I tried to prove or disprove that pp is closed under cook reductions. anyone has a idea or link to a answer?
PH $\subseteq$ PPP is known to hold, so if PP is closed under Cook reductions
then PH $\subseteq$ PPP $\subseteq$ PPPP $\subseteq$ PP . ( PH $\subseteq$ PP is not known to hold.)
If PP = PSPACE then PPPP = PPPSPACE = PSPACE = PP .
( PP ≠ PSPACE is not known to hold.)
See the Counting Hierarchy.