Regular Article
On hausdorff and topological dimensions of the kolmogorov complexity of the real line*

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We investigate the Kolmogorov complexity of real numbers. Let K be the Kolmogorov complexity function; we determine the Hausdorff dimension and the topological dimension of the graph of K. Since these dimensions are different, the graph of the Kolmogorov complexity function of the real line forms a fractal in the sense of Mandelbrot. We also solve an open problem of Razborov using our exact bound on the topological dimension.

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*

A preliminary version of the paper titled “The Real Line is a Fractal” has appeared in “The Structure in Complexity Theory Conference 1989.”

†

Research supported by NSF Grant CGR-8709818, while the author was at Yale Unversity.

‡

Research supported by NSF Grants CCR-8520597 and CCR-8823053.

Copyright © 1994 Published by Elsevier Inc.