# S2P (complexity)

In computational complexity theory, SP
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.

## Relationship to other complexity classes

It is immediate from the definition that SP
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 SP
2
is contained in ${\displaystyle \Sigma _{2}^{P}\cap \Pi _{2}^{P}}$. This inclusion can in fact be strengthened to ZPPNP.[1]

Every language in NP also belongs to SP
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 SP
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 SP
2
.
These straightforward inclusions can be strengthened to show that the class SP
2
contains MA (by a generalization of the Sipser–Lautemann theorem) and ${\displaystyle \Delta _{2}^{P}}$ (more generally, ${\displaystyle P^{{\mathsf {S}}_{2}^{P}}={\mathsf {S}}_{2}^{P}}$).

## 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 SP
2
. This result yields a strengthening of Kannan's theorem: it is known that SP
2
is not contained in Template:San-serif(nk) for any fixed k.

## Symmetric hierarchy

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.

## References

1. ^ Cai, Jin-Yi (2007), "${\displaystyle \mathrm {S} _{2}^{p}\subseteq \mathrm {{ZPP}^{NP}} }$" (PDF), Journal of Computer and System Sciences, 73 (1): 25–35, doi:10.1016/j.jcss.2003.07.015, MR 2279029. A preliminary version of this paper appeared earlier, in FOCS 2001, ECCC TR01-030, MR1948751, doi:10.1109/SFCS.2001.959938.
• Canetti, Ran (1996). "More on BPP and the polynomial-time hierarchy". Information Processing Letters. 57 (5). Elsevier. pp. 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. pp. 152–162. doi:10.1007/s000370050007. ISSN 1016-3328.