Resource-bounded strong dimension versus resource-bounded category
Information Processing Letters
Entropy rates and finite-state dimension
Theoretical Computer Science
Theoretical Computer Science
Dimensions of Copeland--Erdös sequences
Information and Computation
Finite-state dimension and real arithmetic
Information and Computation
Turing degrees of reals of positive effective packing dimension
Information Processing Letters
Constructive Dimension and Weak Truth-Table Degrees
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Effective Dimensions and Relative Frequencies
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
Dimensions of Points in Self-similar Fractals
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
A Divergence Formula for Randomness and Dimension
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Resource-bounded strong dimension versus resource-bounded category
Information Processing Letters
Information measures for infinite sequences
Theoretical Computer Science
Complex network dimension and path counts
Theoretical Computer Science
A divergence formula for randomness and dimension
Theoretical Computer Science
Extracting Kolmogorov complexity with applications to dimension zero-one laws
Information and Computation
On the polynomial depth of various sets of random strings
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Effective dimensions and relative frequencies
Theoretical Computer Science
High-confidence predictions under adversarial uncertainty
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Online learning and resource-bounded dimension: winnow yields new lower bounds for hard sets
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Every sequence is decompressible from a random one
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Two open problems on effective dimension
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Dimensions of copeland-erdös sequences
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Extracting kolmogorov complexity with applications to dimension zero-one laws
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Finite-sate dimension and real arithmetic
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Randomness, computation and mathematics
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
Dimension, halfspaces, and the density of hard sets
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
On the polynomial depth of various sets of random strings
Theoretical Computer Science
Base invariance of feasible dimension
Information Processing Letters
High-confidence predictions under adversarial uncertainty
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
Hi-index | 0.01 |
The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff [Math. Ann., 79 (1919), pp. 157-179], and packing dimension, developed independently by Tricot [Math. Proc. Cambridge Philos. Soc., 91 (1982), pp. 57-74] and Sullivan [Acta Math., 153 (1984), pp. 259-277]. Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz [Proceedings of the 15th IEEE Conference on Computational Complexity, Florence, Italy, 2000, IEEE Computer Society Press, Piscataway, NJ, 2000, pp. 158-169] 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, polynomial-space, polynomial-time, and finite-state 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 of—and every bit as simple as—the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions, which are exact duals of the effective dimensions mentioned above. In general (and in analogy with the classical fractal dimensions), the effective strong dimension of a set or sequence is at least as great as its effective dimension, with equality for sets or sequences that are sufficiently regular. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression. Aside from the above characterization of packing dimension, our two main theorems are the following. 1. If $\vec{\beta} = (\beta_0,\beta_1,\ldots)$ is a computable sequence of biases that are bounded away from 0 and $R$ is random with respect to $\vec{\beta}$, then the dimension and strong dimension of $R$ are the lower and upper average entropies, respectively, of $\vec{\beta}$. 2. For each pair of $\Delta^0_2$-computable real numbers $0