Suppose I have some (increasing, nice asymptotics) function $f(n)$. Does there exist a complexity theoretic problem (e.g. PATH, 3-SAT, GO etc.) that can be solved in time $\Omega(f(n))$ on a deterministic Turing machine. In other words is there a problem proven to be an element of $\mathsf{DTIME}(f(n))$, but not $\mathsf{DTIME}(g(n))$ for any function $g$ such that $\frac{f(n)}{g(n)} \to \infty$ as $n \to \infty$?

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    $\begingroup$ Check out the Time Hierarchy Theorem. $\endgroup$
    – Pål GD
    Feb 1, 2021 at 20:23
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    $\begingroup$ The title of your post didn't match the body of your post. I modified it to reflect your post. If you're also interested in the question in the original title, please ask it separately. $\endgroup$ Feb 2, 2021 at 9:22

1 Answer 1


Consider the language of all binary words which consist only of zeroes. It can be decided on a deterministic Turing machine in time $O(n)$, but not in time $o(n)$.

For general time-constructible functions $f(n)$, the time hierarchy theorem shows that the problem of deciding whether a Turing machine halts in time $f(n)$ can be solved in time $O(f(n)\log f(n))$ but not in time $o(f(n))$. It is not known how to eliminate this logarithmic gap in general.


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