PH (complexity)

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 is contained in the complexity classes P^PP (the class of problems that are decidable by a polynomial time Turing machine with an access to PP oracle) and 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.