Would it be possible for a non-context-free language L to be context-free under iteration?

Question

Language L is not a context-free language.

But could L* be a context-free language?

Answer 1

Yes, this is possible. As an example, consider the alphabet Σ = {1} and let L be the language { 1 ^p | p is a prime number }. You can prove that this language is not context-free by using the pumping lemma.

However, the language L* is the set of all strings except for 1. The reason for this is that

• ε ∈ L*, because ε ∈ N* for any language N.
• 1 ² ∈ L* because 2 is prime.
• 1 ³ ∈ L* because 3 is prime.
• 1 ⁿ ∈ L* for any n ≥ 2, because you can start with either 1 ² or 1 ³ and concatenate an appropriate number of copies of 1 ² to it.

This language is indeed context-free, and you can prove that by writing a grammar for it.

Hope this helps!