Abstract not found

Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resource-bounded extension: if a class X has ∆-strong dimension less than 1, then X is ∆-meager. This has the applications of explaining some of Lutz’s simultaneous ∆-meager, ∆-measure 0 results and providing a new proof of a Gu’s strong dimension result on infinitely-often classes.

ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree ’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans. 1.

Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1-random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.

In this paper we discuss three notions of partial randomness or ε-randomness. ε-randomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [15] proved, Tadaki [22] and Calude et al. [3] proposed at least three different concepts of partial randomness. We show that all of them satisfy the natural requirement that any ε-non-null set contains an ε-random infinite word. This allows us to focus our investigations on the strongest one which is based on a priori complexity. We investigate this concept of partial randomness and show that it allows—similar to the random infinite words—oscillationfree (w.r.t. to a priori complexity) ε-random infinite words if only ε is a computable number. The proof uses the dilution principle. Alternatively, for certain sets of infinite words (ω-languages) we show that their most complex infinite words are oscillation-free ε-random. Here the parameter ε is also computable and depends on the set chosen.

Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.