This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depth and polynomial-time depth. It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth law holds, implying that deep sequences cannot be created easily from shallow sequences. Furthermore, the E analogue of the halting language is shown to be polynomial-time deep, by proving a more general result: every language to which a nonnegligible subset of E can be reduced in uniform exponential time is polynomial-time deep.

In this article, we describe a house price index algorithm which requires only sparse and frugal data, namely house location, date of sale and sale price, as input data. We aim to show that our algorithm is as effective for predicting price changes as more complex models which require detailed or extensive data. Although various methods are employed for determining house price indexes, such as hedonic regression, mix-adjusted median or repeat sales, there is no consensus on how to determine the robustness of an index, and hence no agreement on which method is the best to use. We formalise an objective criterion for what a house price index should achieve, namely consistency between time periods. Using this criterion, we investigate whether it is possible to achieve strong robustness using frugal data covering only 66 months of transactions on the Irish property market. We develop a simple multi-stage algorithm and show that it is more robust than the complex hedonic regression model currently employed by the Irish Central Statistics Office.

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We introduce a new measure notion on small complexity classes (called F-measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0's and 1's. On larger complexity classes (E and above), F-measure is equivalent to Lutz resource-bounded measure. As applications to F-measure, we answer a question raised in [1] by improving their result to: for almost every language A decidable in subexponential time, P A = BPP A. We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F-measure to previous notions and prove that martingale families are strictly stronger than Γ-measure [1], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [15] on P, which meets the intuition behind Lutz's notion. We show that P-dimension lies between finite-state dimension and dimension on E. We prove an analogue of a Theorem of Eggleston in P, i.e. the class of languages whose characteristic sequence contains 1's with frequency α, has dimension the Shannon entropy of α in P.

The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffices to specify x on a general-purpose computer with arbitrarily high precisions 2 −r . The dimension spectrum of a set X in Euclidean space is the subset of [0, n] consisting of the dimensions of all points in X.The dimensions of points have been shown to be geometrically meaningful (Lutz 2003, Hitchcock 2003, and the dimensions of points in self-similar fractals have been completely analyzed (Lutz and Mayordomo 2008). Here we begin the more challenging task of analyzing the dimensions of points in random fractals. We focus on fractals that are randomly selected subfractals of a given self-similar fractal. We formulate the specification of a point in such a subfractal as the outcome of an infinite two-player game between a selector that selects the subfractal and a coder that selects a point within the subfractal. Our selectors are algorithmically random with respect to various probability measures, so our selector-coder games are, from the coder's point of view, games against nature.We determine the dimension spectra of a wide class of such randomly selected subfractals. We show that each such fractal has a dimension spectrum that is a closed interval whose endpoints can be computed or approximated from the parameters of the fractal. In general, the maximum of the spectrum is determined by the degree to which the coder can reinforce the randomness in the selector, while the minimum is determined by the degree to which the coder can cancel randomness in the selector. This constructive and destructive interference between the players' randomnesses is somewhat subtle, even in the simplest cases. Our proof techniques include van Lambalgen's theorem on independent random sequences, measure preserving transformations, an application of network flow theory, a Kolmogorov complexity lower bound argument, and a nonconstructive proof that this bound is tight.