2-EXPTIME

In computational complexity theory, the complexity class 2-EXPTIME (sometimes called 2-EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2^2p(n)) time, where p(n) is a polynomial function of n.

In terms of DTIME,

${\displaystyle {\mathsf {2{\mbox{-}}EXPTIME}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DTIME}}\left(2^{2^{n^{k}}}\right).}$

We know

P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE ⊆ 2-EXPTIME ⊆ ELEMENTARY.

2-EXPTIME can also be reformulated as the space class AEXPSPACE, the problems that can be solved by an alternating Turing machine in exponential space. This is one way to see that EXPSPACE ⊆ 2-EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine.^[1]

2-EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 3-EXPTIME is defined similarly to 2-EXPTIME but with a triply exponential time bound ${\displaystyle 2^{2^{2^{n^{k}}}}}$. This can be generalized to higher and higher time bounds.