(Redirected from EXPTIME)

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/EXPTIME Retrieved:2015-6-7.
• In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n.

In terms of DTIME, : $\displaystyle{ \mbox{EXPTIME} = \bigcup_{k \in \mathbb{N} } \mbox{ DTIME } \left( 2^{ n^k } \right) . }$ We know

:P $\displaystyle{ \subseteq }$ NP $\displaystyle{ \subseteq }$ PSPACE $\displaystyle{ \subseteq }$ EXPTIME $\displaystyle{ \subseteq }$ NEXPTIME $\displaystyle{ \subseteq }$ EXPSPACE

and also, by the time hierarchy theorem and the space hierarchy theorem, that

:P $\displaystyle{ \subsetneq }$ EXPTIMEandNP $\displaystyle{ \subsetneq }$ NEXPTIMEandPSPACE $\displaystyle{ \subsetneq }$ EXPSPACE

so at least one of the first three inclusions and at least one of the last three inclusions must be proper, but it is not known which ones are. Most expertsbelieve all the inclusions are proper. It's also known that if, then, the class of problems solvable in exponential time by a nondeterministic Turing machine. [1] More precisely, EXPTIMENEXPTIME if and only if there exist sparse languages in NP that are not in P. [2] EXPTIME can also be reformulated as the space class APSPACE, the problems that can be solved by an alternating Turing machine in polynomial space. This is one way to see that PSPACE $\displaystyle{ \subseteq }$ EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine. [3]

EXPTIME is one class in an exponential hierarchy of complexity classes with increasingly more complex oracles or quantifier alternations. The class 2-EXPTIME is defined similarly to EXPTIME but with a doubly exponential time bound $\displaystyle{ 2^{2^n} }$ . This can be generalized to higher and higher time bounds.

1. Section 20.1, page 491.
2. Juris Hartmanis, Neil Immerman, Vivian Sewelson. Sparse Sets in NP-P: EXPTIME versus NEXPTIME. Information and Control, volume 65, issue 2/3, pp.158–181. 1985. At ACM Digital Library
3. Papadimitriou (1994), section 20.1, corollary 3, page 495.