Exponential hierarchy

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 for a constant c, and full exponential bounds ), leading to two versions of the exponential hierarchy.[1][2]

EHEdit

EH is the union of the classes   for all k, where   (i.e., languages computable in nondeterministic time   for some constant c with a   oracle). One also defines

 

An equivalent definition is that a language L is in   if and only if it can be written in the form

 

where   is a predicate computable in time   (which implicitly bounds the length of yi). Also equivalently, EH is the class of languages computable on an alternating Turing machine in time   for some c with constantly many alternations.

EXPHEdit

EXPH is the union of the classes  , where   (languages computable in nondeterministic time   for some constant c with a   oracle), and again:

 

A language L is in   if and only if it can be written as

 

where   is computable in time   for some c, which again implicitly bounds the length of yi. Equivalently, EXPH is the class of languages computable in time   on an alternating Turing machine with constantly many alternations.

ComparisonEdit

ENE ⊆ EH ⊆ ESPACE,
EXPNEXP ⊆ EXPH ⊆ EXPSPACE,
EH ⊆ EXPH.

ReferencesEdit

  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.

External linksEdit

Complexity Zoo: Class EH