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.^[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 $\Sigma _{k}^{\mathsf {E}}$ for all k, where $\Sigma _{k}^{\mathsf {E}}={\mathsf {NE}}^{\Sigma _{k-1}^{\mathsf {P}}}$ (i.e., languages computable in nondeterministic time $2^{cn}$ for some constant c with a $\Sigma _{k-1}^{\mathsf {P}}$ oracle) and $\Sigma _{0}^{\mathsf {E}}={\mathsf {E}}$ . One also defines

\Pi _{k}^{\mathsf {E}}={\mathsf {coNE}}^{\Sigma _{k-1}^{\mathsf {P}}}

and

\Delta _{k}^{\mathsf {E}}={\mathsf {E}}^{\Sigma _{k-1}^{\mathsf {P}}}.

An equivalent definition is that a language L is in $\Sigma _{k}^{\mathsf {E}}$ 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},\ldots ,y_{k}),

where $R(x,y_{1},\ldots ,y_{n})$ 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 $\Sigma _{k}^{\mathsf {EXP}}$ , where $\Sigma _{k}^{\mathsf {EXP}}={\mathsf {NEXP}}^{\Sigma _{k-1}^{\mathsf {P}}}$ (languages computable in nondeterministic time $2^{n^{c}}$ for some constant c with a $\Sigma _{k-1}^{\mathsf {P}}$ oracle), $\Sigma _{0}^{\mathsf {EXP}}={\mathsf {EXP}}$ , and again:

\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 $\Sigma _{k}^{\mathsf {EXP}}$ 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},\ldots ,y_{k}),

where $R(x,y_{1},\ldots ,y_{k})$ 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.

References

^ Sarah Mocas, Separating classes in the exponential-time hierarchy from classes in PH, Theoretical Computer Science 158 (1996), no. 1–2, pp. 221–231.
^ ^a ^b Anuj Dawar, Georg Gottlob, Lauri Hella, Capturing relativized complexity classes without order, Mathematical Logic Quarterly 44 (1998), no. 1, pp. 109–122.
^ Hemachandra, Lane A. (1989). "The strong exponential hierarchy collapses". Journal of Computer and System Sciences. 39 (3): 299–322. doi:10.1016/0022-0000(89)90025-1.

External links

Complexity Zoo: Class EH

[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.

[:0-2] Anuj Dawar, Georg Gottlob, Lauri Hella, Capturing relativized complexity classes without order, Mathematical Logic Quarterly 44 (1998), no. 1, pp. 109–122.

[3] Hemachandra, Lane A. (1989). "The strong exponential hierarchy collapses". Journal of Computer and System Sciences. 39 (3): 299–322. doi:10.1016/0022-0000(89)90025-1.

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Complexity classes
Considered feasible	DLOGTIME AC⁰ ACC⁰ TC⁰ L SL RL FL NL NL-complete NC SC CC P P-complete ZPP RP BPP BQP APX FP
Suspected infeasible	UP NP NP-complete NP-hard co-NP co-NP-complete TFNP FNP AM QMA PH ⊕P PP #P #P-complete IP PSPACE PSPACE-complete
Considered infeasible	EXPTIME NEXPTIME EXPSPACE 2-EXPTIME ELEMENTARY PR R RE ALL
Class hierarchies	Polynomial hierarchy Exponential hierarchy Grzegorczyk hierarchy Arithmetical hierarchy Boolean hierarchy
Families of classes	DTIME NTIME DSPACE NSPACE Probabilistically checkable proof Interactive proof system
List of complexity classes

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