CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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 recognizer for {0} that loops on every nonmember

Statement refuted

Whenever a machine recognizes a language, it halts on every input.

Facts & Assumptions

Given: The one-word language ${0} \subseteq {0, 1}^{*}$ .

[L1]

By Decidable and recognizable languages, recognizers are required to accept the members of their language, but they may diverge on nonmembers.

[L2]

By FALSE: recognizable means total halting, the statement claims that such divergence cannot occur.

Counterexample

technique · direct

1.1

Let $M$ accept immediately on input 0 and otherwise move right forever without entering any halting state. Then [L1] shows that $M$ recognizes exactly ${0}$ .

L1given

2.1

Every nonmember of ${0}$ makes $M$ diverge, so $M$ is not total. This contradicts the claim summarized in [L2].

L2step 1.1

3.1

Therefore $M$ is a concrete counterexample to the refuted statement.

step 1.1step 2.1∎

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

John Watrous, Introduction to the Theory of Computing, Lecture 18: Further discussion of computability (standard reference, not scraped)