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 with two different meanings (linear exponential bounds ${\displaystyle 2^{cn}}$ for a constant c, and full exponential bounds ${\displaystyle 2^{n^{c}}}$), leading to two versions of the exponential hierarchy.¹² These hierarchies are sometimes also referred to as the weak exponential hierarchies, to differentiate them from the strong exponential hierarchy, which contains both of the weak hierarchies.²³

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

${\displaystyle \Pi _{k}^{\mathsf {E}}={\mathsf {coNE}}^{\Sigma _{k-1}^{\mathsf {P}}}}$ and ${\displaystyle \Delta _{k}^{\mathsf {E}}={\mathsf {E}}^{\Sigma _{k-1}^{\mathsf {P}}}.}$

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

${\displaystyle x\in L\iff \exists y_{1}\forall y_{2}\dots Qy_{k}R(x,y_{1},\ldots ,y_{k}),}$

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

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

${\displaystyle \Pi _{k}^{\mathsf {EXP}}={\mathsf {coNEXP}}^{\Sigma _{k-1}^{\mathsf {P}}},\Delta _{k}^{\mathsf {EXP}}={\mathsf {EXP}}^{\Sigma _{k-1}^{\mathsf {P}}}.}$

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

${\displaystyle x\in L\iff \exists y_{1}\forall y_{2}\dots Qy_{k}R(x,y_{1},\ldots ,y_{k}),}$

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

The strong exponential hierarchy, denoted SEH, is the union of NE, NP^NE, NP^NPNE, and so on.⁴

The same class is obtained if we replace NE by NEXP.⁴

E ⊆ NE ⊆ EH⊆ ESPACE,

EXP ⊆ NEXP ⊆ EXPH⊆ EXPSPACE,

EH ⊆ EXPH.