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 formula_1 for a constant "c", and full exponential bounds formula_2), leading to two versions of the exponential hierarchy. This hierarchy is sometimes also referred to as the "weak" exponential hierarchy, to differentiate it from the "strong" exponential hierarchy.
EH.
The complexity class EH is the union of the classes formula_3 for all "k", where formula_4 (i.e., languages computable in nondeterministic time formula_1 for some constant "c" with a formula_6 oracle) and formula_7. One also defines 
formula_8 and formula_9
An equivalent definition is that a language "L" is in formula_3 if and only if it can be written in the form
formula_11
where formula_12 is a predicate computable in time formula_13 (which implicitly bounds the length of "yi"). Also equivalently, EH is the class of languages computable on an alternating Turing machine in time formula_1 for some "c" with constantly many alternations.
EXPH.
EXPH is the union of the classes formula_15, where formula_16 (languages computable in nondeterministic time formula_2 for some constant "c" with a formula_6 oracle), formula_19, and again:
formula_20 
A language "L" is in formula_15 if and only if it can be written as
formula_22
where formula_23 is computable in time formula_24 for some "c", which again implicitly bounds the length of "yi". Equivalently, EXPH is the class of languages computable in time formula_2 on an alternating Turing machine with constantly many alternations.
E ⊆ NE ⊆ EH⊆ ESPACE, 
EXP ⊆ NEXP ⊆ EXPH⊆ EXPSPACE, 
EH ⊆ EXPH.
References.
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External links.
"Complexity Zoo": Class EH