The probability distribution as a computational resource for randomness testing

When testing a set of data for randomness according to a probability distribution that depends on a parameter, access to this parameter can be considered as a computational resource. We call a randomness test Hippocratic if it is not permitted to access this resource. In these terms, we show that for Bernoulli measures μp, 0 6 p 6 1 and the Martin-Löf randomness model, Hippocratic randomness of a set of data is the same as ordinary randomness. The main idea of the proof is to first show that from Hippocrates-random data one can Turing compute the parameter p. However, we show that there is no single Hippocratic randomness test such that passing the test implies computing p, and in particular there is no universal Hippocratic randomness test.