S2P (complexity)

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

where size of y and z must be polynomial of x.
 * 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, $$P(x,y,z)=0$$,

Relationship to other complexity classes
It is immediate from the definition that S$P 2$ 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 S$P 2$ is contained in $$\Sigma_{2}^P \cap \Pi_{2}^P$$. This inclusion can in fact be strengthened to ZPPNP.

Every language in NP also belongs to S$P 2$. 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 S$P 2$ 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 S$P 2$. These straightforward inclusions can be strengthened to show that the class S$P 2$ 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$P 2$. This result yields a strengthening of Kannan's theorem: it is known that S$P 2$ 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.