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.

Membership, emptiness, finiteness, equivalence, and containment for regular languages

Definition

The standard DFA decision problems for regular languages are the following.

Membership: given a DFA $D$ over $Σ$ and a word $w \in Σ^{*}$ , decide whether $w \in L (D)$ .
Emptiness: given a DFA $D$ , decide whether $L (D) = \emptyset$ .
Finiteness: given a DFA $D$ , decide whether $L (D)$ is finite.
Equivalence: given DFA's $D_{1}, D_{2}$ over the same alphabet, decide whether $L (D_{1}) = L (D_{2})$ .
Containment: given DFA's $D_{1}, D_{2}$ over the same alphabet, decide whether $L (D_{1}) \subseteq L (D_{2})$ .

By Regular languages, these are decision problems for regular languages because a DFA is a witness for regularity.

Depends on

Used by

DFA membership, emptiness, finiteness, equivalence, and containment are decidable 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

John Watrous, Introduction to the Theory of Computing, Lecture 15: Encodings; examples of decidable languages (standard reference, not scraped)
H. Conrad Cunningham, Notes on Models of Computation, Chapter 4: Properties of Regular Languages (standard reference, not scraped)