In computational complexity theory, S is a complexity class, intermediate between the first and second levels of the polynomial hierarchy. A language is in if there exists a polynomial-time predicate P such that
where size of y and z must be polynomial of x.
It is immediate from the definition that S is closed under unions, intersections, and complements. Comparing the definition with that of and , it also follows immediately that S is contained in . This inclusion can in fact be strengthened to ZPP<sup>NP</sup>.
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 SipserâÂÂLautemann theorem) and (more generally, ).
A version of KarpâÂÂLipton 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<sup>k</sup>) for any fixed k.
As an extension, it is possible to define as an operator on complexity classes; then . Iteration of operator yields a "symmetric hierarchy"; the union of the classes produced in this way is equal to the Polynomial hierarchy.