I am looking for a reference to the fact that if NP is included in BPP then NP is equal to RP. See for instance these links:

I know that this is folklore, but I'd still like to cite something that is published and where this would be properly proved.

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    $\begingroup$ At this point in history, I don't know if this needs a citation, e.g. it is regularly given as an exercise. (I also don't know if there is such a citation, or if it was always just an exercise.) $\endgroup$ Commented Dec 21, 2019 at 16:49

1 Answer 1


An actual factual reference is

K. Ko. Some observations on the probabilistic algorithms and NP-hard problems. Information Processing Letters, 14(1):39–43, 1982.

(When I first saw this result --- I don't remember where it was now --- it was called "Ko's Theorem". Googling suggests that another theorem has that name as well...)

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    $\begingroup$ Nice - I didn't know it actually existed in the literature! $\endgroup$ Commented Dec 23, 2019 at 4:57

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