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I'm trying to study for an exam and having difficulty with the following practice questions. Any help would be appreciated.

  1. Give a language $L$ such that $L$ is not recursive but $\text{prefix}(L)$ is regular.
  2. Give two languages $L_1$ and $L_2$, such that $L_1$ and $L_2$ are not recursive, but $L_1\cap L_2$ is recursive.
  3. Give a language $L$ such that $L$ is regular but $\text{unary}(L)$ is not context-free.
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Tips.

  1. Any (infinite) unary language has regular prefix language.

  2. $L_1 = a \cdot K$ and similary $L_2 = b \cdot K$ where $K$ is a well-chosen horrible language.

Question.

What is unary($L$)? If it is what I think, there is an easy example. Powers of two are not context-free: $\{ a^{2^n} \mid n\ge 1\}$.

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