# Complexity Zoo:Y

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##### YACC: Yet Another Complexity Class

A term of derision, used against a complexity class.

##### YP: Your Polynomial-Time or Yaroslav-Percival

The class of decision problems for which there exists a polynomial-time machine M such that:

- For all input sizes n, there exists a polynomial-size advice string s
_{n}such that M(x,s_{n}) outputs the correct answer for all inputs x of size n. - For all inputs x and advice strings s, M(x,s) outputs either the correct answer or "I don't know."

Defined in a recent post of the blog Shtetl-Optimized. See there for an explanation of the class's name.

Contains ZPP by the same argument that places BPP in P/poly.

Also contains P with TALLY ∩ NP ∩ coNP oracle.

Is contained in NP ∩ coNP and YPP.

##### YPP: Yaroslav BPP

The probabilistic analogue of YP; it is to YP what MA is to NP. Formally, the class of decision problems for which there exists a syntactic BPP machine M such that:

- For all input sizes n, there exists a polynomial-size advice string a
_{n}such that for all inputs x of size n, M(x,a_{n}) outputs the correct answer with probability at least 2/3. - For all inputs x and advice strings a, the probability that M(x,a) outputs the incorrect answer is at most 1/3. In other words, the sum of the probabilities of the correct answer and "I don't know" is at least 2/3.

To amplify a YPP machine, one can run it multiple times, then accept if a majority of runs accept, reject if a majority reject, and otherwise output "I don't know."

Contains BPP and YP, and is contained in MA and P/poly.

##### YQP: Yaroslav BQP

Is to YPP as BQP is to BPP, and QMA is to MA. The machine is now a quantum computer and the advice is a quantum state |ψ_n>.

Contains BQP and YPP, and is contained in QMA and BQP/qpoly.