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 ${\displaystyle {\mathsf {S}}_{2}^{P}}$ if there exists a polynomial-time predicate P such that

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

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

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

Every language in NP also belongs to 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 ${\displaystyle \Delta _{2}^{P}}$ (more generally, ${\displaystyle P^{{\mathsf {S}}_{2}^{P}}={\mathsf {S}}_{2}^{P}}$).

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 SIZE(n^k) for any fixed k.

As an extension, it is possible to define ${\displaystyle {\mathsf {S}}_{2}}$ as an operator on complexity classes; then ${\displaystyle {\mathsf {S}}_{2}P={\mathsf {S}}_{2}^{P}}$. Iteration of ${\displaystyle S_{2}}$ operator yields a "symmetric hierarchy"; the union of the classes produced in this way is equal to the Polynomial hierarchy.

• Canetti, Ran (1996). "More on BPP and the polynomial-time hierarchy". Information Processing Letters. 57 (5). Elsevier: 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: 152–162. doi:10.1007/s000370050007. ISSN 1016-3328. S2CID 15331219.

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