CorollaryStatement: Literature-sourcedProof: AI-generatedverified 2026-09-23 (gpt-6-sol)

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.

DFAs and NFAs recognize the same languages

Statement

Let $Σ$ be an alphabet and let $L \subseteq Σ^{*}$ .

Then $L$ is recognized by some NFA over $Σ$ if and only if $L$ is regular.

Facts & Assumptions

Given: A language $L \subseteq Σ^{*}$ .

[L1]

By Regular languages, $L$ is regular exactly when some DFA over $Σ$ recognizes $L$ .

[L2]

Every total DFA embeds as an epsilon-NFA with the same recognized language (A DFA embeds as an epsilon-free NFA).

[L3]

The exact subset-construction invariant produces a total DFA recognizing the language of each epsilon-NFA (Subset construction and its exact reachability invariant).

[L4]

The NFA recognition convention is the finite epsilon-closure and word-transition construction (Epsilon-NFAs, finite epsilon closure, and extended word transition).

Proof

technique · direct

1.1L1L2L4given

If $L$ is regular, then [L1] gives a DFA recognizing $L$ , and [L2] turns that DFA into an epsilon-NFA recognizing the same language under [L4].

1.2L1L3L4given

If some NFA recognizes $L$ , interpret it with [L4]. Then [L3] gives a total DFA recognizing the same language, and [L1] therefore shows that $L$ is regular.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove that $L$ is recognized by an NFA if and only if $L$ is regular.

Depends on

Used by

Regular languages are closed under left and right quotient with regular languages Theorem

Dependency tree · two levels

18 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)