Some results on effective randomness

Abstract

We investigate the characterizations of effective randomness in terms of Martin-Löf covers and martingales. First, we address a question of Ambos-Spies and Kučera [1], who asked for a characterization of computable randomness in terms of tests. We argue that computable randomness can be characterized in term of Martin-Löf tests and effective probability distributions on Cantor space. Second, we show that the class of Martin-Löf random sets coincides with the class of sets of reals that are random with respect to computable martingale processes. This improves on results of Hitchcock and Lutz [8], who showed that the latter class is contained in the class of Martin-Löf random sets and is a strict superset of the class of rec-random sets. Third, we analyze the sequence of measures of the components of a universal Martin-Löf test. Kucera and Slaman [12] showed that any compo, nent of a universal Martin-Löf test defines a class of Martin-Löf random measure. Further, since the sets in a Martin-Löf test are uniformly computably enumerable, so is the corresponding sequence of measures. We prove an exact converse and hence a characterization. For any uniformly computably enumerable sequence r1, r2, . . . of reals such that each ri is Martin-Löf random and less than 2-i there is a universal Martin-Löf test U1, U2, . . . such that Ui[0, 1}∞ has measure ri. © Springer-Verlag Berlin Heidelberg 2004.