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 = PPPPPP ... 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.
References
Further reading Torán, Jacobo (1991). "Complexity classes defined by counting quantifiers". Journal of the ACM. 38 (3): 753–774. doi:10.1145/116825.116858. MR 1125929.