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Let me state the problem again:

Suppose PSPACE $\ne$ EXP. Is there a language in EXP - PSPACE that is not PSPACE-hard?

Context

I have a problem that's in EXP. Currently I don't think it's in PSPACE. Besides proving it's EXP-complete, what else can I do? Since it seem not to be in PSPACE, I start to think it's PSPACE-hard. But is this necessary? That's why I asked this question.

Actually I have a related question, that is, given PSPACE $\ne$ EXP, whether there is a language in EXP - PSPACE that is not EXP-hard. A yes answer to the main question will imply a yes answer to this question. I think answering this question will also help me somehow.

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  • $\begingroup$ $PSPACE-hard$ functions include $PSPACE-complete$ languages, that are of course in $PSPACE$. I believe that you mean $PSPACE-hard \backslash PSPACE-complete$? $\endgroup$ – chazisop Aug 7 '11 at 15:05
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    $\begingroup$ @chazisop: "Since it seem not to be in PSPACE, I start to think it's PSPACE-hard. But is this necessary? That's why I asked this question." $\endgroup$ – Daniel Apon Aug 7 '11 at 15:19
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The proof of Ladner's theorem doesn't use any special properties of P and NP and the same proof unchanged will show, assuming EXP<>PSPACE, there is a language L in EXP-PSPACE and not EXP-complete under either P-time or PSPACE-reductions.

You need the full Landner look-back trick to keep L in EXP.

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The answer to your questions depends on what kind of reductions you are using for your notion of hardness. If you are using polynomial-space reductions, then I believe Daniel's answer is correct. If you are using polynomial-time reductions, however, just the opposite is true.

Namely, assuming $EXP \neq PSPACE$, there is a problem in $EXP$ which is neither in $PSPACE$ nor hard for $PSPACE$ under polynomial-time reductions. This can essentially be constructed by diagonalizing against all possible polynomial-time reductions to $QBF$ (preventing the constructed language from being in $PSPACE$) and from $QBF$ (preventing the constructed language from being $PSPACE$-hard).

Also, by the general version of Ladner's Theorem, if $EXP \neq PSPACE$ then there are problems in $EXP \backslash PSPACE$ which are not $EXP$-hard under polynomial-time reductions.

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  • $\begingroup$ Oops, deleted my comment because I couldn't edit it. Thanks though, Kaveh :) Also - What I'm thinking of is going to branch into an entirely different direction, so I'll ask a new question later if so $\endgroup$ – Daniel Apon Aug 7 '11 at 17:19
  • $\begingroup$ @Daniel Apon: :) $\endgroup$ – Kaveh Aug 7 '11 at 17:22
  • $\begingroup$ @Joshua: Why do you consider polynomial-time reduction? I think for PSPACE-hardness, polynomial-space reduction is sufficient. Why do you restrict yourself to polynomial time? $\endgroup$ – Zirui Wang Aug 8 '11 at 1:38
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    $\begingroup$ @Zirui: under polynomial-space reductions, all problems in PSPACE are PSPACE-complete... If you are only interested in problems "above" PSPACE, then you could use polynomial-space reductions. $\endgroup$ – Joshua Grochow Aug 8 '11 at 15:05
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Please see comments.


No to the first question. Assuming that $\mathsf{EXP}$ - $\mathsf{PSPACE}$ is not empty, since $\mathsf{PSPACE} \subseteq \mathsf{EXP}$, all languages in $\mathsf{EXP}$ - $\mathsf{PSPACE}$ are $\mathsf{PSPACE}$-hard.

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  • $\begingroup$ And I would believe Yes to the second question as well, for similar reasons as before. $\endgroup$ – Daniel Apon Aug 7 '11 at 14:44
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    $\begingroup$ I don't understand this argument. Are you saying that if X and Y are complexity classes such that $X\subseteq Y$ and $X \neq Y$ then everything in Y\X is X-hard? $\endgroup$ – Robin Kothari Aug 7 '11 at 15:31
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    $\begingroup$ That was my implication -- Do you have a counterexample in mind? $\endgroup$ – Daniel Apon Aug 7 '11 at 16:09
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    $\begingroup$ Example 1: Let's say your implication is true. Let X be NP and Y be PSPACE. If X is not equal to Y this means NP = co-NP (if it's not then a co-NP-complete problem will be NP-hard by your claim and thus NP = co-NP). If X is equal to Y, then NP = PSPACE, and thus NP = co-NP. So we have proved that NP = co-NP without any assumptions. Example 2: Let X = AC0[3] and Y = L. We know PARITY is not in AC0[3], so PARITY must be AC0[3]-hard. But that will put AC0[3] inside AC0[2] (which contains PARITY) contradicting the known fact that AC0[3] is not contained in AC0[2]. $\endgroup$ – Robin Kothari Aug 7 '11 at 16:42
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    $\begingroup$ I stand corrected! $\endgroup$ – Daniel Apon Aug 7 '11 at 16:53

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