DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

The extended transition function of an NFA

Definition

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

Its extended transition function is the map $\hat{δ} : P (Q) \times Σ^{*} \to P (Q)$ defined recursively by $\hat{δ} (S, ε) : = ECl (S)$ and $\hat{δ} (S, w a) : = ECl ⁣ (⋃_{q \in \hat{δ} (S, w)} δ (q, a))$ for every state set $S \subseteq Q$ , every word $w \in Σ^{*}$ , and every letter $a \in Σ$ .

For a single state $q \in Q$ we abbreviate $\hat{δ} (q, w) : = \hat{δ} ({q}, w) .$

Thus $\hat{δ} (S, w)$ is the set of states that can be reached by starting in some state of $S$ , taking any number of $ε$ -moves before and between letter moves and after the final letter move, and reading the word $w$ .

Depends on

The epsilon-closure of a set of NFA states

Used by

Subset construction tracks exactly the reachable NFA state set Lemma

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)