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A k-query oracle Turing machine can query its oracle for at most $k$ times. How could I show that assuming $\text{NP} \not= \text{coNP}$, we have $\text{NP} \cup \text{coNP} \subsetneq \text{P}^{\text{SAT},1}$ (query only once)? I tried to show this strict inclusion directly or its counterexample but to no avail... There was such a question on this site, but the top vote isn't a direct answer. Could somebody provide me more insight into this again? Thanks!

More specifically, I wonder how should I think of these $k$-query machines from machines with unlimited number of queries. I can see that they should have limited computation power, but I'm not sure how to utilize that for the proof.

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  • $\begingroup$ Hint: there is a very simple way to construct a problem which is both NP-hard and coNP-hard. $\endgroup$ – user114966 Dec 5 '20 at 6:47
  • $\begingroup$ So any PSPACE-complete problem like TQBF will work, since NP and coNP are in PSPACE so the languages inside reduce to it? $\endgroup$ – Macrophage Dec 5 '20 at 6:53
  • $\begingroup$ No, TQBF almost surely doesn't lie in $P^{SAT,1}$. What I mean is that given two complexity classes $C_1,C_2$ and languages $L_1,L_2$ such that $L_1$ is $C_1$-hard and $L_2$ is $C_2$-hard, you can construct a language which is both $C_1$-hard and $C_2$-hard. $\endgroup$ – user114966 Dec 5 '20 at 7:21
  • $\begingroup$ Oh I see what you mean, that's kind of what I tried to do but didn't quite figure out. $\endgroup$ – Macrophage Dec 5 '20 at 16:14
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Let's try to show the contrapositive. Since $\mathsf{NP} \cup \mathsf{coNP} \subseteq \mathsf{P}^{\mathsf{SAT},1}$, the contrapositive states that if $\mathsf{P}^{\mathsf{SAT},1} \subseteq \mathsf{NP} \cup \mathsf{coNP}$ then $\mathsf{NP} = \mathsf{coNP}$. Let us assume, therefore, that every problem in $\mathsf{P}^{\mathsf{SAT},1}$ is both in $\mathsf{NP}$ and in $\mathsf{coNP}$. We would like to show that this implies that $\mathsf{NP} = \mathsf{coNP}$.

We can imagine that the argument will have the following structure:

  1. Come up with some language $L \in \mathsf{P}^{\mathsf{SAT},1}$.
  2. The assumption implies that $L \in \mathsf{NP}$ or $L \in \mathsf{coNP}$.
  3. Use these facts to say something about the relation between $\mathsf{NP}$ and $\mathsf{coNP}$.

Our goal is to show that $\mathsf{NP} = \mathsf{coNP}$, or equivalently $\mathsf{NP} \subseteq \mathsf{coNP}$. This, in turn, is equivalent to showing that $\mathsf{SAT} \in \mathsf{coNP}$. Considering the plan above, what we need is to find a language $L \in \mathsf{P}^{\mathsf{SAT},1}$ such that both $L \in \mathsf{NP}$ and $L \in \mathsf{coNP}$ imply $\mathsf{SAT} \in \mathsf{coNP}$.

We can take as $L$ the following language: $$ L = \{(x,0) : x \in \mathsf{SAT}\} \cup \{(x,1) : x \notin \mathsf{SAT}\}. $$ Details left to you.

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  • $\begingroup$ Er, I’m not super familiar with complexity theory, but doesnt $$A \subseteq B \cup C$$ only imply that elements in A are in at least one of B or C, but not necessarily both? $\endgroup$ – D. Ben Knoble Dec 5 '20 at 15:26
  • $\begingroup$ Right, thanks. Let me modify the proof. $\endgroup$ – Yuval Filmus Dec 5 '20 at 15:26
  • $\begingroup$ Indeed, this makes it a bit harder to demystify the proof. $\endgroup$ – Yuval Filmus Dec 5 '20 at 15:36
  • $\begingroup$ That's an interesting language! I remembered using a similar trick to construct a language neither Turing recognizable nor co-Turing recognizable...Is there a name for this disjoint union trick, and how does it work in general? $\endgroup$ – Macrophage Dec 5 '20 at 16:42
  • $\begingroup$ Sometimes this kind of combination is denoted by $\oplus$. So $L = \mathsf{SAT}\oplus\mathsf{coSAT}$. $\endgroup$ – Yuval Filmus Dec 5 '20 at 16:46

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