# Can P=PSPACE and PSPACE problems be formulated as $\Pi_1$ formula?

To say simply, can PSPACE problems be written as $\Pi_1$ formula? Or how can these problems be written in terms of (first-order) arithmetic hierarchy?

edit:also currently, by what arithmetic hierarchy formula can P=PSPACE be written?

and what would be the consequence of being able to write P=PSPACE as $\Pi_1$ formula?

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What do you mean by $P=PSPACE$? They're two different complexity classes and it's unknown whether they're equal or not. –  John Moeller Feb 25 '13 at 5:24
Are you familiar with descriptive complexity? –  JSchlather Feb 25 '13 at 5:34
I am learning it-and that's why I am asking this question. –  mars Feb 25 '13 at 7:34

## migrated from cstheory.stackexchange.comFeb 25 '13 at 14:27

This question came from our site for theoretical computer scientists and researchers in related fields.

You can get a $\Sigma_2$ sentence for $\mathsf{P}=\mathsf{PSPACE}$, using the same technique as in the blog post, only replace $SAT$ by $QBF$. I doubt there is a known sentence of lower complexity.

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$PSPACE \subset R \subset \text{co-}RE = \Pi_1$. $PSPACE$ problems are decidable in finite time, since $PSPACE \subset EXP$. Languages in $R$ are decidable by a Turing machine, and $R=RE\cap\text{co-}RE$, and $\text{co-}RE$ is $\Pi_1$. This implies that any language in $PSPACE$ is also in $\Pi_1$, so the basic answer to your question is "yes."

If you're also asking about $P$, then the answer is also "yes" because $P \subseteq PSPACE$.

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but what about expressing P=PSPACE as first-order formula? like rjlipton.wordpress.com/2009/05/27/arithmetic-hierarchy-and-pnp –  mars Feb 25 '13 at 7:37
As the question is worded, this is a complete answer. –  András Salamon Feb 25 '13 at 9:31
@mars Ah, ok. I'm not really qualified to answer that. However, if you wanted to know about something from a blog post, you probably should have provided the link in the question. When you had asked this over on math.sx, I asked for clarification on a couple different points. You could have referenced the blog post then too. –  John Moeller Feb 25 '13 at 17:55
As already answered by sdcvvc, $\mathrm P=\mathrm{PSPACE}$ can be written as a $\Sigma^0_2$-sentence. On the other hand, if it is equivalent to a $\Pi^0_2$-sentence (let alone $\Pi^0_1$), then this fact does not relativize, by the same argument as in http://mathoverflow.net/questions/57348. A combinatorial proof is given in http://cstheory.stackexchange.com/a/16644/6959.