DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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 extended transition function of a DFA

Definition

Let $D = (Q, Σ, δ, q_{0}, F)$ be a DFA. An extended transition function for $D$ is a function $δ^{*} : Q \times Σ^{*} \to Q$ satisfying the two clauses $δ^{*} (q, ε) = q$ for every $q \in Q$ , and $δ^{*} (q, u a) = δ (δ^{*} (q, u), a)$ for every state $q \in Q$ , every word $u \in Σ^{*}$ , and every letter $a \in Σ$ .

Thus $δ^{*} (q, w)$ is intended to be the state reached from $q$ after reading the whole word $w$ . The next theorem proves that such a function exists and is unique.

Depends on

Used by

Acceptance of a word by a DFA and the recognized language Definition
Reachable states of a DFA Definition
The extended transition function exists and is unique 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)