A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

Given a set X of sequences over a nite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X.

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the Kolmogorov-Loveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the Kolmogorov-Loveland stochastic sequences form a proper subclass of the Mises-Wald-Church stochastic sequences.

We observe that known results on the Kolmogorov complexity of pre xes of eectively stochastic sequences extend to corresponding random sequences. First, there are recursively random random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the uniform complexity of the length n pre x of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partial-recursively random sequences.

The sequence selected from a sequence R(0)R(1)... by a language L is the subsequence of all bits R(n + 1) such that the prefix R(0)... R(n) is in L. By a result of Agafonoff [1], a sequence is normal if and only if any subsequence selected by a regular language is again normal. Kamae and Weiss [11] and others have raised the question of how complex a language must be such that selecting according to the language does not preserve normality. We show that there are such languages that are only slightly more complicated than regular ones, namely, normality is neither preserved by linear languages nor by deterministic one-counter languages. In fact, for both types of languages it is possible to select a constant sequence from a normal one.