26 lines
943 B
Markdown
26 lines
943 B
Markdown
---
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title: "Counting hierarchy"
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chunk: 1/1
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source: "https://en.wikipedia.org/wiki/Counting_hierarchy"
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category: "reference"
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tags: "science, encyclopedia"
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date_saved: "2026-05-05T11:32:19.950193+00:00"
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instance: "kb-cron"
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---
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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.
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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:
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C0P = P
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C1P = PP
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C2P = PPPP
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C3P = PPPPPP
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...
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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.
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== References ==
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== Further reading ==
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Torán, Jacobo (1991). "Complexity classes defined by counting quantifiers". Journal of the ACM. 38 (3): 753–774. doi:10.1145/116825.116858. MR 1125929. |