von Neumann’s biased coin revisited

International audienceSuppose you want to generate a random sequence of zeros and ones and all you have at your disposal is a coin which you suspect to be biased (but do not know the bias). Can "perfect" randomness be produced with this coin? The answer is positive, thanks to a little trick discovered by von Neumann. In this paper, we investigate a generalization of this question: if we have access to a source of bits produced according to some probability measure µ in some class C of measures, and suppose we know C but not µ (in the above example, C would be the class of all Bernoulli measures), can perfect randomness be produced? We will look at this question from the viewpoint of constructive mathematics and in particular the theory of effective randomness