PH (complexity)

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In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy:

$\mbox{PH} = \bigcup_{k\in\mathbb{N}} \Delta_k\mbox{P}$

PH was first defined by Larry Stockmeyer. It is contained in P^PP (the class of problems that are decidable by a polynomial time Turing machine with access to a PP oracle), P^#P (by Toda's theorem), and also in PSPACE.

PH has a simple logical characterization: it is the set of languages expressible by second-order logic.

PH contains almost all well-known complexity classes inside PSPACE; in particular, it contains P, NP, and co-NP. It even contains probabilistic classes such as BPP and RP.

P = NP if and only if P = PH. This may simplify a potential proof of P ≠ NP, since it's only necessary to separate P from the more general class PH.

[edit] References

L. J. Stockmeyer, "The polynomial hierarchy", Theoretical Computer Science, Vol. 3 (1976), pp. 1–22.
The Complexity Zoo: PH

v • d • e Important complexity classes (more)

P • NP • co-NP • NP-C • co-NP-C • NP-hard • UP • #P • #P-C • L • NL • NC • P-C • PSPACE • PSPACE-C • EXPTIME • NEXPTIME • EXPSPACE • 2-EXPTIME • PR • RE • Co-RE • RE-C • Co-RE-C • R • BQP • BPP • RP • ZPP • PCP • IP • PH