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[Solved]: Why is $BPP$ closed under complement?

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Problem Detail: 

Why is $\text{BPP}=\text{co-BPP}$?
I tried to find a proof online but couldn't.
Can anyone please provide a quick explanation (if it's trivial and I just can't see it) or a link to a proof?

Asked By : so.very.tired

Answered By : D.W.

It follows directly from the definition of BPP.

The definition says that a problem with a yes-or-no answer is in BPP if there is a randomized algorithm that gets the correct answer with probability $\ge 2/3$ on all possible instances. This means that on all instances where the correct answer is YES, the algorithm has to output YES with probability at least $2/3$; and on all instances where the correct answer is NO, the algorithm has to output NO with probability at least $2/3$.

Therefore, if a problem $P$ is in BPP, its negation is also in BPP. Take the algorithm for $P$, and just flip its output. That will be a valid algorithm for the negation of the problem.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/28036

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