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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.

The zeta-dimension of a set A of positive integers isZeta-dimension serves as a fractal dimension on Z + that extends naturally and usefully to discrete lattices such as Z d , where d is a positive integer. This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory. New results presented include extended connections between zeta-dimension and classical fractal dimensions, a gale characterization of zeta-dimension, and a theorem on the zeta-dimensions of pointwise sums and products of sets of positive integers. which are the fractal dimensions (especially Hausdorff dimension [14,12] and packing dimension [30,29,12]). Theoretical computer scientists have recently developed effective fractal dimensions [22,20,21,7,4] that work in complexity classes and other countable settings, but these, too, are best regarded as continuous, albeit effective, mathematical methods.Some fractal structures are inherently discrete and best modeled that way. To some extent this is already true for structures created by cellular automata. For the nascent theory of nanostructure self-assembly [1,25], the case is even more compelling. This theory models the bottom-up selfassembly of molecular structures. The tile assembly models that achieve this cannot be regarded as discrete approximations of continuous phenomena (as cellular automata often are), because their bottom-level units (tiles) correspond directly to discrete objects (molecules). Fractal structures assembled by such a model are best analyzed using discrete tools.This paper concerns a discrete fractal dimension, called zeta-dimension, that works in discrete lattices such as Z d , where d is a positive integer. Curiously, although our work is motivated by twenty-first century concerns in theoretical computer science, zeta-dimension has its mathematical origins in eighteenth and nineteenth century number theory. Specifically, zeta-dimension is defined in terms of a generalization of Euler's 1737 zeta-function [11] ζ(s) = ∞ n=1 n −s , defined for nonnegative real s (and extended in 1859 to complex s by Riemann [24], after whom the zeta-function is now named). Moreover, this generalization can be formulated in terms of Dirichlet series [9], which were developed in 1837, and one of the most important properties of zeta-dimension (in modern terms, the entropy characterization) was proven in these terms by Cahen [5] in 1894.Our objectives here are twofold. First, we present zeta-dimension and its basic theory, citing its origins in scattered references, but stating things in a unified framework emphasizing zetadimension's role as a discrete fractal dimension in theoretical computer science. Second, we present several new results on zeta-dimension and its interactions with classical fractal geometry and algorithmic information theory.Our presentation is organized as follows. In section 2, we give an intuitive development of zeta-di...

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension 0. We also show that $\Ppoly^\io$ has $\pthree$-dimension 1/2, $\pthree$-strong dimension 1. Our results improve previous measure results of Lutz (1992) and dimension results of Hitchcock and Vinodchandran (2004).Comment: 11 page

We exhibit a polynomial time computable plane curve Γ that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization f of Γ and every positive integer m, there is some positive-length subcurve of Γ that f retraces at least m times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem.

Abstract. This paper presents the following results on sets that are complete for NP.(i) If there is a problem in NP that requires 2n Ω(1) time at almost all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. (ii) If there is a problem in co-NP that cannot be solved by polynomial-size nondeterministic circuits, then every many-one complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. (iii) If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in NP∩co-NP, then there is a Turing complete language for NP that is not many-one complete. Our first two results use worst-case hardness hypotheses whereas earlier work that showed similar results relied on average-case or almost-everywhere hardness assumptions. The use of average-case and worst-case hypotheses in the last result is unique as previous results obtaining the same consequence relied on almost-everywhere hardness results.

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The base-k Copeland-Erdös sequence given by an infinite set A of positive integers is the infinite sequence CE k (A) formed by concatenating the base-k representations of the elements of A in numerical order. This paper concerns the following four quantities.• The finite-state dimension dim FS (CE k (A)), a finite-state version of classical Hausdorff dimension introduced in 2001.• The finite-state strong dimension Dim FS (CE k (A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of dim FS (CE k (A)) satisfying Dim FS (CE k (A)) ≥ dim FS (CE k (A)).• The zeta-dimension Dim ζ (A), a kind of discrete fractal dimension discovered many times over the past few decades. • The lower zeta-dimension dim ζ (A), a dual of Dim ζ (A) satisfying dim ζ (A) ≤ Dim ζ (A).We prove the following.1. dim FS (CE k (A)) ≥ dim ζ (A). This extends the 1946 proof by Copeland and Erdös that the sequence CE k (PRIMES) is Borel normal. Dim3. These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in [0, 1] satisfying the four above-mentioned inequalities.