The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of-and every bit as simple as-the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions, which are exact duals of the effective dimensions mentioned above. In general (and in analogy with the classical fractal dimensions), the effective strong dimension of a set or sequence is at least as great as its effective dimension, with equality for sets or sequences that are sufficiently regular. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression. Aside from the above characterization of packing dimension, our two main theorems are the following. 1. If β = (β 0 , β 1 ,. . .) is a computable sequence of biases that are bounded away from 0 and R is random with respect to β, then the dimension and strong dimension of R are the lower and upper average entropies, respectively, of β. 2. For each pair of ∆ 0 2-computable real numbers 0 < α ≤ β ≤ 1, there exists A ∈ E such that the polynomial-time many-one degree of A has dimension α in E and strong dimension β in E. Our proofs of these theorems use a new large deviation theorem for self-information with respect to a bias sequence β that need not be convergent.

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-Löf) have dimension 1, while sequences that are decidable, Σ 0 1 , or Π 0 1 have dimension 0. It is shown that for every ∆ 0 2 -computable real number α in [0,1] there is a ∆ 0 2 sequence S such that dim(S) = α. 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 β in (0, 1) is shown to have dimension H(β), the binary entropy of β.

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We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T . The simulation is "intrinsic" in the sense that the self-assembly process carried out by U is exactly that carried out by T , with each tile of T represented by an m×m "supertile" of U . Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T .Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each supertile of U, starting with those in the seed assembly, carries the "genome" of the simulated system T . At each location of a potential supertile in the self-assembly of U , a decision is made whether and how to express this genome, i.e., whether to generate a supertile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct global outcome(s).

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A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound ∆ (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter ∆ yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X | C) ∈ [0, 1]. Along with the elements of this theory, two preliminary applications are presented:1. For every real number 0 ≤ α ≤ 1 2 , the set FREQ(≤ α), consisting of all languages that asymptotically contain at most α of all strings, has dimension H(α) -the binary entropy of α -in E and in E 2 .2. For every real number 0 ≤ α ≤ 1, the set SIZE(α 2 n n ), consisting of all languages decidable by Boolean circuits of at most α 2 n n gates, has dimension α in ESPACE.

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Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004).Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else.We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2007). IntroductionStructures that self-assemble in naturally occurring biological systems are often fractals of low dimension, by which we mean that they are usefully modeled as fractals and that their fractal dimensions are less than the dimension of the space or surface that they occupy. The advantages of such fractal geometries for materials transport, heat exchange, information processing, and robustness imply that structures engineered by nanoscale self-assembly in the near future will also often be fractals of low dimension. The simplest mathematical model of nanoscale self-assembly is the Tile Assembly Model (TAM), an extension of Wang tiling [17,18] that was introduced by Winfree [20] and refined by Rothemund and Winfree [13,12]. (See also [1,11,16].) This elegant model, which is described in section 2, uses tiles with various types and strengths of "glue" on their edges as abstractions of molecules adsorbing to a growing structure. (The tiles are squares in the two-dimensional TAM, which is most widely used, cubes in the three-dimensional TAM, etc.) Despite the model's deliberate oversimplification of molecular geometry and binding, Winfree [20] proved that the TAM is computationally universal in two or more dimensions. Self-assembly in the TAM can thus be directed algorithmically.This paper concerns the self-assembly of fractal structures in the Tile Assembly Model. The typical test bed for a new research topic involving fractals is the Sierpinski triangle, and this is certainly the case for fractal self-assembly. Specifically, Winfree [20] showed that the standard discrete Sierpinski triangle S,