Class Description

SPP: Stoic PP

The class of decision problems solvable by an NP machine such that

  1. If the answer is "no," then the number of accepting computation paths exactly equals the number of rejecting paths.
  2. If the answer is "yes," then these numbers differ by 1.

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.

Linked From

Few: FewP With Flexible Acceptance Mechanism

The class of decision problems solvable by an NP machine such that

  1. The number of accepting paths a is bounded by a polynomial in the size of the input x.
  2. For some polynomial-time predicate Q, Q(x,a) is true if and only if the answer is 'yes.'

Also called FewPaths.

Defined in [CH89].

Contains FewP, and is contained in PFewP [Kob89] and in SPP [FFK94].

See also the survey [Tor90].

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LWPP: Length-Dependent Wide PP

The class of decision problems solvable by an NP machine such that

  1. If the answer is "no," then the number of accepting computation paths exactly equals the number of rejecting paths.
  2. If the answer is "yes," then the difference of these numbers equals a function f(|x|) computable in polynomial time (i.e. FP). Here |x| is the length of the input x, and ``polynomial time means polynomial in |x|, the length of x, rather than the length of |x|.

Defined in [FFK94], where it was also shown that LWPP is low for PP and C=P. (I.e. adding LWPP as an oracle does not increase the power of these classes.)

Contained in WPP and AWPP.

Contains SPP.

Also, contains the graph isomorphism problem [KST92].

Contains a whole litter of problems for solvable black-box groups: group intersection, group factorization, coset intersection, and double-coset membership [Vin04].

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SPL: Stoic PL

Has the same relation to PL as SPP does to PP.

Contains the maximum matching and perfect matching problems under a pseudorandom assumption [ARZ99].

Contains UL.

Contained in C=L and ModkL.

Equals the set of problems low for GapL.

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UAP: Unambiguous Alternating Polynomial-Time

Same as AP, except we are promised that each existential quantifier has at most one 'yes' path, and each universal quantifier has at most one 'no' path.

Contains UP.

Defined in [NR98], where it was also shown that, even though AP = PSPACE, it is unlikely that the same is true for UAP, since UAP is contained in SPP.

[CGR+04] have also shown that UAPUAP = UAP, and that UAP contains Graph Isomorphism problem.

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WPP: Wide PP

The class of decision problems solvable by an NP machine such that

  1. If the answer is "no," then the number of accepting computation paths exactly equals the number of rejecting paths.
  2. If the answer is "yes," then their difference exactly equals a function f(x) computable in polynomial time (i.e. FP).

Defined in [FFK94].

Contained in C=P ∩ coC=P, as well as AWPP.

Contains SPP and LWPP.

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