Effective Fractal Dimension in Algorithmic Information Theory

Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that smooth curves have dimension 1 and smooth surfaces have dimension 2, but from its introduction in 1918 [23] Hausdorff noted that many sets have noninteger dimension, what he called “fractional dimension”. The development and applications of Fractal Geometry quickly outgrew the field of Geometry and spread through many other areas [19, 56, 15, 16, 17, 13, 12, 49]. In the 1980s Tricot [73] and Sullivan [71] independently developed a dual of Hausdorff dimension called packing dimension that is now widely used. In this paper we will focus on the use of fractal dimensions in the Cantor space of infinite sequences over a finite alphabet. The results obtained since the 1990’s, and in particular the effectivizations of dimension that we will review in this paper, have introduced the powerful tools of fractal geometry into Computational Complexity and Information Theory. In 2000 Lutz [45] proved a new characterization of Hausdorff dimension for the case of Cantor space that was based on gales. This characterization was the beginning of a whole range of effective versions of dimensions naturally based on bounding the computing power of the gale. Gales are a generalization of martingales which are strategies for betting on the successive bits of infinite binary sequences with fair payoffs. Martingales were introduced by Ville [74] in 1939 (also implicit in [38, 39]) and used by Schnorr [61, 62, 63, 64] in his work on randomness. In the 1990s, Ryabko [59, 60] and Staiger [69] proved several connections of Hausdorff dimension and martingales, that included relating the Hausdorff dimension of a set X of binary sequences to the growth rates achievable by computable martingales betting on the sequences in X (see section 4 for more details). The introduction of resource-bounded dimension by Lutz [45] had the immediate motivation of overcoming the limitations of resource-bounded measure, a