Is $DSPACE(f) \subseteq DTIME(f)$ always?

For example, if we have a language $A\in DSPACE(log^2(n))$ can we say that $A\in P$ (and subsequently in NP and coNP) since $DSPACE(log^2(n))\subseteq DSPACE(n^3) \subseteq DTIME(n^3)$?

I know about $ DTIME(f) \subseteq DSPACE(f) \subseteq DTIME(2^{O(f)})$ but $DTIME(2^{O(f)})$ is too big in this case.


1 Answer 1


We don't know that $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$ is ever true and we suspect that it's never true.

A Turing machine that uses space $f(n)$ can run for $2^{O(f(n))}$ steps without getting stuck in an infinite loop, so it should be able to do much more than a Turing machine that can only run for $f(n)$ steps. As you've observed, if $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$, then $\mathrm{DSPACE}[f(n)]=\mathrm{DTIME}[f(n)]$ and we have no reason to think that would ever be true for reasonable functions.

  • $\begingroup$ So it's an open question? is it related to P=NP? like if $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$ then P=NP? $\endgroup$
    – shinzou
    Commented Jun 27, 2017 at 7:52
  • 1
    $\begingroup$ @kuhaku, it depends on $f(n)$. If it is polynomial, then - yes, it will imply $\mathsf{P = NP}$. $\endgroup$
    – rus9384
    Commented Jun 27, 2017 at 8:25
  • $\begingroup$ You use Savitch to prove that @rus9384? $\endgroup$
    – shinzou
    Commented Jun 27, 2017 at 8:26
  • $\begingroup$ @kuhaku, padding argument, and the fact that $\mathsf{NTIME(f(n)) \subseteq DSPACE(f(n))}$. I'm not sure if last is true for every $f(n))$, but surely it is for polynomials and larger functions. $\endgroup$
    – rus9384
    Commented Jun 27, 2017 at 8:29
  • $\begingroup$ @rus9384 No, Savitch is for proving things about nondeterministic space classes; everything here is deterministic. $\endgroup$ Commented Jun 27, 2017 at 8:29

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