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

  • If x ∈ L {\displaystyle x\in L}, then there exists a y such that for all z, P ( x , y , z ) = 1 {\displaystyle P(x,y,z)=1},
  • If x ∉ L {\displaystyle x\notin L}, then there exists a z such that for all y, P ( x , y , z ) = 0 {\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 Σ 2 P {\displaystyle \Sigma _{2}^{P}} and Π 2 P {\displaystyle \Pi _{2}^{P}}, it also follows immediately that SP 2 is contained in Σ 2 P ∩ Π 2 P {\displaystyle \Sigma _{2}^{P}\cap \Pi _{2}^{P}}. This inclusion can in fact be strengthened to ZPPNP.

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 Δ 2 P {\displaystyle \Delta _{2}^{P}} (more generally, P S 2 P = S 2 P {\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 SIZE(nk) for any fixed k.

Symmetric hierarchy

As an extension, it is possible to define S 2 {\displaystyle {\mathsf {S}}_{2}} as an operator on complexity classes; then S 2 P = S 2 P {\displaystyle {\mathsf {S}}_{2}P={\mathsf {S}}_{2}^{P}}. Iteration of S 2 {\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:.
  • Russell, Alexander; Sundaram, Ravi (1998). "Symmetric alternation captures BPP". Computational Complexity. 7 (2). Birkhäuser Verlag: 152–162. doi:. ISSN . S2CID .

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