Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
A Kolmogorov complexity characterization of constructive Hausdorff dimension
Information Processing Letters
MAX3SAT is exponentially hard to approximate if NP has positive dimension
Theoretical Computer Science
Recursive Automata on Infinite Words
STACS '93 Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science
Gales suffice for constructive dimension
Information Processing Letters
Hausdorff Dimension in Exponential Time
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
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
Constructive dimension equals Kolmogorov complexity
Information Processing Letters
Journal of Computer and System Sciences - Special issue on COLT 2002
Connectivity Properties of Dimension Level Sets
Electronic Notes in Theoretical Computer Science (ENTCS)
Arithmetic complexity via effective names for random sequences
ACM Transactions on Computational Logic (TOCL)
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Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) ∈ [0,1] and a strong dimension Dim(A) ∈ [0,1]. Let DIMα and DIMαstr be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM0 is properly Π02, and that for all Δ02-computable α ∈ (0, 1], DIMα is properly Π03. To classify the strong dimension classes, we use a more powerful effective Borel hierarchy where a coenumerable predicate is used rather than an enumerable predicate in the definition of the Σ01 level. For all Δ02-computable α ∈ [0, 1), we show that DIMαstr is properly in the Π03 level of this hierarchy. We show that DIM1str is properly in the Π02 level of this hierarchy. We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly Π03.