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
for a constant c, and full exponential bounds
), 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] The complexity class EH is the union of the classes
(i.e., languages computable in nondeterministic time
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.
EXPH is the union of the classes
(languages computable in nondeterministic time
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.
Complexity Zoo: Class EH