# Difference between revisions of "Complexity Zoo:O"

Complexity classes by letter: Symbols - A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z

Lists of related classes: Communication Complexity - Hierarchies - Nonuniform

OIP - OMA - ONP - OptP - O2P

##### ONP: Oblivious NP

The class of functions computable in polynomial time with a shared, untrusted witness for each input size. The input-oblivious version of NP.

L is in ONP if there exists a polynomial time verifier V taking an input and a witness, so that:

1. There is a witness for each n of polynomial size, so that for any input of size n, if the answer is "yes," then the verifier accepts on that input and the witness.
2. If the answer for an input is "no," then for any witness, the verifier rejects on that input.

Defined in [FSW09], where it was shown NP has size nk circuits for some constant k if and only if ONP/1 has size nj circuits for some constant j.

ONP is in P/poly [FSW09]

ONP = NP iff NP is in P/poly [FSW09].

If NE is not E then ONP is not P [GM15].

##### OptP: Optimum Polynomial-Time

The class of functions computable by taking the maximum of the output values over all accepting paths of an NP machine.

Defined in [Kre88].

Contrast with FNP.

##### O2P: Second Level of the Oblivious Symmetric Hierarchy

The class of decision problems for which there is a polynomial-time predicate P such that, for each length n, there exists y* and z* of length poly(n) such that for all x of length n

1. If the answer is 'yes,' for all z, P(x,y*,z) is true.
2. If the answer is 'no,' then for all y, P(x,y,z*) is false.

Note that this differs from S2P in that the witnesses in each case must only depend on the length of the input, and not on the input itself.

Less formally, O2P is the class of one-round games in which a prover and a disprover submit simultaneous moves to a deterministic, polynomial-time referee, and furthermore, there is a single winning move the prover can make that works for all x of length n that are yes-instances, and there is a single winning move the disprover can make that works for all x of length n that are no-instances.

Defined in [CR06], where it was shown (among other properties) that O2P is self-low, and that the Karp-Lipton collapse goes all the way down to O2P: if NP is contained in P/poly then PH = O2P.

Contrast with S2P.