I'm studying for my exam from Logic and Computability and we have to face following kind of examples:
$$f(x) = \begin{cases} \ 1 &\text{if }\Phi_x(x+1)\!\downarrow\text{ and }x\leq50\\ \ 0 &\text{otherwise.} \end{cases}$$
Unfortunatelly I'm not really clear on how to tackle those. $\Phi_x$ is the $x$-th computable function, thus it's also Turing computable. This implies, there exists a TM (we have $N$ of them) which halts and returns the output.
The task is to determine whether it's computable. In order to prove it we either provide an informal algorithm to show it's computable or a formal proof that it's not. Can any of you solve this example?
If $f(x)$ is computable, then $\Phi_x$ needs to halt for every $x \in\{1, \dots, 51\}$. However $\Phi_x$ could be undefined on one of the elements from $\{1, \dots, 51\}$ and keep looping forever.
At the other hand, there is an upper bound on $x$, which indicates its computability.
Please, can you show me the right direction?
Thanks!