Give one example where it takes Non- deterministically exponential time to solve the problem?

Question

I am a starter in complexity theory though I have fair knowledge in Turing machine. I know what it means to be non-deterministically polynomial time solvable but I am trying to understand where the time goes beyond polynomial time even using non-deterministic concept. Is it possible or we can do everything in polynomial time using non-deterministic concept though in reality it can not be implemented. Therefore, I am looking for an example where we can't do it in polynomial time rather it takes exponential time even after using non-deterministic concept. Please explain clearly why it takes exponential time(non-deterministically).

Answer 1

There are problems that cannot be solved in polynomial time on a nondeterministic machine. Indeed, there are problems that cannot be solved on any Turing machine, such as the halting problem.

We know from the nondeterministic version of the time hierarchy theorem that there are things that can be done in exponential time on a nondeterministic TM that cannot be done in polynomial time. That is, $\mathrm{NP}\subsetneq\mathrm{NEXP}$.

Wikipedia gives several examples of $\mathrm{NEXP}$-complete problems: these are examples of problems that are provably not in $\mathrm{NP}$.

Answer 2

According to the Nondeterministic time hierarchy theorem, $\mathsf{NEXP\neq NP}$, which means there are languages decidable in nondeterministic exponential time, but not with nondeterministic polynomial time. A simple consequence is that no $\mathsf{NEXP}$ complete problem can lie inside $\mathsf{NP}$, which means that the following language requires exponential nondeterministic time to solve:

$L=\left\{\left(n,\langle M\rangle,x\right) \big|\hspace{1mm} \exists w\in\{0,1\}^{\le n} : M(x,w)\right\}$

Note that if in the above you replace the binary representation of $n$ with its unary representation $1^n$ then the problem becomes $\mathsf{NP}$ complete rather then $\mathsf{EXP}$-complete. This is an example of a more general phenomena, where many $\mathsf{NP}$ complete problems can be turned into $\mathsf{EXP}$-complete problems by giving a succinct description of the inputs. More details about this and some examples of more natural $\mathsf{EXP}$-complete problems can be found in this question from cstheory.