Editing NEXPTIME (section)

== Alternative characterizations ==
In [[descriptive complexity]], the sets of natural numbers that can be recognized in NEXPTIME are exactly those that form the [[spectrum of a sentence]], the set of sizes of [[finite model]]s of some logical sentence.<ref>{{citation | last1=Jones | first1=Neil D. | last2=Selman | first2=Alan L.|author2-link=Alan Selman | title=Turing machines and the spectra of first-order formulas | zbl=0288.02021 | journal=J. Symb. Log. | volume=39 | issue=1 | pages=139–150 | year=1974 | doi=10.2307/2272354 | jstor=2272354 }}</ref>

'''NEXPTIME''' often arises in the context of [[interactive proof system]]s, where there are two major characterizations of it. The first is the '''MIP''' proof system, where we have two all-powerful provers which communicate with a randomized polynomial-time verifier (but not with each other). If the string is in the language, they must be able to convince the verifier of this with high probability. If the string is not in the language, they must not be able to collaboratively trick the verifier into accepting the string except with low probability. The fact that '''MIP''' proof systems can solve every problem in '''NEXPTIME''' is quite impressive when we consider that when only one prover is present, we can only recognize all of '''PSPACE'''; the verifier's ability to "cross-examine" the two provers gives it great power. See [[interactive proof system#MIP]] for more details.

Another interactive proof system characterizing '''NEXPTIME''' is a certain class of [[probabilistically checkable proof]]s. Recall that '''NP''' can be seen as the class of problems where an all-powerful prover gives a purported proof that a string is in the language, and a deterministic polynomial-time machine verifies that it is a valid proof. We make two changes to this setup:

* Add randomness, the ability to flip coins, to the verifier machine.
* Instead of simply giving the purported proof to the verifier on a tape, give it random access to the proof. The verifier can specify an index into the proof string and receive the corresponding bit. Since the verifier can write an index of polynomial length, it can potentially index into an exponentially long proof string.

These two extensions together greatly extend the proof system's power, enabling it to recognize all languages in '''NEXPTIME'''. The class is called '''PCP'''(poly, poly). What more, in this characterization the verifier may be limited to read only a constant number of bits, i.e. '''NEXPTIME''' = '''PCP'''(poly, 1). See [[probabilistically checkable proof]]s for more details.