing transformation changes the target to its complement and is not a valid reduction to $B$ ; this refutes [L2].

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)