Results 1 
8 of
8
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... 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 ..."
Abstract

Cited by 93 (10 self)
 Add to MetaCart
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
Prediction and Dimension
 Journal of Computer and System Sciences
, 2002
"... 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 polynomialtime randomized predictor can achieve on all sequences in X. ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
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 polynomialtime randomized predictor can achieve on all sequences in X.
Constructive dimension and weak truthtable degrees
 In Computation and Logic in the Real World  Third Conference of Computability in Europe. SpringerVerlag Lecture Notes in Computer Science #4497
, 2007
"... Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1−ɛ. 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 wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the wtt 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.
Every sequence is decompressible from a random one
 In Logical Approaches to Computational Barriers, Proceedings of the Second Conference on Computability in Europe, Springer Lecture Notes in Computer Science, volume 3988 of Computability in Europe
, 2006
"... ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of b ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. It is shown that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, a new characterization of constructive dimension is given in terms of Turing reduction compression ratios.
Scaled dimension and the Kolmogorov complexity of Turinghard sets
 In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science
, 2004
"... We study constructive and resourcebounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
We study constructive and resourcebounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/polyTuring reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rdorder scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turinghard sets for ESPACE. 1
Effective dimension in some general metric spaces
, 2012
"... We introduce the concept of effective dimension for a general metric space. Effective dimension was defined by Lutz in (Lutz 2003) for Cantor space and has also been extended to Euclidean space. Our extension to other metric spaces is based on a supergale characterization of Hausdorff dimension. We ..."
Abstract
 Add to MetaCart
We introduce the concept of effective dimension for a general metric space. Effective dimension was defined by Lutz in (Lutz 2003) for Cantor space and has also been extended to Euclidean space. Our extension to other metric spaces is based on a supergale characterization of Hausdorff dimension. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity. Further research directions are indicated. 1