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 right language of a DFA state and state equivalence

Definition

Let $D = (Q, Σ, δ, q_{0}, F)$ be a DFA with extended transition function $δ^{*}$ .

For a state $q \in Q$ , the right language of $q$ is $R_{D} (q) : = {w \in Σ^{*} : δ^{*} (q, w) \in F} .$

Two states $p, q \in Q$ are state-equivalent when they have the same right language: $p \equiv_{D} q ⟺ R_{D} (p) = R_{D} (q) .$ Equivalently, $p \equiv_{D} q ⟺ \forall w \in Σ^{*}, δ^{*} (p, w) \in F ⟺ δ^{*} (q, w) \in F .$

Remarks

State-equivalent states have the same future accepting behavior under every continuation.
Distinct states may still be equivalent; minimization identifies exactly such redundancies.

Depends on

Used by

Dependency tree · two levels

9 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)
Eric Blais, Models of Computation, 20. Nonregular Languages (standard reference, not scraped)