We show that a wide variety of lowness notions can be expressed in a similar way, i.e., via the ability to cover open sets of a certain type by opensets of some other type.Expand

Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of Martin-Lof randomness, Schnorr randomness and Kurtz randomness.Expand

We show that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f (x) = K(x) for infinitely many x.Expand

Martingales played an important role in the study of randomness in the twentieth century. Jean Ville invented martingales in the 1930s in order to improve Richard von Mises' concept of a collective,… Expand

This paper offers some new results on randomness with respect to classes of measures, along with a didactic exposition of their context based on results that appeared elsewhere.Expand

A theorem of Kucera states that given a Martin-Lof random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A.Expand

We determine all the implications that hold between the equivalence relations induced by Martin-Lof Randomness, computable randomness, Schnorr randomness and weak randomness.Expand

Suppose 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… Expand

We prove the effective version of [email protected]s ergodic theorem for Martin-Lof random points and effectively open sets, improving the results previously obtained in this direction (in particular those of Vyugin and Hoyrup, Rojas).Expand