Counting hierarchy

In complexity theory, the counting hierarchy is a hierarchy of complexity classes. It is analogous to the polynomial hierarchy, but with NP replaced with PP. It was defined in 1986 by Klaus Wagner.

More precisely, the zero-th level is C0P = P, and the (n+1)-th level is Cn+1P = PPCnP (i.e., PP with oracle Cn). Thus:


 * C0P = P
 * C1P = PP
 * C2P = PPPP
 * C3P = PPPP PP

The counting hierarchy is contained within PSPACE. By Toda's theorem, the polynomial hierarchy PH is entirely contained in PPP, and therefore in C2P = PPPP.