PH (complexity)

Inclusions of complexity classes including P, NP, co-NP, BPP, P/poly, PH, and PSPACE

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

\mathrm {PH} =\bigcup _{k\in \mathbb {N} }\Delta _{k}^{\mathrm {P} }

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 = P^PP and PSPACE.

PH has a simple logical characterization: it is the set of languages expressible by second-order logic.

Relationship to other classes

Unsolved problem in computer science:

⁠

{\mathsf {P}}{\overset {?}{=}}{\mathsf {NP}}

⁠

(more unsolved problems in computer science)

Unsolved problem in computer science:

⁠

{\mathsf {PH}}{\overset {?}{=}}{\mathsf {PSPACE}}

⁠

(more unsolved problems in computer science)

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^[2] (this is the Sipser–Lautemann theorem) 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.^[3]^[4]

P = NP if and only if P = PH.^[5] This may simplify a potential proof of P ≠ NP, since it is only necessary to separate P from the more general class PH.

PH is a subset of P^#P = P^PP 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.

Examples

References

^ 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.
^ Lautemann, Clemens (1983-11-08). "BPP and the polynomial hierarchy". Information Processing Letters. 17 (4): 215–217. doi:10.1016/0020-0190(83)90044-3. ISSN 0020-0190.
^ 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.
^ "Finally, a Problem That Only Quantum Computers Will Ever be Able to Solve". 21 June 2018.
^ Hemaspaandra, Lane (2018). "17.5 Complexity classes". In Rosen, Kenneth H. (ed.). Handbook of Discrete and Combinatorial Mathematics. Discrete Mathematics and Its Applications (2nd ed.). CRC Press. pp. 1308–1314. ISBN 9781351644051.

General references

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

P ≟ NP

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[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] Lautemann, Clemens (1983-11-08). "BPP and the polynomial hierarchy". Information Processing Letters. 17 (4): 215–217. doi:10.1016/0020-0190(83)90044-3. ISSN 0020-0190.

[3] 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.

[4] "Finally, a Problem That Only Quantum Computers Will Ever be Able to Solve". 21 June 2018.

[5] Hemaspaandra, Lane (2018). "17.5 Complexity classes". In Rosen, Kenneth H. (ed.). Handbook of Discrete and Combinatorial Mathematics. Discrete Mathematics and Its Applications (2nd ed.). CRC Press. pp. 1308–1314. ISBN 9781351644051.

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PH (complexity)

Relationship to other classes

Examples

References

General references

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