The language L is ?

Question

let $L =\begin{cases} \Phi & \text{ if } Recursive set(RS) = RES\\ \epsilon & \text{ if } RS \neq RES \end{cases}$

then $L$ is:

a) Regular B) CFL but not regular

c)Regular but not CSL  D)none

Answer 1

I feel the answer should be Recursive enumerbale .

All REGULAR,CFL,CSL are recursive and hence recursive enumerable.

But there the languages that are Recursive enumerable but not recursive  .hence for those languages if RE=RS L would be ∅ .

AND it is said when RE= RS then  L=  ∊ and we know it is undecidable whether TM accepts ∊ ,so it falls in the category of Recursive enumerable even as TM may or may not halt for ∊ as input  

so  option d

Answer 2

Answer is A, regular.

There are two choices for $L$ but both are regular sets. Now, RS != RES, and this is a known fact. So, $L = \phi$ which is a regular set.

Now, even if RE = RES? is not a proven fact, still $L$ is a regular set. Just that we don't know which regular set it is.