Exponential hierarchy

In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes that is an exponential time analogue of the polynomial hierarchy. As elsewhere in complexity theory, “exponential” is used in two different meanings (linear exponential bounds  2 c n {\displaystyle 2^{cn}} for a constant c, and full exponential bounds  2 n c {\displaystyle 2^{n^{c}}} ), leading to two versions of the exponential hierarchy.[1][2] This hierarchy is sometimes also referred to as the weak exponential hierarchy, to differentiate it from the strong exponential hierarchy.[2][3]

## EH

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The complexity class EH is the union of the classes  Σ k E {\displaystyle \Sigma _{k}^{\mathsf {E}}} for all k, where  Σ k E = N E Σ k − 1 P {\displaystyle \Sigma _{k}^{\mathsf {E}}={\mathsf {NE}}^{\Sigma _{k-1}^{\mathsf {P}}}} (i.e., languages computable in nondeterministic time  2 c n {\displaystyle 2^{cn}} for some constant c with a  Σ k − 1 P {\displaystyle \Sigma _{k-1}^{\mathsf {P}}} oracle) and  Σ 0 E = E {\displaystyle \Sigma _{0}^{\mathsf {E}}={\mathsf {E}}} . One also defines

Π k E = c o N E Σ k − 1 P {\displaystyle \Pi _{k}^{\mathsf {E}}={\mathsf {coNE}}^{\Sigma _{k-1}^{\mathsf {P}}}} and  Δ k E = E Σ k − 1 P . {\displaystyle \Delta _{k}^{\mathsf {E}}={\mathsf {E}}^{\Sigma _{k-1}^{\mathsf {P}}}.}

An equivalent definition is that a language L is in  Σ k E {\displaystyle \Sigma _{k}^{\mathsf {E}}} if and only if it can be written in the form

x ∈ L ⟺ ∃ y 1 ∀ y 2 … Q y k R ( x , y 1 , … , y k ) , {\displaystyle x\in L\iff \exists y_{1}\forall y_{2}\dots Qy_{k}R(x,y_{1},\ldots ,y_{k}),}

where  R ( x , y 1 , … , y n ) {\displaystyle R(x,y_{1},\ldots ,y_{n})} is a predicate computable in time  2 c | x | {\displaystyle 2^{c|x|}} (which implicitly bounds the length of yi). Also equivalently, EH is the class of languages computable on an alternating Turing machine in time  2 c n {\displaystyle 2^{cn}} for some c with constantly many alternations.

## EXPH

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EXPH is the union of the classes  Σ k E X P {\displaystyle \Sigma _{k}^{\mathsf {EXP}}} , where  Σ k E X P = N E X P Σ k − 1 P {\displaystyle \Sigma _{k}^{\mathsf {EXP}}={\mathsf {NEXP}}^{\Sigma _{k-1}^{\mathsf {P}}}} (languages computable in nondeterministic time  2 n c {\displaystyle 2^{n^{c}}} for some constant c with a  Σ k − 1 P {\displaystyle \Sigma _{k-1}^{\mathsf {P}}} oracle),  Σ 0 E X P = E X P {\displaystyle \Sigma _{0}^{\mathsf {EXP}}={\mathsf {EXP}}} , and again:

Π k E X P = c o N E X P Σ k − 1 P , Δ k E X P = E X P Σ k − 1 P . {\displaystyle \Pi _{k}^{\mathsf {EXP}}={\mathsf {coNEXP}}^{\Sigma _{k-1}^{\mathsf {P}}},\Delta _{k}^{\mathsf {EXP}}={\mathsf {EXP}}^{\Sigma _{k-1}^{\mathsf {P}}}.}

A language L is in  Σ k E X P {\displaystyle \Sigma _{k}^{\mathsf {EXP}}} if and only if it can be written as

x ∈ L ⟺ ∃ y 1 ∀ y 2 … Q y k R ( x , y 1 , … , y k ) , {\displaystyle x\in L\iff \exists y_{1}\forall y_{2}\dots Qy_{k}R(x,y_{1},\ldots ,y_{k}),}

where  R ( x , y 1 , … , y k ) {\displaystyle R(x,y_{1},\ldots ,y_{k})} is computable in time  2 | x | c {\displaystyle 2^{|x|^{c}}} for some c, which again implicitly bounds the length of yi. Equivalently, EXPH is the class of languages computable in time  2 n c {\displaystyle 2^{n^{c}}} on an alternating Turing machine with constantly many alternations.

## Comparison

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E ⊆ NE ⊆ EH⊆ ESPACE,
EXP ⊆ NEXP ⊆ EXPH⊆ EXPSPACE,
EH ⊆ EXPH.

## References

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1. ^ Sarah Mocas, Separating classes in the exponential-time hierarchy from classes in PH, Theoretical Computer Science 158 (1996), no. 1–2, pp. 221–231.
2. ^ a b Anuj Dawar, Georg Gottlob, Lauri Hella, Capturing relativized complexity classes without order, Mathematical Logic Quarterly 44 (1998), no. 1, pp. 109–122.
3. ^ Hemachandra, Lane A. (1989). "The strong exponential hierarchy collapses". Journal of Computer and System Sciences. 39 (3): 299–322\. doi:10.1016/0022-0000(89)90025-1.

## External links

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Complexity Zoo: Class EH

* v
* t
* e

Complexity classes

Considered feasible|

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

Suspected infeasible|

* UP
* NP
* NP-complete
* NP-hard
* co-NP
* co-NP-complete
* TFNP
* FNP
* 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

List of complexity classes
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*[t]: Discuss this template
*[e]: Edit this template