CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-12

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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.

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Context-free languages are not closed under complement

Statement

Context-free languages are not closed under complement.

Facts & Assumptions

Given: The class of context-free languages.

[L1]

By Context-free languages are not closed under intersection, context-free languages are not closed under intersection.

[L2]

By Context-free languages are closed under union, concatenation, Kleene star, and homomorphism, context-free languages are closed under union.

[A1]

De Morgan's law gives $A \cap B = (A^{c} \cup B^{c})^{c}$ for languages over one alphabet.

Proof

technique · direct

1.1

Assume for contradiction that context-free languages were closed under complement. Then the complements $A^{c}$ and $B^{c}$ of context-free languages would again be context-free.

given

2.1

By [L2], the union $A^{c} \cup B^{c}$ would then be context-free, and by the contradiction hypothesis its complement would be context-free as well. By [A1], that complement is exactly $A \cap B$ . So the contradiction hypothesis would force closure under intersection.

L2A1step 1.1

3.1

This contradicts [L1]. Therefore context-free languages are not closed under complement.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Alfred V. Aho, COMS W3261 CS Theory, Lecture 11 (standard reference, not scraped)
H. Conrad Cunningham, CSci 311, Models of Computation, Chapter 8 (standard reference, not scraped)