---
title: "Counting hierarchy"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Counting_hierarchy"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:32:19.950193+00:00"
instance: "kb-cron"
---

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.