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Consider the language $A=\{(\phi_1, \phi_2) | \phi_1 \in SAT, \phi_2\in \overline{SAT} \}$. What is the smallest class that $A$ is known to belong to?

Apparently, the answer is $\Pi_2$, although I thought it was $NP \cap coNP$, since if we have a non-deterministic machine $M_1$ that decides $SAT$ and another non-deterministic machine $M_2$ that decides $\overline {SAT}$, then we can define a machine $M_3$ that on input $(\phi_1, \phi_2)$, runs $M_1$ on $\phi_1$ and $M_2$ on $\phi_2$, and only accepts if both machines return True. If the input does belong to $A$, then for some set of nondeterministic choices, we know $M_3$ will accept. If the input doesn't belong to $A$, then either $M_1$ or $M_2$ always return False, and so $M_3$ always rejects.

My first question is whether my explanation correctly shows $A \in NP \cap coNP$. More importantly though, why is $\Pi_2$ smaller than $NP\cap coNP$!?

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  • $\begingroup$ You can't find nondeterministic $M_2$ that decides $\overline{SAT}$. For this purpose you should find a polynomial guess that shows there is no assignment for SAT, just one witness needed not all of them and you have no access to the other branches of nondeterministic machine! $\endgroup$ – Mohsen Ghorbani Jun 25 at 14:18
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You seem to be using the following definition of $\mathsf{NP} \cap \mathsf{coNP}$: it consists of all languages of the form $L_1 \cap L_2$, where $L_1 \in \mathsf{NP}$ and $L_2 \in \mathsf{coNP}$. However, in reality $\mathsf{NP} \cap \mathsf{coNP}$ consists of all languages $L$ such that $L \in \mathsf{NP}$ and $L \in \mathsf{coNP}$.

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  • $\begingroup$ Okay, thanks. So $A$ may not even belong to $NP \cap coNP$ at all, correct? $\endgroup$ – Adam G Jun 25 at 15:31
  • $\begingroup$ Right. Of course we don't know for sure – for all we, know, the polynomial hierarchy collapses to $\mathsf{P}$. $\endgroup$ – Yuval Filmus Jun 25 at 16:25
  • $\begingroup$ The set of all languages $L$ where $L=L_1\bigcap L_2$ and $L_1\in NP,L_2\in coNP$ is called $DP$. It is sometimes confusing, but it is not $NP\bigcap coNP$. $\endgroup$ – nir shahar Jun 25 at 18:27

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