In this paper we analyze Ko's Theorem 3.4 in \cite{Ko87}. We extend point b) of Ko's Theorem by showing that $\poneh(\upcoup)=\upcoup$. As a corollary, we get the equality $\ph(\upcoup) = \poneh(\upcoup)$, which is, to our knowledge, a unique result of type $\poneh({\cal C})=\ph({\cal C})$, for a class $\cal C$ that would not be equal to $\DP$. With regard to point a) of Ko's Theorem, we observe that it also holds for the classes $\upk{k}$ and for $\fewp$. In spite of this, we prove that point b) of Theorem 3.4 fails for such classes in a relativized world. This is obtained by showing the relativized separation of $\upcoupk{2}$ from $\poneh(\npconp)$. Finally, we suggest a natural line of research arising from these facts.

### Revisiting a result of Ko

#### Abstract

In this paper we analyze Ko's Theorem 3.4 in \cite{Ko87}. We extend point b) of Ko's Theorem by showing that $\poneh(\upcoup)=\upcoup$. As a corollary, we get the equality $\ph(\upcoup) = \poneh(\upcoup)$, which is, to our knowledge, a unique result of type $\poneh({\cal C})=\ph({\cal C})$, for a class $\cal C$ that would not be equal to $\DP$. With regard to point a) of Ko's Theorem, we observe that it also holds for the classes $\upk{k}$ and for $\fewp$. In spite of this, we prove that point b) of Theorem 3.4 fails for such classes in a relativized world. This is obtained by showing the relativized separation of $\upcoupk{2}$ from $\poneh(\npconp)$. Finally, we suggest a natural line of research arising from these facts.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11581/116684
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