Equals the union of DSPACE(2p(n)) over all polynomials p.
See also: ESPACE.
Given a first-order statement about real numbers, involving only addition and comparison (no multiplication), we can decide in EXPSPACE whether it's true or not [Ber80].
Literally, the class of ALL languages.
ALL is a gargantuan beast that's been wreaking havoc in the Zoo of late.
First [Aar04b] observed that PP/rpoly (PP with polynomial-size randomized advice) equals ALL, as does PostBQP/qpoly (PostBQP with polynomial-size quantum advice).
Then [Raz05] showed that QIP/qpoly, and even IP(2)/rpoly, equal ALL.
Nor is it hard to show that MAEXP/rpoly = ALL.
Also, per [Aar18], PDQP/qpoly = ALL.
On the other hand, even though PSPACE contains PP, and EXPSPACE contains MAEXP, it's easy to see that PSPACE/rpoly = PSPACE/poly and EXPSPACE/rpoly = EXPSPACE/poly are not ALL.
So does ALL have no respect for complexity class inclusions at ALL? (Sorry.)
It is not as contradictory as it first seems. The deterministic base class in all of these examples is modified by computational non-determinism after it is modified by advice. For example, MAEXP/rpoly means M(AEXP/rpoly), while (MAEXP)/rpoly equals MAEXP/poly by a standard argument. In other words, it's only the verifier, not the prover or post-selector, who receives the randomized or quantum advice. The prover knows a description of the advice state, but not its measured values. Modification by /rpoly does preserve class inclusions when it is applied after other changes.
Equals DSPACE(2O(n)).
If E = ESPACE then P = BPP [HY84].
Indeed if E has nonzero measure in ESPACE then P = BPP [Lut91].
ESPACE is not contained in P/poly [Kan82].
Is not contained in BQP/mpoly [NY03].
See also: EXPSPACE.
Nondeterministic exponential time (i.e. NTIME(2p(n)) for p a polynomial).
Equals MIP [BFL91] (but not relative to all oracles).
NEXP is in P/poly if and only if NEXP = MA [IKW01].
[KI02] show the following:
Does not equal EXP if and only if there is a sparse set in NP that is not in P.
There exists an oracle relative to which EXP = NEXP but still P does not equal NP [Dek76].
The theory of reals with addition (see EXPSPACE) is hard for NEXP [FR74].