The geometry of fractal sets
Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Incompleteness theorems for random reals
Advances in Applied Mathematics
Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
Elements of information theory
Elements of information theory
Kolmogorov complexity and Hausdorff dimension
Information and Computation
The complexity and effectiveness of prediction algorithms
Journal of Complexity
Proceedings of the 30th IEEE symposium on Foundations of computer science
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
On the Length of Programs for Computing Finite Binary Sequences
Journal of the ACM (JACM)
On the Length of Programs for Computing Finite Binary Sequences: statistical considerations
Journal of the ACM (JACM)
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
A Kolmogorov complexity characterization of constructive Hausdorff dimension
Information Processing Letters
Correspondence Principles for Effective Dimensions
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Gales suffice for constructive dimension
Information Processing Letters
Dimension in Complexity Classes
SIAM Journal on Computing
Gales suffice for constructive dimension
Information Processing Letters
Theoretical Computer Science
Scaled dimension and nonuniform complexity
Journal of Computer and System Sciences
Constructive dimension equals Kolmogorov complexity
Information Processing Letters
SIGACT news complexity theory column 48
ACM SIGACT News
Journal of Computer and System Sciences - Special issue on COLT 2002
Dimension, entropy rates, and compression
Journal of Computer and System Sciences
Entropy rates and finite-state dimension
Theoretical Computer Science
A note on dimensions of polynomial size circuits
Theoretical Computer Science
Natural halting probabilities, partial randomness, and zeta functions
Information and Computation
The arithmetical complexity of dimension and randomness
ACM Transactions on Computational Logic (TOCL)
Theoretical Computer Science
The Kolmogorov complexity of infinite words
Theoretical Computer Science
Finite-state dimension and real arithmetic
Information and Computation
Generic density and small span theorem
Information and Computation
Connectivity Properties of Dimension Level Sets
Electronic Notes in Theoretical Computer Science (ENTCS)
A Characterization of Constructive Dimension
Electronic Notes in Theoretical Computer Science (ENTCS)
Martingale families and dimension in P
Theoretical Computer Science
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
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
Algorithmically Independent Sequences
DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
On Oscillation-free ε-random Sequences
Electronic Notes in Theoretical Computer Science (ENTCS)
A Divergence Formula for Randomness and Dimension
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
Information and Computation
Constructive dimension equals Kolmogorov complexity
Information Processing Letters
Algorithmically independent sequences
Information and Computation
Kolmogorov-Loveland stochasticity and Kolmogorov complexity
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Complex network dimension and path counts
Theoretical Computer Science
Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
A divergence formula for randomness and dimension
Theoretical Computer Science
Extracting Kolmogorov complexity with applications to dimension zero-one laws
Information and Computation
Connectedness properties of dimension level sets
Theoretical Computer Science
Constructive dimension and Hausdorff dimension: the case of exact dimension
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Effective dimensions and relative frequencies
Theoretical Computer Science
High-confidence predictions under adversarial uncertainty
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
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
Martingale families and dimension in p
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Finite-sate dimension and real arithmetic
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Lempel-ziv dimension for lempel-ziv compression
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Generic density and small span theorem
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
The dimension of a point: computability meets fractal geometry
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
Characterizing languages by normalization and termination in string rewriting
DLT'12 Proceedings of the 16th international conference on Developments in Language Theory
A correspondence principle for exact constructive dimension
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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 |
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 are random (in the sense of Martin-Löf) have dimension 1, while sequences that are decidable, Σ10, or Π10, have dimension 0. It is shown that for every Δ20-computable real number α in [0, 1] there is a Δ20 sequence S such that dim(S) = α. A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit inferior of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number β in (0, 1) is shown to have dimension H(β), the binary entropy of β.