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Hi,

Is there any difference between the classes $NP^{SAT} \cap CoNP^{SAT}$ (i.e. $\Sigma_2^p \cap \Pi_2^p $) and $(NP \cap CoNP)^{SAT}$? If so, what is the structural difference between the two?

Searching around, I found this paper which shows a relativized world where $NP_{poly} \cap CoNP_{poly}$ is not contained in $(NP \cap CoNP)_{poly}$. Is there any similar result for the question I've asked?

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 $\let\M\mathrm$How do you define $(\M{NP}\cap\M{coNP})^{SAT}$, other than $\M{NP}^{SAT}\cap\M{coNP}^{SAT}$? Note that there is no general way of assigning $C^A$ to any given complexity class $C$. There is no concept of “$\M{NP}\cap\M{coNP}$ Turing machines” that you could just relativize. (This is in contrast to nonuniformity, which you mention in the second paragraph: $C/\M{poly}$ is well defined for any class $C$. The result you quote is actually that there is an oracle $A$ such that $\M{NP}^A/\M{poly}\cap\M{coNP}^A/\M{poly}\nsubseteq(\M{NP}^A\cap\M{coNP}^A)/\M{poly}$.) – Emil Jeřábek Sep 5 at 13:01

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