A simple fact is that $P = NP \to P = coNP$, which follows from the observation that $P$ is closed under complement.

I am having trouble seeing that an analogous statement is true at higher levels of $PH$. For example, is it known that $NP = \Sigma_2 P$ implies $NP = \Pi_2 P$? If so, is there an easy proof? Would such a statement have any other interesting implications (For example $NP = coNP$)?

It seems somewhat likely to me that this is true, based on the observation that $NP = \Sigma_2 P$ means that $\exists \forall$ quantifier patterns can be replaced with $\exists$ quantifier patterns, and so all higher levels of $PH$ would at least collapse to $\Pi_2 P$.


1 Answer 1


The answer is yes. notice that $NP=\Sigma_1$ and for all $i$ you have $\Sigma_i=\Sigma_{i+1} \Rightarrow \Sigma_i=\Pi_i$. This is true since $\Pi_i\subseteq \Sigma_{i+1}=\Sigma_i$. The inclusion holds since if $\bar{L}\in\Sigma_i$ then you have $x\notin L \iff \exists v_1\in \left\{0,1\right\}^{p(|v_1|)}...Q_iv_i\in\left\{0,1\right\}^{p(|v_i|)}M(x,v_1,...,v_i)=1 $ or equivalently $x\in L \iff \forall v_1\in \left\{0,1\right\}^{p(|v_1|)}...\bar{Q_i}v_i\in\left\{0,1\right\}^{p(|v_i|)}M(x,v_1,...,v_i)=0$. So now your $\Sigma_{i+1}$ verifier for $L$ will ignore $v_1$, run $M(v_2,...,v_{i+1})$ and flip the answer. It remains to show $\Sigma_i\subseteq\Pi_i$. Let $L\in\Sigma_i, \bar{L}\in \Sigma_{i+1}$ and since $\Sigma_i=\Sigma_{i+1}$ you get $\bar{L}\in\Sigma_i$.

Back to your question. If $NP=\Sigma_2$ (or $\Sigma_1=\Sigma_2)$ then $\Sigma_1=\Pi_1$. Let $L\in\Pi_2$ then $\bar{L}\in\Sigma_2\Rightarrow \bar{L}\in\Sigma_1\Rightarrow \bar{L}\in\Pi_1\Rightarrow L\in\Sigma_1$, So $L\in NP$ as required. The opposite direction is clear(for all $i$ $\Sigma_i\subseteq\Pi_{i+1}$).


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