Why does $\Pi_i^p \subseteq \Sigma_i^p$ imply $\Pi_i^p = \Sigma_i^p$?


Hint: $L \in \Sigma_i^p$ iff $\overline{L} \in \Pi_i^p$. (Here $\overline{L}$ is the complement of $L$.)

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  • $\begingroup$ Well, this shows that $L \in \Sigma_i^p \text{iff} \bar{L} \in \Sigma_i^p$. And since by definition $\Pi_i^p = co\Sigma_i^p$, I guess it follows? (I was trying to show that $\Pi_i^p \subseteq \Sigma_i^p \implies \Sigma_i^p \subseteq \Pi_i^p$ but I couldn't do this. $\endgroup$ – ComplexityStudent Aug 12 '15 at 18:17
  • $\begingroup$ Keep thinking. You'll manage to solve this eventually. $\endgroup$ – Yuval Filmus Aug 12 '15 at 18:18
  • $\begingroup$ :D Okay! My approach is this : for some $x$ whose membership in $L \in \Sigma_i^p$ is being tested we have a decision predicate of the form $\exists y_1 .. Qy_i f(x,\vec{y})$ which is equivalent to its double negated form $\neg(\forall y_i .. \bar{Q} y_i \bar{f}(x,\vec{y}))$. But since $\Pi_i^p \subseteq \Sigma_i^p$ for a predicate like $\forall y_i .. \bar{Q} y_i \bar{f}(x,\vec{y})$ there exists an equivalent predicate like $\exists z_1.. Q z_i g(x,\vec{z})$ and then $\neg(\exists z_i .. Q z_i g(x,\vec{z}))$ is a predicate in $\Pi_i^p$. $\endgroup$ – ComplexityStudent Aug 12 '15 at 18:29
  • $\begingroup$ Yes, that's the idea, though it can be stated more succinctly. $\endgroup$ – Yuval Filmus Aug 12 '15 at 19:25

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