PH (complexity)

Short description: Class in computational complexity theory

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:

PH=⋃k∈ℕΔkP

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 and PSPACE.

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

## Contents

* 1 Relationship to other classes
* 2 Examples
* 3 References
* 4 General references

## Relationship to other classes

Further information: Polynomial hierarchy § Relationships to other classes

| Unsolved problem in computer science:

P=?NP

(more unsolved problems in computer science)

|

| Unsolved problem in computer science:

PH=?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 = 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.

## Examples

## References

1. ↑ Stockmeyer, Larry J. (1977). "The polynomial-time hierarchy". Theor. Comput. Sci. 3: 1–22. doi:10.1016/0304-3975(76)90061-X.
2. ↑ Lautemann, Clemens (1983-11-08). "BPP and the polynomial hierarchy" (in en). Information Processing Letters 17 (4): 215–217. doi:10.1016/0020-0190(83)90044-3. ISSN 0020-0190. https://dx.doi.org/10.1016/0020-0190(83)90044-3.
3. ↑ Aaronson, Scott (2009). "Proc. 42nd Symposium on Theory of Computing (STOC 2009)". Association for Computing Machinery. pp. 141–150. 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. https://www.quantamagazine.org/finally-a-problem-that-only-quantum-computers-will-ever-be-able-to-solve-20180621/.
5. ↑ Hemaspaandra, Lane (2018). "17.5 Complexity classes". in Rosen, Kenneth H.. 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. 7. Berlin: Springer-Verlag. p. 66. ISBN 3-540-66752-0.
* Complexity Zoo: PH

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Important complexity classes (more)

Considered feasible|

* DLOGTIME
* AC0
* ACC0
* TC0
* L
* SL
* RL
* NL
* NC
* SC
* CC
* P
* P-complete
* ZPP
* RP
* BPP
* BQP
* APX

Suspected infeasible|

* UP
* NP
* NP-complete
* NP-hard
* co-NP
* co-NP-complete
* AM
* QMA
* PH
* ⊕P
* PP
* #P
* #P-complete
* IP
* PSPACE
* PSPACE-complete

Considered infeasible|

* EXPTIME
* NEXPTIME
* EXPSPACE
* 2-EXPTIME
* ELEMENTARY
* PR
* R
* RE
* ALL

Class hierarchies|

* Polynomial hierarchy
* Exponential hierarchy
* Grzegorczyk hierarchy
* Arithmetical hierarchy
* Boolean hierarchy

Families of classes|

* DTIME
* NTIME
* DSPACE
* NSPACE
* Probabilistically checkable proof
* Interactive proof system

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