S2P (complexity)

In computational complexity theory, SP
2 is a complexity class, intermediate between the first and second levels of the polynomial hierarchy. A language $L$ is in $S_2^P$ if there exists a polynomial-time predicate P such that

• If $x \in L$, then there exists a y such that for all z, $P(x,y,z)=1$,
• If $x \notin L$, then there exists a z such that for all y, $P(x,y,z)=0$,

where size of y and z must be polynomial of x.

Relationship to other complexity classes

It is immediate from the definition that SP
2 is closed under unions, intersections, and complements. Comparing the definition with that of $\Sigma_{2}^P$ and $\Pi_{2}^P$, it also follows immediately that SP
2 is contained in $\Sigma_{2}^P \cap \Pi_{2}^P$. This inclusion can in fact be strengthened to ZPP^NP.^[1]

Every language in NP also belongs to SP
2. For by definition, a language L is in NP, if and only if there exists a polynomial-time verifier V(x,y), such that for every x in L there exists y for which V answers true, and such that for every x not in L, V always answers false. But such a verifier can easily be transformed into an SP
2 predicate P(x,y,z) for the same language that ignores z and otherwise behaves the same as V. By the same token, co-NP belongs to SP
2. These straightforward inclusions can be strengthened to show that the class SP
2 contains MA (by a generalization of the Sipser–Lautemann theorem) and $\Delta_{2}^P$ (more generally, $P^{S_2^P}=S_2^P$).

Karp–Lipton theorem

A version of Karp–Lipton theorem states that if every language in NP has polynomial size circuits then the polynomial time hierarchy collapses to SP
2. This result yields a strengthening of Kannan's theorem: it is known that SP
2 is not contained in SIZE(n^k) for any fixed k.

Symmetric hierarchy

As an extension, it is possible to define $S_2$ as an operator on complexity classes; then $S_2 P = S_2^P$. Iteration of $S_2$ operator yields a "symmetric hierarchy"; the union of the classes produced in this way is equal to the Polynomial hierarchy.

References

1. Cai, Jin-Yi (2007), $S_2^P \subseteq \mathsf{ZPP}^{\mathsf{NP}}$ (PDF), Journal of Computer and System Sciences 73 (1): 25–35, doi:10.1016/j.jcss.2003.07.015, MR 2279029. A preliminary version of this paper appeared earlier, in FOCS 2001, ECCC TR01-030, MR 1948751, doi:10.1109/SFCS.2001.959938.

• Canetti, Ran (1996). "More on BPP and the polynomial-time hierarchy". Information Processing Letters 57 (5) (Elsevier). pp. 237–241. doi:10.1016/0020-0190(96)00016-6.
• Russell, Alexander; Sundaram, Ravi (1998). "Symmetric alternation captures BPP". computational complexity 7 (2) (Birkhäuser Verlag). pp. 152–162. doi:10.1007/s000370050007. ISSN 1016-3328.

External links

• Complexity Zoo: S₂P
• Complexity Class of the Week: S₂^P, Lance Fortnow, August 28, 2002.