Results 1  10
of
12
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 94 (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
Effective strong dimension in algorithmic information and computational complexity
 SIAM Journal on Computing
, 2004
"... The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded exten ..."
Abstract

Cited by 81 (30 self)
 Add to MetaCart
The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomialspace, polynomialtime, and finitestate dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual
A Tight Upper Bound on Kolmogorov Complexity by Hausdorff Dimension and Uniformly Optimal Prediction
 Theory of Computing Systems
, 1995
"... The present paper links the concepts of Kolmogorov complexity (in Complexity theory) and Hausdorff dimension (in Fractal geometry) for a class of recursive (computable) !languages. ..."
Abstract

Cited by 43 (5 self)
 Add to MetaCart
The present paper links the concepts of Kolmogorov complexity (in Complexity theory) and Hausdorff dimension (in Fractal geometry) for a class of recursive (computable) !languages.
Hausdorff dimension in exponential time
 Computational Complexity, IEEE Computer Society
"... In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including ..."
Abstract

Cited by 35 (3 self)
 Add to MetaCart
In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resourcebounded dimension we show that the class of pmcomplete sets for E has dimension 1 in E. Moreover, we show that there are pmlower spans in E of dimension H(β) for any rational β between 0 and 1, where H(β) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz’s concept of weak completeness. Finally we characterize resourcebounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions. 1.
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.
How Much Can You Win When Your Adversary is Handicapped?
 PRESENTED AT THE SYMPOSIUM "NUMBERS, INFORMATION AND COMPLEXITY", BIELEFELD, OCTOBER 8  11, 1998
, 2000
"... We consider infinite games where a gambler plays a cointossing game against an adversary. The gambler puts stakes on heads or tails, and the adversary tosses a fair coin, but has to choose his outcome according to a previously given law known to the gambler. In other words, the adversary is not all ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
We consider infinite games where a gambler plays a cointossing game against an adversary. The gambler puts stakes on heads or tails, and the adversary tosses a fair coin, but has to choose his outcome according to a previously given law known to the gambler. In other words, the adversary is not allowed to play all infinite headstailssequences, but only a certain subset F of them. We present an algorithm for the player which, depending on the structure of the set F , guarantees an optimal exponent of increase of the player's capital, independently on which one of the allowed headstailssequences the adversary chooses. Using the known upper bound on the exponent provided by the maximum Kolmogorov complexity of sequences in F we show the optimality of our result.
Effective Fractal Dimension in Algorithmic Information Theory
, 2006
"... Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that
Schnorr dimension
 in exponential time, Computational Complexity 2001, 210217, IEEE Computer Society
, 2001
"... ABSTRACT. Following Lutz’s approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr’s concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
ABSTRACT. Following Lutz’s approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr’s concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, i.e. the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterization of Schnorr dimension, based on prefixfree machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets which are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0 there are c.e. sets of computable packing dimension 1, a property impossible in the case of effective (constructive) dimension, due to Barzdiņˇs’ Theorem. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1. 1.