I would like to know if there is a known decision problem with the following characteristics:

  • Represented in unary, the problem is decidable in polynomial time.
  • Represented in binary, the problem is not decidable in polynomial time (and this fact has been proved, not just conjectured).

For example, Subset-Sum is in $P$ when represented in unary, but it is $NP$-complete in binary. However, this problem does not satisfy my second requirement because we do not know whether $P=NP$.

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    $\begingroup$ $\{ \langle M, s, x \rangle \mid M \text{ accepts } x \text{ within } s \text{ steps}\}$ $\endgroup$ – GMB Jun 10 '14 at 3:18
  • $\begingroup$ @GMB: Assuming you can show that there is no faster way but simulating $M$ for $s$ steps. $\endgroup$ – Raphael Jun 11 '14 at 6:26
  • $\begingroup$ @Raphael - the language GMB have mentioned is known to be EXPTIME-complete. $\endgroup$ – R B Jun 11 '14 at 6:35
  • $\begingroup$ @RB: Ah, that's important to note then. Too many beginners fall into the trap of thinking along the lines of "I can't find a better algorithm, therefore there is none!". $\endgroup$ – Raphael Jun 11 '14 at 6:37

Every single-exponential (i.e. known to be solvable in $O(c^n)$ for some constant $c$) EXPTIME-complete problem will answer your requirements.

For example, see checking thorough re refinement on finite modal transition systems.

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    $\begingroup$ "solvable in ... for some constant $c$" $\: \mapsto \:$ "in E" $\;\;\;\;$ $\endgroup$ – user12859 Jun 10 '14 at 6:04
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    $\begingroup$ One should note that P $\subsetneq$ EXPTIME is known. $\endgroup$ – Raphael Jun 11 '14 at 6:38

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