DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 quotient DFA by state equivalence

Definition

Let $D = (Q, Σ, δ, q_{0}, F)$ be a DFA, and let $\equiv_{D}$ be the state-equivalence relation of The right language of a DFA state and state equivalence. Its quotient set $Q / \equiv_{D}$ is available by State-equivalent DFA states stay equivalent after every input word together with Equivalence relation, equivalence class, and the quotient set $A / \sim$ .

The quotient DFA by state equivalence is the automaton $D / \equiv_{D} : = (Q / \equiv_{D}, Σ, \overline{δ}, [q_{0}], \overline{F}),$ where $\overline{δ} ([q], a) : = [δ (q, a)]$ and $\overline{F} : = {[q] \in Q / \equiv_{D} : q \in F} .$

The next item proves that these definitions do not depend on the chosen representatives and that the quotient recognizes the same language as $D$ .

Depends on

Used by

Dependency tree · two levels

17 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)