DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Acceptance of a word by an NFA and the recognized language

Definition

Let $N = (Q, Σ, δ, q_{0}, F)$ be an NFA with epsilon-moves.

A word $w \in Σ^{*}$ is accepted by $N$ when $\hat{δ} (q_{0}, w) \cap F \neq \emptyset .$

The language recognized by $N$ is $L (N) : = {w \in Σ^{*} : \hat{δ} (q_{0}, w) \cap F \neq \emptyset} .$ This is a language over $Σ$ in the sense of Languages over an alphabet.

Remarks

NFA acceptance is existential: one accepting computation path is enough.
Rejection means that no computation path ends in an accepting state after the whole input has been read.

Depends on

Languages over an alphabet

Used by

The subset construction preserves the recognized language Theorem

Dependency tree · two levels

3 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)
John Watrous, Introduction to the Theory of Computing, Lecture 3: Nondeterministic finite automata (standard reference, not scraped)