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.[1] It is a special case of hierarchy of bounded alternating Turing machine. It is contained in P#P = PPP (by Toda's theorem; the class of problems that are decidable by a polynomial time Turing machine with access to a #P or equivalently PP oracle), 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.[2][3]

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

References

  1. ↑ Stockmeyer, Larry J. (1977). "The polynomial-time hierarchy". Theor. Comput. Sci. 3: 1–22. doi:10.1016/0304-3975(76)90061-X. Zbl 0353.02024.
  2. ↑ Aaronson, Scott (2009). "BQP and the Polynomial Hierarchy". Proc. 42nd Symposium on Theory of Computing (STOC 2009). Association for Computing Machinery. pp. 141–150. arXiv:0910.4698. doi:10.1145/1806689.1806711. ECCC TR09-104.
  3. ↑ https://www.quantamagazine.org/finally-a-problem-that-only-quantum-computers-will-ever-be-able-to-solve-20180621/

General references

  • Bürgisser, Peter (2000). Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics. 7. Berlin: Springer-Verlag. p. 66. ISBN 3-540-66752-0. Zbl 0948.68082.
  • Complexity Zoo: PH


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