Constructive Hausdorff Dimension Research Articles

Dimension spectra of random subfractals of self-similar fractals

The dimension of a point x in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}) is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffice to specify x on a general-purpose computer with arbitrarily high precision 2−r. The dimension spectrum of a set X in Euclidean space is the subset of [0,n] consisting of the dimensions of all points in X.The dimensions of points have been shown to be geometrically meaningful (Lutz 2003 [16], Hitchcock 2005 [12]), and the dimensions of points in self-similar fractals have been completely analyzed (Lutz and Mayordomo 2008 [18]). Here we begin the more challenging task of analyzing the dimensions of points in random fractals. We focus on fractals that are randomly selected subfractals of a given self-similar fractal. We formulate the specification of a point in such a subfractal as the outcome of an infinite two-player game between a selector that selects the subfractal and a coder that selects a point within the subfractal. Our selectors are algorithmically random with respect to various probability measures, so our selector–coder games are, from the coder's point of view, games against nature.We determine the dimension spectra of a wide class of such randomly selected subfractals. We show that each such fractal has a dimension spectrum that is a closed interval whose endpoints can be computed or approximated from the parameters of the fractal. In general, the maximum of the spectrum is determined by the degree to which the coder can reinforce the randomness in the selector, while the minimum is determined by the degree to which the coder can cancel randomness in the selector. This constructive and destructive interference between the players' randomnesses is somewhat subtle, even in the simplest cases. Our proof techniques include van Lambalgen's theorem on independent random sequences, measure preserving transformations, an application of network flow theory, a Kolmogorov complexity lower bound argument, and a nonconstructive proof that this bound is tight.

Normality and Finite-State Dimension of Liouville Numbers

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano (1993) and Bugeaud (2002) have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k ≥ 2, we give a new construction of a Liouville number which is normal to the base k. This construction is combinatorial, and is based on de Bruijn sequences. A real number in the unit interval is normal if and only if its finite-state dimension is 1. We generalize our construction to prove that for any rational r in the closed unit interval, there is a Liouville number with finite state dimension r. This refines Staiger’s result [19] that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state dimension.

Kolmogorov-Loveland Stochasticity and Kolmogorov Complexity

Merkle et al. (Ann. Pure Appl. Logic 138(1–3):183–210, 2006) showed that all Kolmogorov-Loveland stochastic infinite binary sequences have constructive Hausdorff dimension 1. In this paper, we go even further, showing that from an infinite sequence of dimension less than $\mathcal {H}(\frac {1}{2}+\delta)$(ℋ being the Shannon entropy function) one can extract by an effective selection rule a biased subsequence with bias at least δ. We also prove an analogous result for finite strings.

Algorithmically independent sequences

Two objects are independent if they do not affect each other. Independence is well-understood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence x, the set of sequences that are independent with x has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, it is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have super-logarithmic complexity. Finally, a few conjectures and open questions are discussed.

Constructive Dimension and Turing Degrees

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim H(S) and constructive packing dimension dim P(S) is Turing equivalent to a sequence R with dim H(R)≥(dim H(S)/dim P(S))−ε, for arbitrary ε>0. Furthermore, if dim P(S)>0, then dim P(R)≥1−ε. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension.A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim H(S)/dim P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim H(S)=dim P(S)) such that dim H(S)>0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1.Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.

A characterization of constructive dimension

AbstractIn the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin‐Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a new characterization of constructive dimensions, in particular, the constructive Hausdorff dimension, the constructive packing dimension, and their generalizations, the constructive scaled dimension and the constructive scaled strong dimension (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Characterization of Constructive Dimension

In the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a new characterization of constructive dimensions, in particular, the constructive Hausdorff dimension and the constructive packing dimension.