Can we determine whether the following language is recursively enumerable?
$$L = \{\langle M \rangle \mid L(M) \text{ is a regular language}\}$$
Here, $\langle M \rangle$ denotes the encoding of a Turing machine $M$, and $L(M)$ is the language recognized by $M$.
I understand that if $L$ were recursively enumerable (RE), then by Rice's theorem, it would not be decidable. However, I am unable to determine whether $L$ is RE.
- I can't construct a Turing machine that recognizes $L$.
- I also can't reduce any known non-RE language (e.g., the diagonalization language, the set of Turing machines with empty languages, etc.) to $L$ to prove that $L$ is not RE.
Additionally, I attempted to show that $\overline{L}$ is RE. If this were true, then by Rice's theorem, $\overline{L}$ would not be decidable, implying that $L$ is non-RE. However, I couldn't establish the first step because I am unsure whether $\overline{L}$ is actually RE.
Any insights or hints on how to approach this problem would be greatly appreciated!
