Context free grammar is not closed under :

Answer: Complementation is not a closure property of context-free grammars.

Which operations are CFLs closed under?

• Concatenation — closed. You can combine grammars to produce strings from L1 followed by L2.

• Union — closed. Use a new start symbol with productions to each grammar's start symbol.

• Kleene star — closed. Add rules that allow repetition of the language's start symbol (including the empty string).

Why complementation is not closed:

1. Assume, for contradiction, that context-free languages were closed under complementation.

2. Since CFLs are closed under union, De Morgan's law would imply they are closed under intersection: A ∩ B = complement( complement(A) ∪ complement(B) ).

3. But intersection of two context-free languages need not be context-free. For example, let L1 = { a^n b^n c^m | n,m ≥ 0 } and L2 = { a^m b^n c^n | m,n ≥ 0 }. Both L1 and L2 are context-free, and L1 ∩ L2 = { a^n b^n c^n | n ≥ 0 }, which is not context-free.

4. This contradiction shows context-free languages cannot be closed under complementation.

Therefore, among the listed operations, complementation is the one under which context-free grammars (languages) are not closed.