# Complexity Zoo:O

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##### 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:

- 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.
- 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 n^{k} circuits for some constant k if and only if ONP/1 has size n^{j} circuits for some constant j.

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

##### 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.

##### O_{2}P: 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

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

Note that this differs from S_{2}P in that the witnesses in each case must only depend on the length of the input, and not on the input itself.

Less formally, O_{2}P 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 O_{2}P is self-low, and that the Karp-Lipton collapse goes all the way down to O_{2}P: if NP is contained in P/poly then PH = O_{2}P.

Contrast with S_{2}P.