# 2-EXPTIME

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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(22p(n)) time, where p(n) is a polynomial function of n.

In terms of DTIME,

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

We know

P $\subseteq$ NP $\subseteq$ PSPACE $\subseteq$ EXPTIME $\subseteq$ NEXPTIME $\subseteq$ EXPSPACE $\subseteq$ 2-EXPTIME $\subseteq$ 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 $\subseteq$ 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 $2^{2^{2^{n^k}}}$. This can be generalized to higher and higher time bounds.

## 2-EXPTIME-complete problems

Generalizations of many fully observable games are EXPTIME-complete. These games can be viewed as particular instance of a class of transition systems defined in terms of a set of state variables and actions/events that change the values of the state variables, together with the question of whether a winning strategy exists.

A generalization of this class of fully observable problems to partially observable problems lifts the complexity from EXPTIME-complete to 2-EXPTIME-complete.[2]

## References

1. ^ Christos Papadimitriou, Computational Complexity (1994), ISBN 978-0-201-53082-7. Section 20.1, corollary 3, page 495.
2. ^ Jussi Rintanen (2004). "Complexity of Planning with Partial Observability". Proceedings of International Conference on Automated Planning and Scheduling (AAAI Press): 345–354.