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 $2^{cn}$ for a constant c, and full exponential bounds $2^{n^c}$), leading to two versions of the exponential hierarchy. This hierarchy is sometimes also referred to as the weak exponential hierarchy, to differentiate it from the strong exponential hierarchy.

EH

The complexity class EH is the union of the classes ${\mathrm{\Sigma }}_k^{𝖤}$ for all k, where ${\mathrm{\Sigma }}_k^{𝖤}={𝖭𝖤}^{{\mathrm{\Sigma }}_{k-1}^{𝖯}}$ (i.e., languages computable in nondeterministic time $2^{cn}$ for some constant c with a ${\mathrm{\Sigma }}_{k-1}^{𝖯}$ oracle) and ${\mathrm{\Sigma }}_0^{𝖤}=𝖤$. One also defines

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

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

$x\in L⟺\mathrm{\exists }{y}_1\mathrm{\forall }{y}_2\dots Q{y}_{k}R(x,y_1,\dots ,y_k),$

where $R(x,y_1,\dots ,y_n)$ is a predicate computable in time $2^{c|x|}$ (which implicitly bounds the length of y_i). Also equivalently, EH is the class of languages computable on an alternating Turing machine in time $2^{cn}$ for some c with constantly many alternations.

EXPH

EXPH is the union of the classes ${\mathrm{\Sigma }}_k^{𝖤𝖷𝖯}$, where ${\mathrm{\Sigma }}_k^{𝖤𝖷𝖯}={𝖭𝖤𝖷𝖯}^{{\mathrm{\Sigma }}_{k-1}^{𝖯}}$ (languages computable in nondeterministic time $2^{n^c}$ for some constant c with a ${\mathrm{\Sigma }}_{k-1}^{𝖯}$ oracle), ${\mathrm{\Sigma }}_0^{𝖤𝖷𝖯}=𝖤𝖷𝖯$, and again:

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

A language L is in ${\mathrm{\Sigma }}_k^{𝖤𝖷𝖯}$ if and only if it can be written as

$x\in L⟺\mathrm{\exists }{y}_1\mathrm{\forall }{y}_2\dots Q{y}_{k}R(x,y_1,\dots ,y_k),$

where $R(x,y_1,\dots ,y_k)$ is computable in time $2^{|x{|}^c}$ for some c, which again implicitly bounds the length of y_i. Equivalently, EXPH is the class of languages computable in time $2^{n^c}$ on an alternating Turing machine with constantly many alternations.

Comparison

E ⊆ NE ⊆ EH⊆ ESPACE,

EXP ⊆ NEXP ⊆ EXPH⊆ EXPSPACE,

EH ⊆ EXPH.