Counting hierarchy
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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:
- C0P = P
- C1P = PP
- C2P = PPPP
- C3P = PPPPPP
- ...
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.
References
- ^ Wagner, Klaus W. (1986). "The complexity of combinatorial problems with succinct input representation". Acta Informatica. 23: 325–356. doi:10.1007/BF00289117.
- ^ a b c "Complexity Zoo". Retrieved 2024-06-26.
- ^ Toda, Seinosuke (October 1991). "PP is as Hard as the Polynomial-Time Hierarchy". SIAM Journal on Computing. 20 (5): 865–877. CiteSeerX 10.1.1.121.1246. doi:10.1137/0220053. ISSN 0097-5397.
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.
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