Is this language depending on P = NP recursive? Nobody yet knows if ${\sf P}={\sf NP}$. Let us consider the following language

$$L = \begin{cases} (0+1)^* &amp; \text{ if ${\sf P}$ = ${\sf NP}$} \\ \emptyset &amp;\text{ otherwise}. \end{cases}$$

A language is said to be recursive if there exists any rule to determine whether a string belong to language or not. We have a rule here, but the rule itself depends upon an unknown equation. So can we say $L$ is recursive?
If P=NP, then $L = (0+1)^*$ which is regular grammar. Hence, it is recursive.

If P!=NP, then $L = \phi $   which is also a regular grammar. Hence, recursive.

In both the cases, $L$ will be a recursive language.

PS I see a lotta GATE problems these days :D