PH (complexity)

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

:

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.

References

*L. J. Stockmeyer, "The polynomial hierarchy", "Theoretical Computer Science", Vol. 3 (1976), pp. 1–22.
* [http://qwiki.caltech.edu/wiki/Complexity_Zoo#ph The Complexity Zoo: PH]