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 was first defined by Larry Stockmeyer. It is a special case of hierarchy of bounded alternating Turing machine. 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. However, there is some evidence that BQP, the class of problems solvable in polynomial time by a quantum computer, is not contained in PH (Aaronson 2010).

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

• Larry J. Stockmeyer, "The polynomial hierarchy", Theoretical Computer Science, Vol. 3 (1976), pp. 1–22.
• Scott Aaronson, BQP and the Polynomial Hierarchy, ACM STOC (2010), arXiv:0910.4698, ECCC TR09-104.
• Complexity Zoo: PH