Almost everywhere high nonuniform complexity
Journal of Computer and System Sciences
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
Note: fractal dimension and logarithmic loss unpredictability
Theoretical Computer Science
Dimension in Complexity Classes
SIAM Journal on Computing
The dimensions of individual strings and sequences
Information and Computation
Theoretical Computer Science
Dimension, entropy rates, and compression
Journal of Computer and System Sciences
Entropy rates and finite-state dimension
Theoretical Computer Science
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Effective Strong Dimension in Algorithmic Information and Computational Complexity
SIAM Journal on Computing
Theory of Computing Systems
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Effective fractal dimensions were introduced by Lutz (2003) in order to study the dimensions of individual sequences and quantitatively analyze the structure of complexity classes. Interesting connections of effective dimensions with information theory were also found, implying that constructive dimension as well as polynomial-space dimension are invariant under base change while finite-state dimension is not. We consider the intermediate case, polynomial-time dimension, and prove that it is indeed invariant under base change by a nontrivial argument which is quite different from the Kolmogorov complexity ones used in the other cases. Polynomial-time dimension can be characterized in terms of prediction loss rate, entropy, and compression algorithms. Our result implies that in an asymptotic way each of these concepts is invariant under base change. A corollary of the main theorem is any polynomial-time dimension 1 number (which may be established in any base) is an absolutely normal number, providing an interesting source of absolute normality.