With the new FOCS paper by Arvind and Kurur, "Graph Isomorphism in SPP", people have asked me why they should be interested in SPP, a class first defined by a paper by Steve Fenner, Stuart Kurtz and myself. I thought I would discuss how this class was developed and why we feel it is important.
Gill, in his seminal paper on probabilistic complexity classes, defined the class PP and asked whether the class was closed under intersection. In 1990, Fenner and Kurtz and later myself, decided to try a new approach to the question: Consider a class defined like PP but with additional restrictions, show that this class is closed under intersection and then show the class was really the same as PP. Kurtz had a philosophical approach to the problem and defined three variants of PP, Epicurean-PP, Cynical-PP and Stoic-PP.
Recall that PP is the set of languages L accepted by probabilistic machines such that x is in L exactly when the probability of accepting is greater than the probability of rejecting. Epicurean-PP machines were happy to accept but only rejected by barely rejecting--having one more rejecting paths than accepting paths. Cynical-PP machines were the opposite, willing to reject in any way but would only barely accept. Stoic-PP machines stood their ground and would just barely accept or barely reject. Cynical-PP turned out to be the same as the well-studied class C=P and Epicurean-P was co-C=P. Stoic-PP or SPP was new and thus a complexity class was born.
While it was easy to show SPP was closed under intersection it is unlikely to be the same as PP and thus we failed in this attempt to show PP was closed under intersection. While we were having this discussion, sitting on the printer was a paper Richard Beigel had emailed me earlier, his paper with Nick Reingold and Daniel Spielman entitled "PP is Closed Under Intersection". Their successful approach was completely different the ours. They used rational functions to approximate the sign function.
Not to be deterred we started studying SPP and related classes which also led to GapP functions. Valiant had defined the class #P, functions f such that there was some nondeterministic polynomial-time Turing machine M such that f(x) was the number of accepting paths of M(x). GapP functions were the closure of #P functions under subtraction, or equivalently the difference or gap of the number of accepting and rejecting computation paths of an NP machine.
GapP functions are closed under many of the same properties as #P functions such as polynomial products and exponential sums as well as subtraction of course. The power of subtraction made GapP a much cleaner approach to studying counting classes and the study of GapP showed the great importance of the class SPP.
Independently of us, Ogihara and Hemachandra defined a class XP and Gupta defined a class ZUP, both of which were equivalent to SPP.
I will stop here and in a later post describe the actual properties of SPP that make it such an interesting class.