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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:^[1]^[2]

EH is the union of the classes $\Sigma _{k}^{E}$ for all k, where $\Sigma _{k}^{E}=\mathrm {NE} ^{\Sigma _{k-1}^{P}}$ (i.e., languages computable in nondeterministic time $2^{cn}$ for some constant c with a $\Sigma _{k-1}^{P}$ oracle). One also defines $\Pi _{k}^{E}=\mathrm {coNE} ^{\Sigma _{k-1}^{P}}$ , $\Delta _{k}^{E}=\mathrm {E} ^{\Sigma _{k-1}^{P}}$ . An equivalent definition is that a language L is in $\Sigma _{k}^{E}$ if and only if it can be written in the form

x\in L\iff \exists y_{1}\,\forall y_{2}\dots Qy_{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 is the union of the classes $\Sigma _{k}^{EXP}$ , where $\Sigma _{k}^{EXP}=\mathrm {NEXP} ^{\Sigma _{k-1}^{P}}$ (languages computable in nondeterministic time $2^{n^{c}}$ for some constant c with a $\Sigma _{k-1}^{P}$ oracle), and again $\Pi _{k}^{EXP}=\mathrm {coNEXP} ^{\Sigma _{k-1}^{P}}$ , $\Delta _{k}^{EXP}=\mathrm {EXP} ^{\Sigma _{k-1}^{P}}$ . A language L is in $\Sigma _{k}^{EXP}$ if and only if it can be written as

x\in L\iff \exists y_{1}\,\forall y_{2}\dots Qy_{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.

References

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↑ Sarah Mocas, Separating classes in the exponential-time hierarchy from classes in PH, Theoretical Computer Science 158 (1996), no. 1–2, pp. 221–231.
↑ Anuj Dawar, Georg Gottlob, Lauri Hella, Capturing relativized complexity classes without order, Mathematical Logic Quarterly 44 (1998), no. 1, pp. 109–122.

[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] Anuj Dawar, Georg Gottlob, Lauri Hella, Capturing relativized complexity classes without order, Mathematical Logic Quarterly 44 (1998), no. 1, pp. 109–122.

[1]

[2]

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+In [[computational complexity theory]], the '''exponential hierarchy''' is a hierarchy of [[complexity class]]es, which is an [[EXPTIME|exponential time]] analogue of the [[polynomial hierarchy]]. As elsewhere in complexity theory, “exponential” is used in two different meanings (linear exponential bounds <math>2^{cn}</math> for a constant ''c'', and full exponential bounds <math>2^{n^c}</math>), leading to two versions of the exponential hierarchy:<ref>Sarah Mocas, Separating classes in the exponential-time hierarchy from classes in ''PH'', Theoretical Computer Science 158 (1996), no.&nbsp;1–2, pp.&nbsp;221–231.</ref><ref>Anuj Dawar, Georg Gottlob, Lauri Hella, Capturing relativized complexity classes without order, Mathematical Logic Quarterly 44 (1998), no.&nbsp;1, pp.&nbsp;109–122.</ref>
+*EH is the union of the classes <math>\Sigma^E_k</math> for all ''k'', where <math>\Sigma^E_k=\mathrm{NE}^{\Sigma^P_{k-1}}</math> (i.e., languages computable in [[nondeterministic Turing machine|nondeterministic]] time <math>2^{cn}</math> for some constant ''c'' with a <math>\Sigma^P_{k-1}</math> [[oracle Turing machine|oracle]]). One also defines <math>\Pi^E_k=\mathrm{coNE}^{\Sigma^P_{k-1}}</math>, <math>\Delta^E_k=\mathrm E^{\Sigma^P_{k-1}}</math>. An equivalent definition is that a language ''L'' is in <math>\Sigma^E_k</math> if and only if it can be written in the form
+::<math>x\in L\iff\exists y_1\,\forall y_2\dots Qy_k\,R(x,y_1,\dots,y_k),</math>
+:where <math>R(x,y_1,\dots,y_n)</math> is a predicate computable in time <math>2^{c|x|}</math> (which implicitly bounds the length of ''y<sub>i</sub>''). Also equivalently, EH is the class of languages computable on an [[alternating Turing machine]] in time <math>2^{cn}</math> for some ''c'' with constantly many alternations.
+*EXPH is the union of the classes <math>\Sigma^{EXP}_k</math>, where <math>\Sigma^{EXP}_k=\mathrm{NEXP}^{\Sigma^P_{k-1}}</math> (languages computable in nondeterministic time <math>2^{n^c}</math> for some constant ''c'' with a <math>\Sigma^P_{k-1}</math> oracle), and again <math>\Pi^{EXP}_k=\mathrm{coNEXP}^{\Sigma^P_{k-1}}</math>, <math>\Delta^{EXP}_k=\mathrm{EXP}^{\Sigma^P_{k-1}}</math>. A language ''L'' is in <math>\Sigma^{EXP}_k</math> if and only if it can be written as
+::<math>x\in L\iff\exists y_1\,\forall y_2\dots Qy_k\,R(x,y_1,\dots,y_k),</math>
+:where <math>R(x,y_1,\dots,y_k)</math> is computable in time <math>2^{|x|^c}</math> for some ''c'', which again implicitly bounds the length of ''y<sub>i</sub>''. Equivalently, EXPH is the class of languages computable in time <math>2^{n^c}</math> on an alternating Turing machine with constantly many alternations.
+We have [[E (complexity)|E]] ⊆ [[NE (complexity)|NE]] ⊆ EH ⊆ [[ESPACE]], [[EXPTIME|EXP]] ⊆ [[NEXPTIME|NEXP]] ⊆ EXPH ⊆ [[EXPSPACE]], and EH ⊆ EXPH.
+== References ==
+{{reflist}}
+==External links==
+{{CZoo|Class EH|E#eh}}
+{{ComplexityClasses}}
+{{DEFAULTSORT:Exponential Hierarchy}}
+[[Category:Complexity classes]]

Parallel (geometry): Difference between revisions

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