Given a language $L$ over the alphabet $\{0,1\}$. Let $L^*= \{ w_1w_2...w_n | n \ge 0, w_1,...,w_n \in L\}$.
Prove:
- If $L$ is recursive, then $L^*$ is recursive as well.
- If $L^*$ is recursive, then $L$ is recursive as well.
For the first proof I have this idea:
If $L$ is recursive, then a turing machine $M_L$ can decide all the words $\{w|w \in L\}$. So $M_L$ can decide $\{w_1,...,w_n | w_1w_2...w_n \in L^*\}$ and appearently it can decide $L^*$ as well.
For the second proof I tried the same but actually it does not work that way. Any hints and ideas?