NEXPTIME

NEXPTIME is a complexity class in computational theory that describes the set of decision problems solvable by a nondeterministic Turing machine in exponential time relative to the input size. It is defined as the class of problems decidable by a nondeterministic Turing machine in time bounded by *O*(2^(p(n))), where *p(n)* is a polynomial function of the input size *n*. This class is analogous to the deterministic complexity class EXPTIME, which consists of problems solvable in exponential time by a deterministic Turing machine.

NEXPTIME is a superset of NP, the class of problems verifiable in polynomial time by a nondeterministic Turing machine. While NP problems can be solved efficiently by a nondeterministic machine, NEXPTIME problems require exponential time even with nondeterminism. The relationship between NEXPTIME and NP is not fully understood, but it is known that NEXPTIME contains NP unless NP = EXPTIME, which is widely considered unlikely.

A key feature of NEXPTIME is its closure under complementation, meaning that if a problem is in NEXPTIME, its complement (the problem with the answer "yes" and "no" swapped) is also in NEXPTIME. This contrasts with NP, where complementation is not guaranteed unless NP = co-NP. NEXPTIME is also closed under Boolean operations such as union, intersection, and complement, but it is not known whether it is closed under other operations like reduction.

NEXPTIME is believed to be strictly larger than NP, but the exact boundaries of the class remain an open question in complexity theory. Many naturally occurring problems, such as certain types of optimization or constraint satisfaction problems, are conjectured to lie within NEXPTIME. However, proving membership in or exclusion from NEXPTIME is generally difficult due to the lack of known polynomial-time reductions between NEXPTIME and other well-studied complexity classes.