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 ${\displaystyle 2^{cn))$ for a constant c, and full exponential bounds ${\displaystyle 2^{n^{c))}$), 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

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

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.

Comparison

E ⊆ NE ⊆ EH⊆ ESPACE,

EXP ⊆ NEXP ⊆ EXPH⊆ EXPSPACE,

EH ⊆ EXPH.