In computational complexity theory, S is a complexity class, intermediate between the first and second levels of the polynomial hierarchy.
A language  is in \mathsf 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 S 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 S is contained in \Sigma_{2}^P \cap \Pi_{2}^P.
This inclusion can in fact be strengthened to ZPPNP..
A preliminary version of this paper appeared earlier, in FOCS 2001, , , .
Every language in NP also belongs to  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  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  These straightforward inclusions can be strengthened to show that the class  contains MA (by a generalization of the Sipser–Lautemann theorem) and \Delta_{2}^P (more generally, P^{\mathsf S_2^P}=\mathsf 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 S.
This result yields a strengthening of Kannan's theorem: it is known that S is not contained in (nk) for any fixed k.
Symmetric hierarchy
As an extension, it is possible to define \mathsf S_2 as an operator on complexity classes; then \mathsf S_2 P = \mathsf 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
External links
Complexity Class of the Week: S2P, Lance Fortnow, August 28, 2002.
