DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Ambiguous grammars, unambiguous grammars, and inherently ambiguous languages

Definition

Let $G$ be a context-free grammar.

The grammar $G$ is ambiguous when there exists a word $w \in L (G)$ having two distinct leftmost derivations in $G$ . By Derivations and parse trees correspond, this is equivalent to requiring two distinct finite parse trees for $G$ with yield $w$ .

The grammar $G$ is unambiguous when it is not ambiguous.

A context-free language $L$ is inherently ambiguous when every context-free grammar $G$ with $L (G) = L$ is ambiguous.

Remarks

Ambiguity is a property of a grammar, while inherent ambiguity is a property of a language.
A language may admit both ambiguous and unambiguous grammars; in that case the ambiguous grammar is not evidence that the language is inherently ambiguous.

Depends on

Used by

An ambiguous expression grammar with an unambiguous grammar for the same language Counterexample
FALSE: an ambiguous CFG always generates an inherently ambiguous language False statement

Dependency tree · two levels

5 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 8 (standard reference, not scraped)