The class of decision problems solvable by an NP machine such that
Defined in [AR88].
There exists an oracle relative to which P, UP, FewP, and NP are all distinct [Rub88].
Also, there exists an oracle relative to which FewP does not have a Turing-complete set [HJV93].
Contained in Few.
See also the survey [Tor90].
The class of decision problems solvable by an NP machine such that
Defined in [PZ83].
Contains Graph Isomorphism; indeed, Graph Isomorphism is low for ⊕P [AK02].
There exists an oracle relative to which P = ⊕P but P is not equal to NP (and indeed NP = EXP) [BBF98].
Equals Mod2^mP for every positive integer m.
Roughly, the class of decision problems for which the following holds. For all polynomials p(n), there exist GapP functions f and g such that for all inputs x with n=|x|,
Defined in [Li93], where the following was also shown:
The abbreviation APP is also used for Approximable in Probabilistic Polynomial Time, see AxPP.
Contains FewP [Li93] and contains YQP*, YMA*, and YP* [Yir24].
The class of decision problems solvable by an NP machine such that
Also called FewPaths.
Defined in [CH89].
Contains FewP, and is contained in PFewP [Kob89] and in SPP [FFK94].
See also the survey [Tor90].
Same as FewP but for logspace-bounded (i.e. NL) machines.
Defined in [BDH+92], where it was also shown that LogFewNL is contained in ModZkL for all k>1.
The class of decision problems solvable by an NP machine such that
We may additionally require that all paths make the same number of binary nondeterministic choices, but then the second condition has to be modified so that if the answer is "yes", the number of accepting and rejecting paths differ by 2. (If the total number of paths is even then the numbers can't differ by 1.)
Defined in [FFK94], where it was also shown that SPP is low for PP, C=P, ModkP, and SPP itself. (I.e. adding SPP as an oracle does not increase the power of these classes.)
Independently defined in [OH93], who called the class XP.
Contained in LWPP, C=P, and WPP among other classes.
Contains FewP; indeed, FewP is low for SPP, so that SPPFewP = SPP [FFK94].
Contains the problem of deciding whether a graph has any nontrivial automorphisms [KST92].
Indeed, contains graph isomorphism [AK02].
Contains a whole gaggle of problems for solvable black-box groups: solvability testing, membership testing, subgroup testing, normality testing, order verification, nilpotetence testing, group isomorphism, and group intersection [Vin04]
[AK02] also showed that the Hidden Subgroup Problem for permutation groups, of interest in quantum computing, is in FPSPP.