Last week we gave the history of the complexity class SPP and described GapP functions. This week we will give a definition of SPP and many of the class' amazing properties.

A language L is in SPP if there is a GapP function f such that

- If x is in L then f(x)=1.
- If x is not in L then f(x)=0.

If we used #P functions instead of GapP functions we have the definition of UP. SPP contains UP since every #P function is a GapP function. In fact SPP contains FewP and even Few where we don't believe such languages are in UP.

SPP is the smallest Gap-definable class, i.e., the smallest class that
can be defined by GapP functions as above. There are a number of
common Gap-definable classes, for example from the Zoo:
⊕P, AWPP, C_{=}P, ModP, Mod_{k}P, MP, AmpMP, PP,
WPP and of course SPP. SPP is contained in all of these classes. AWPP
is the smallest classical class known to contain BQP, the class of
problems with efficient quantum algorithms, though it is not known if
BQP is itself Gap-definable.

SPP is exactly equal to the low sets for GapP, i.e., SPP is exactly
the set of oracles A such that for any NP machine M, the number of
accepting minus the number of rejecting paths of M^A(x) is still an
(unrelativized) GapP function. This means that SPP is low for all of
the Gap-definable classes, for example that ⊕P^{SPP} =
⊕P. This also means that SPP is self-low: SPP^{SPP} =
SPP which means SPP is closed under union, complement and in fact any
Turing-reduction.

Kobler, Schoning and Toran showed that graph automorphism is in SPP and very recently Arvind and Kurur have show that graph isomorphism is in SPP. This means that graph isomorphism sits in and is in fact low for every Gap-definable class.

The decision tree version of SPP is interesting. A function f on n
bits is in this class if there is a polynomial g with polylog degree
such that f(x)=g(x) on all x in {0,1}^{*}. All such functions
have low deterministic decision tree complexity--the first complexity
application of a combinatorial lemma of Nisan and
Szegedy. Applications of this result include relativized worlds where
SPP does not have complete sets or where P = SPP and the
polynomial-time hierarchy is infinite.

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