Is everything that is turing computable an element of NPSPACE?

Question

So, if there is something that is turing computable, I know it is in NP (is that right?).

If M is computable, does it mean M is in NP?

M computable IFF in NP (is that true)?

And does all computable machines exist under NPSPACE?

Answer 1

So, if there is something that is turing computable, I know it is in NP (is that right?).

No!!! The time hierarchy theorem says that, for example, there are problems in $\mathrm{NEXP} = \mathrm{NTime}[2^{\mathrm{poly}(n)}]$ that aren't in $\mathrm{NP}$. And that there are problems in $\mathrm{2NEXP} = \mathrm{NTime}[2^{2^{\mathrm{poly}(n)}}]$ that aren't in $\mathrm{NEXP}$. And so on, ad infinitum. In particular, this means that any $\mathrm{NEXP}$-complete problem is decidable but not in $\mathrm{NP}$.

To put it another way, there's no function $f$ such that every decidable problem can be solved in time (or space) $f(n)$, deterministically or nondeterministically.

In particular, there are perfectly reasonable decidable problems that aren't in $\mathrm{NP}$. For example, suppose you want to work with a graph on $n$ vertices. The graph is really large, so you can't store it. But, if I give you two vertices, it's easy for you to compute whether or not there's an edge between them (e.g., you can do that in deterministic polynomial time). Any problem that's $\mathrm{NP}$-complete when you're just given the full adjacency matrix as the input becomes $\mathrm{NEXP}$-complete when the graph is encoded in this way.