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1 LIAFA - Laboratoire d-informatique Algorithmique : Fondements et Applications 2 Boston University - Computer Science Department Boston 3 CARTE - Theoretical adverse computations, and safety Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods 4 Department of Mathematics - University of Toronto 5 ESCAPE - Systèmes complexes, automates et pavages LIRMM - Laboratoire d-Informatique de Robotique et de Microélectronique de Montpellier

Abstract : This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness with respect to computable measures in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given in two forms. Some approaches that go in another direction from deficiency to complexity are considered. The notion of Bernoulli randomness independent coin tosses for an asymmetric coin with some probability p of head is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli measure Bp. A notion of -uniform test- for Bernoulli sequences is introduced which allows a quantitative strengthening of this result. Uniform tests are then generalized to arbitrary measures. Bernoulli measures Bp have the important property that p can be recovered from each random sequence of Bp. The paper studies some important consequences of this orthogonality property as well as most other questions mentioned above also in the more general setting of constructive metric spaces.

Autor: Laurent Bienvenu - Peter Gacs - Mathieu Hoyrup - Cristobal Rojas - Alexander Shen -



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