20.2.09

EXP

"EXP" redirects here; for other uses, see exp.
In computational complexity theory, the complexity class EXPTIME (sometimes called EXP) 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,


We know

P NP PSPACE EXPTIME NEXPTIME EXPSPACE
and also, by the time hierarchy theorem and the space hierarchy theorem, that

P EXPTIME and NP NEXPTIME and PSPACE 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 experts believe all the inclusions are proper. It's also known that if P = NP, then EXPTIME = NEXPTIME, the class of problems solvable in exponential time by a nondeterministic Turing machine.[1] More precisely, EXPTIME ≠ NEXPTIME 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 EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine.[3]

EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 2-EXPTIME is defined similarly to EXPTIME but with a doubly-exponential time bound . This can be generalized to higher and higher time bounds.


[edit] EXPTIME-complete
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