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:

<math>\mbox{PH} = \bigcup_{k\in\mathbb{N}} \Delta_k\mbox{P}</math>

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.