I have some troubles understanding why the halting problem is in EXP.
In Wikipedia the following is written:

It is in EXPTIME because a trivial simulation requires O(k) time, and the input k is encoded using O(log k) bits which causes exponential number of simulations. It is EXPTIME-complete because, roughly speaking, we can use it to determine if a machine solving an EXPTIME problem accepts in an exponential number of steps; it will not use more.

Why does encoding in $O(\log k)$ makes it exponential?
How can we reduce the other problems to our halting problem?


You are referring to the following problem: $$H = \{ (\langle M \rangle, w, t) \mid \text{$M$ accepts $w$ in time at most $t$} \}$$ where $t$ is binary encoded. You should note this is not the (classical) halting problem, but, rather, the bounded version of the halting problem. The halting problem itself cannot be complete for $\mathsf{EXP}$ (or any other subclass of the recursive languages).

Now, to answer your question. First, notice that, because $t$ is encoded in binary, the input has length $|\langle M \rangle| + |w| + \log t$. To simulate $M$ on $w$ for $t$ steps we can use a universal TM, which takes time $O(t \log t)$. Since we are doing a worst-case analysis, we can afford to view $|\langle M \rangle|$ as constant. Also, because $t \in O(2^{p(|w|)})$ for some polynomial $p$, $\log t \in O(p(|w|))$. Hence, the input has size $O(|w| + p(|w|))$ while our simulation takes time $O(t \log t) \subset O(p(|w|) 2^{p(|w|)})$ and, in turn, $p(|w|) 2^{p(|w|)} \in 2^{\textrm{poly}(|w|)}$. This proves $H \in \mathsf{EXP}$.

Reducing a problem $P \in \mathsf{EXP}$ to $H$ is also very simple: If $P$ is decidable by $M$ in time bounded by a function $f \in 2^{\textrm{poly}(n)}$, then any instance $x$ of $P$ is a yes-instance if and only if $(\langle M \rangle, x, f(|x|)) \in H$, where $f(|x|)$ is given binary encoded. The time taken by this reduction is dominated by computing $f(|x|)$, which can be performed in time polynomial in $|x|$ (by first converting it to binary and then computing $f(|x|)$ in polynomial time).

  • $\begingroup$ To prove this problem is in $EXP$, is it not sufficient to state that, in the worst case, the problem would take $t$ time, which is exponential to the number of bits used to store $t$? $\endgroup$
    – J. Schmidt
    Aug 12 '19 at 9:17
  • 1
    $\begingroup$ @J.Schmidt That is the gist of it. However, note we cannot really expect to solve instances of $H$ always in time $t$ with a single Turing machine. There is always the $\log t$ slack factor involved because of the simulation of $M$. (If we could avoid this slack factor, then we could also strengthen the time hierarchy theorem, for example.) $\endgroup$
    – dkaeae
    Aug 12 '19 at 9:26
  • $\begingroup$ Might I ask why t is in O(2^p(|w|))? @dkaeae $\endgroup$
    – dfsh
    Aug 12 '19 at 12:09
  • $\begingroup$ @dfsh That follows from the definition of $\mathsf{EXP}$. It contains all problems solvable in $2^{\text{poly}(n)}$ time (by definition). $\endgroup$
    – dkaeae
    Aug 12 '19 at 13:10
  • $\begingroup$ For the reduction, what if P takes more than t steps ? @dkaeae $\endgroup$
    – Perl
    Apr 22 at 2:20

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