S2P (complexity)

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.{{citation

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 (n**k) 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