The geometry of fractal sets
Elements of information theory
Elements of information theory
Gales and the Constructive Dimension of Individual Sequences
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
Dimension in Complexity Classes
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Note: fractal dimension and logarithmic loss unpredictability
Theoretical Computer Science
The dimensions of individual strings and sequences
Information and Computation
Theoretical Computer Science
A note on dimensions of polynomial size circuits
Theoretical Computer Science
Dimensions of Copeland--Erdös sequences
Information and Computation
Effective Strong Dimension in Algorithmic Information and Computational Complexity
SIAM Journal on Computing
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Consider the problem of calculating the fractal dimension of a set Xconsisting of all infinite sequences Sover a finite alphabet Σthat satisfy some given condition Pon the asymptotic frequencies with which various symbols from Σappear in S. Solutions to this problem are known in cases where(i) the fractal dimension is classical (Hausdorff or packing dimension), or(ii) the fractal dimension is effective (even finite-state) and the condition Pcompletelyspecifies an empirical distribution 驴over Σ, i.e., a limiting frequency of occurrence for everysymbol in Σ.In this paper we show how to calculate the finite-state dimension (equivalently, the finite-state compressibility) of such a set Xwhen the condition Ponly imposes partialconstraints on the limiting frequencies of symbols. Our results automatically extend to less restrictive effective fractal dimensions (e.g., polynomial-time, computable, and constructive dimensions), and they have the classical results (i) as immediate corollaries. Our methods are nevertheless elementary and, in most cases, simpler than those by which the classical results were obtained.