Dimension, entropy rates, and compression
Journal of Computer and System Sciences
Entropy rates and finite-state dimension
Theoretical Computer Science
The Kolmogorov complexity of infinite words
Theoretical Computer Science
Connectivity Properties of Dimension Level Sets
Electronic Notes in Theoretical Computer Science (ENTCS)
Dimensions of Points in Self-similar Fractals
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Connectedness properties of dimension level sets
Theoretical Computer Science
A correspondence principle for exact constructive dimension
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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We show that the classical Hausdorff and constructive dimensions of any union of $\Pi^0_1$-definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is $\Sigma^0_2$-definable, then the computable dimension also equals the Hausdorff dimension. This second result is implicit in the work of Staiger (1998). Staiger also proved related results using entropy rates of decidable languages. We show that Staiger’s computable entropy rate provides an equivalent definition of computable dimension. We also prove that a constructive version of Staiger’s entropy rate coincides with constructive dimension.