Wednesday, August 28, 2002

Complexity Class of the Week: S2P

CCW Intro

Suppose a polynomial-time computable judge has to decide whether a string is in a language. Two lawyers submit written arguments to convince the judge, one arguing for the string in the language and the other arguing for the string to be out of the language. Neither lawyer can see the others arguments. For what languages can we have the judge always convinced?

Russell and Sundaram define the class S2P to capture this notion. Formally a language L is in S2P if there is a polynomial-time predicate A and a polynomial q such that

  1. If x is in L then there is a y such that for all z, A(x,y,z).
  2. If x is not in L then there is a z such that for all y, not A(x,y,z).
where |y| and |z| are bounded by q(|x|).

Personally I like to think of S2P as for every input x defining an exponential binary matrix A where the (i,j) entry of A is computable in polynomial time from i,j and x. If x is in the language then A has a row of all ones. If x is not in the language then A has a column of all zeros.

NP∪coNP is in S2P is in Σ2P∩Π2P. Russell and Sundaram show that S2P is closed under Turing reduction and relativizations to BPP. This implies BPP, MA and PNP are contained in S2P.

A big breakthrough for S2P comes from Cai who shows that S2P is contained in ZPPNP. His proof is builds on the learning algorithm of Bshouty, et. al. Sengupta noticed that a variation of the Karp-Lipton result shows that if NP has polynomial-size circuits then the polynomial-time hierarchy collapses to S2P. This also improves Kannan's result to show that for any fixed k, there is a language in S2P that does not have nk-size circuits. S2P is the smallest class known to have these properties.

S2P has come up recently in several of my research projects. Beigel, Buhrman, Fejer, Fortnow, Longpre, Stephan and Torenvliet show that a language L is in S2P if and only if there is a function f mapping Σ* to {1,2,3} such that L is polynomial-time Turing reducible to all 2-enumerators for f. Buhrman and Fortnow give an oracle separating ZPPNP from Σ2P∩Π2P which by Cai's result gives the first relativized world separating S2P from Σ2P∩Π2P. Fortnow, Pavan and Sengupta show that if PNP[2] = PNP[1] then the polynomial-time hierarchy collapses to S2P improving on earlier collapses. You will have to trust me on the latter two results as they are in the process of being written up.

Whether S2P contains ZPPNP is open even for relativized worlds. Since AM∩coAM is in ZPPNP, one could try to prove that AM∩coAM is in S2P or a specific language in AM∩coAM such as graph isomorphism.

There are variations on the class S2P and I recommend reading the papers of Russell and Sundaram and Cai for these and other results about this interesting class.

1 comment:

  1. I could't solve the problem.
    Anyway, it's very tricky to determine the difficulty of the problem.
    I once saw a similar problem wich couldn't be solved by several computerscience students. 2 friends of me wich are literature teachers were able to solve the problem within minutes. At least one of those friends couldn't solve this problem:

    Thinking of numbers as words is not
    "outside the box thinking" for someone with a literature background. For a computerscience
    person it needs an ingenious moment to do so.