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. [1] [2]
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). [2] Thus:
The counting hierarchy is contained within PSPACE. [2] By Toda's theorem, the polynomial hierarchy PH is entirely contained in PPP, [3] and therefore in C2P = PPPP.