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.^[1][2]

More precisely, the zero-th level is C₀P = P, and the (n+1)-th level is C_n+1P = PP^C_nP (i.e., PP with oracle C_n).^[2] Thus:

• C₀P = P
• C₁P = PP
• C₂P = PP^PP
• C₃P = PP^PPPP
• ...

The counting hierarchy is contained within PSPACE.^[2] By Toda's theorem, the polynomial hierarchy PH is entirely contained in P^PP,^[3] and therefore in C₂P = PP^PP.