Parallel (geometry)

In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes, which 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:

• EH is the union of the classes ${\mathrm{\Sigma }}_k^E$ for all k, where ${\mathrm{\Sigma }}_k^E={\mathrm{N}\mathrm{E}}^{{\mathrm{\Sigma }}_{k-1}^P}$ (i.e., languages computable in nondeterministic time $2^{cn}$ for some constant c with a ${\mathrm{\Sigma }}_{k-1}^P$ oracle). One also defines ${\mathrm{\Pi }}_k^E={\mathrm{c}\mathrm{o}\mathrm{N}\mathrm{E}}^{{\mathrm{\Sigma }}_{k-1}^P}$, ${\mathrm{\Delta }}_k^E={\mathrm{E}}^{{\mathrm{\Sigma }}_{k-1}^P}$. 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 $2^{c\left|x\right|}$ (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 is the union of the classes ${\mathrm{\Sigma }}_k^{EXP}$, where ${\mathrm{\Sigma }}_k^{EXP}={\mathrm{N}\mathrm{E}\mathrm{X}\mathrm{P}}^{{\mathrm{\Sigma }}_{k-1}^P}$ (languages computable in nondeterministic time $2^{n^c}$ for some constant c with a ${\mathrm{\Sigma }}_{k-1}^P$ oracle), and again ${\mathrm{\Pi }}_k^{EXP}={\mathrm{c}\mathrm{o}\mathrm{N}\mathrm{E}\mathrm{X}\mathrm{P}}^{{\mathrm{\Sigma }}_{k-1}^P}$, ${\mathrm{\Delta }}_k^{EXP}={\mathrm{E}\mathrm{X}\mathrm{P}}^{{\mathrm{\Sigma }}_{k-1}^P}$. 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.

We have E ⊆ NE ⊆ EH ⊆ ESPACE, EXP ⊆ NEXP ⊆ EXPH ⊆ EXPSPACE, and EH ⊆ EXPH.

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