I was given a question to prof that a language is not in $RE$ using $RICE-theorem$
The question:
$S$ - property of turning machine language
$L$ - language
$S \subseteq \{L : 7 \leq |L| \leq 10 \}$ is not an empty set.
prof that: The language $L_1 = \{ <M> : L(M)\in S\} \notin RE$ using $RICE-theorem$
As I know the uses of RICE-theorem is to prof that a language is not in $R$ and only if $\Phi\in S$ then we can say that $L_1 \notin RE$ From what I see here $\Phi \notin S$ so how do I use $RICE-theorem$ to prof...?
What I tried:
Prof that $L_1 \notin R$.
We will show that $S$ is non-trivial.
- Let $L(M_1) = L(M_2)$, languages of the turning machins $M_1,M_2$ and both $\in S$ so from the information given we can see that if $L(M_1) \notin S$ it means that $L(M_1) < 7 $ or $ 10< L(M_1)$ and because $L(M_1) = L(M_2)$ we know $L(M_2) < 7 $ or $ 10< L(M_2)$ that gives us $L(M_2)\notin S$.
If $L(M_1) \in S$ it means that $7 \leq L(M_1) \leq 10$ and because $L(M_1) = L(M_2)$ we know $7 \leq L(M_2) \leq 10$ that gives us $L(M_2)\in S$. - $\Phi \notin S$ ($\Phi$ = the empty languages) because the size of $| \Phi | =0 < 7$, therefor $S\ne RE.$
- $L_0=\{0,0^2,0^3,... , 0^7 \} \in S$ because $ 7 \leq |L_0|=7 \leq 10 $, therefor $S\ne \Phi.$
So from that be can determine that $L_1 \notin R$, but given the fact that $\Phi \notin S$ I can't say anything regarding $RE$.