Editing Exponential hierarchy (section)

==EXPH==
EXPH is the union of the classes <math>\Sigma^{\mathsf{EXP}}_k</math>, where <math>\Sigma^{\mathsf{EXP}}_k=\mathsf{NEXP}^{\Sigma^\mathsf{P}_{k-1}}</math> (languages computable in nondeterministic time <math>2^{n^c}</math> for some constant ''c'' with a <math>\Sigma^\mathsf{P}_{k-1}</math> oracle), <math>\Sigma^{\mathsf{EXP}}_0 = \mathsf{EXP}</math>, and again:

:<math>\Pi^{\mathsf{EXP}}_k=\mathsf{coNEXP}^{\Sigma^\mathsf{P}_{k-1}}, \Delta^{\mathsf{EXP}}_k=\mathsf{EXP}^{\Sigma^\mathsf{P}_{k-1}}.</math> 

A language ''L'' is in <math>\Sigma^{\mathsf{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,\ldots,y_k),</math>

where <math>R(x,y_1,\ldots,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.