CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

Nerode classes form the canonical minimal DFA

Statement

If $L$ is regular, then the DFA whose states are the Nerode classes of $L$ is a minimal DFA for $L$ . Its state set and transition structure depend only on $L$ , so it is canonical.

Facts & Assumptions

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

[L1]

By A language is regular if and only if its Nerode equivalence has finite index, finite-index Nerode classes build a DFA recognizing $L$ .

[L2]

By A DFA can realize at most one Nerode class per reachable state, every DFA recognizing $L$ has distinct reachable states for distinct Nerode classes.

Proof

technique · direct

1.1

Because $L$ is regular, [L1] provides a DFA $D_{L}$ whose states are exactly the Nerode classes of $L$ and which recognizes $L$ .

L1given

1.2

Let $D$ be any DFA recognizing $L$ . Every Nerode class contains some word $u$ , and the state reached by $D$ on $u$ is reachable. By [L2], two distinct Nerode classes cannot reach the same state of $D$ , so $D$ has at least as many reachable states as $D_{L}$ has states.

L2given

2.1

Step 1.2 shows that no DFA for $L$ can use fewer reachable states than $D_{L}$ , so $D_{L}$ is minimal. It is canonical because its states are defined directly from the language $L$ rather than from an auxiliary machine.

step 1.1step 1.2∎

Depends on

Used by

Minimal DFAs are unique up to start-state-preserving isomorphism Theorem

Dependency tree · two levels

8 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)