DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

How statement and proof provenance work

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The epsilon-closure of a set of NFA states

Definition

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

The epsilon-closure of $S$ , written $ECl (S)$ , is the set of all states $q \in Q$ for which there exist a state $s \in S$ and a finite sequence of states $s = r_{0}, r_{1}, \dots, r_{n} = q$ such that $r_{i + 1} \in δ (r_{i}, ε)$ for every $0 \leq i < n$ .

Equivalently, $ECl (S)$ is the least subset $T \subseteq Q$ with $S \subseteq T$ and $δ (t, ε) \subseteq T$ for every $t \in T$ .

For a single state $q \in Q$ we abbreviate $ECl (q) : = ECl ({q}) .$

Depends on

Used by

The extended transition function of an NFA Definition
The subset-construction DFA Definition
Epsilon-closure is a closure operator Lemma
Epsilon-moves can be eliminated from NFAs Theorem

Dependency tree · two levels

7 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)