CounterexampleConstruction: AI-generatedVerification: 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.

One accepting branch is enough for an NFA to accept

Statement refuted

The statement "an NFA accepts a word only if every branch accepts" is false.

Facts & Assumptions

Given: The NFA and input word $a b b$ from FALSE: an NFA accepts only if every branch accepts.

[A1]

The statement refuted is: an NFA accepts a word only if every branch accepts.

[L1]

The false statement's refutation already exhibits one accepting branch and one rejecting branch on the same input, by FALSE: an NFA accepts only if every branch accepts.

Counterexample

technique · direct

1.1

On the input $a b b$ , one branch follows the path $q_{0} \overset{a}{\to} q_{1} \overset{b}{\to} q_{2} \overset{b}{\to} q_{3}$ and accepts, while another branch stays on the start-state loop too long and rejects.

L1given

2.1

Therefore the same machine and the same word witness that acceptance can occur even when not every branch accepts.

step 1.1

3.1

This contradicts [A1], so the displayed NFA on $a b b$ is a counterexample.

A1step 2.1∎

Depends on

FALSE: an NFA accepts only if every branch accepts

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)