The Dimensions of Individual Strings and Sequences

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0,1]. Sequences that are random (in the sense of Martin-Lof) have dimension 1, while sequences that are decidable, \Sigma^0_1, or \Pi^0_1 have dimension 0. It is shown that for every \Delta^0_2-computable real number \alpha in [0,1] there is a \Delta^0_2 sequence S such that \dim(S) = \alpha. A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit infimum of the dimensions of its prefixes. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit infimum of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number \beta in (0,1) is shown to have dimension \H(\beta), the binary entropy of \beta.Comment: 31 page

Dimension Extractors and Optimal Decompression

A *dimension extractor* is an algorithm designed to increase the effective dimension -- i.e., the amount of computational randomness -- of an infinite binary sequence, in order to turn a "partially random" sequence into a "more random" sequence. Extractors are exhibited for various effective dimensions, including constructive, computable, space-bounded, time-bounded, and finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is examined from the perspective of decompression, by showing that every infinite sequence S is Turing reducible to a Martin-Loef random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S, which is shown to be the optimal ratio of query bits to computed bits achievable with Turing reductions. The extractors and decompressors that are developed lead directly to new characterizations of some effective dimensions in terms of optimal decompression by Turing reductions.Comment: This report was combined with a different conference paper "Every Sequence is Decompressible from a Random One" (cs.IT/0511074, at http://dx.doi.org/10.1007/11780342_17), and both titles were changed, with the conference paper incorporated as section 5 of this new combined paper. The combined paper was accepted to the journal Theory of Computing Systems, as part of a special issue of invited papers from the second conference on Computability in Europe, 200

Effective fractal dimension: foundations and applications

Lutz characterized Hausdorff dimension using gales, gambling functions which generalize martingales. Imposing computability and complexity constraints on the gales in this characterization gives the constructive and resource-bounded dimensions. These effective dimensions are useful measures of complexity for theoretical computer science and are closely related to other measures of complexity including Kolmogorov complexity, Boolean circuit-size complexity, and predictability.;This thesis makes several foundational contributions to effective fractal dimension. We show that packing dimension, despite the greater complexity of its definition, can also be effectivized using gales. This is used to introduce constructive strong dimension and resource-bounded strong dimension. The theory of effective strong dimension is presented alongside that of effective dimension, emphasizing the duality between the two types of dimension. We show that gales and supergales are equivalent for defining the constructive dimensions, and that the constructive dimensions can be characterized using constructive entropy rates. The resource-bounded dimensions are characterized by log-loss unpredictability in a standard model of prediction. Characterizations of the computability and polynomial-space dimensions involving resource-bounded Kolmogorov complexity and entropy rates are given.;Relationships between constructive dimension and the arithmetical hierarchy are investigated. We identify the levels of the arithmetical hierarchy in which the Hausdorff and constructive dimensions of a set are guaranteed to be equal. The constructive dimension classes are precisely located in the arithmetical hierarchy.;We use resource-bounded dimension to investigate complexity classes involving polynomial-time reductions. Resource-bounded scaled dimension is applied to extend the small span theorem of resource-bounded measure. We show that degrees of arbitrary dimension and strong dimension exist within exponential time. The dimensions of classes that are reducible to nondense languages are investigated.</p

Effective fractal dimension: foundations and applications

Lutz characterized Hausdorff dimension using gales, gambling functions which generalize martingales. Imposing computability and complexity constraints on the gales in this characterization gives the constructive and resource-bounded dimensions. These effective dimensions are useful measures of complexity for theoretical computer science and are closely related to other measures of complexity including Kolmogorov complexity, Boolean circuit-size complexity, and predictability.;This thesis makes several foundational contributions to effective fractal dimension. We show that packing dimension, despite the greater complexity of its definition, can also be effectivized using gales. This is used to introduce constructive strong dimension and resource-bounded strong dimension. The theory of effective strong dimension is presented alongside that of effective dimension, emphasizing the duality between the two types of dimension. We show that gales and supergales are equivalent for defining the constructive dimensions, and that the constructive dimensions can be characterized using constructive entropy rates. The resource-bounded dimensions are characterized by log-loss unpredictability in a standard model of prediction. Characterizations of the computability and polynomial-space dimensions involving resource-bounded Kolmogorov complexity and entropy rates are given.;Relationships between constructive dimension and the arithmetical hierarchy are investigated. We identify the levels of the arithmetical hierarchy in which the Hausdorff and constructive dimensions of a set are guaranteed to be equal. The constructive dimension classes are precisely located in the arithmetical hierarchy.;We use resource-bounded dimension to investigate complexity classes involving polynomial-time reductions. Resource-bounded scaled dimension is applied to extend the small span theorem of resource-bounded measure. We show that degrees of arbitrary dimension and strong dimension exist within exponential time. The dimensions of classes that are reducible to nondense languages are investigated