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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>
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*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
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::<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]]

Revision as of 08:07, 27 February 2014

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