Alphabeta MathDefinitionDefinition: 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 language generated by a CFG
Definition
Let be a context-free grammar. The language generated by is
A language over is context-free when for some context-free grammar over .
Depends on
Used by
- Ambiguous grammars, unambiguous grammars, and inherently ambiguous languages Definition
- Eliminating immediate left recursion preserves the language Lemma
- Substituting a variable's productions into a leading variable occurrence preserves the language Lemma
- Context-free languages are closed under union, concatenation, Kleene star, and homomorphism Theorem
- Eliminating useless symbols preserves the generated language Theorem
- Epsilon-productions can be eliminated except possibly at the start symbol Theorem
- Unit productions can be eliminated Theorem
Dependency tree · two levels
3 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)
- John Watrous, Introduction to the Theory of Computing, Lecture 7 (standard reference, not scraped)