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.

The effective fractal dimensions at the polynomial-space level and above can all be equivalently defined as the C-entropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspace-dimension is equivalent to the PSPACE-entropy rate.At lower levels of complexity the equivalence proofs break down. In the polynomial-time case, the P-entropy rate is a lower bound on the p-dimension. Equality seems unlikely, but separating the P-entropy rate from p-dimension would require proving P = NP.We show that at the finite-state level, the opposite of the polynomial-time case happens: the REG-entropy rate is an upper bound on the finite-state dimension. We also use the finite-state genericity of Ambos-Spies and Busse [Automatic forcing and genericity: On the diagonalization strength of finit automata, in: Proc. fourth Int. Conf. on Discrete Mathematics and Theoretical Computer Science, 2003, Springer, Berlin, pp. 97-108] to separate finite-state dimension from the REG-entropy rate. However, we point out that a block-entropy rate characterization of finite-state dimension follows from the work of Ziv and Lempel [Compression of individual sequences via variable rate coding, IEEE Trans. Inform. Theory 24 (1978) 530-536] on finitestate compressibility and the compressibility characterization of finite-state dimension by Dai et al. [Finite-state dimension, Theoret. Comput. Sci. 310(1-3) (2004) 1-33].As applications of the REG-entropy rate upper bound and the block-entropy rate characterization, we prove that every regular language has finite-state dimension 0 and that normality is equivalent to finite-state dimension 1.

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Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (in: Proceedings of the 15th Annual IEEE Conference on Computational Complexity, 2000, pp. 158-169) in order to investigate the fractal structure of sets that have resource-bounded measure 0. For example, while it has long been known that the Boolean circuit-size complexity class SIZEða 2 n n Þ has measure 0 in ESPACE for all 0pap1; we now know that SIZEða 2 n n Þ has dimension a in ESPACE for all 0pap1: The present paper furthers this program by developing a natural hierarchy of ''rescaled'' resourcebounded dimensions. For each integer i and each set X of decision problems, we define the ith-order dimension of X in suitable complexity classes. The zeroth-order dimension is precisely the dimension of Hausdorff (Math. Ann. 79 (1919) 157-179) and Lutz (2000). Higher and lower orders are useful for various sets X : For example, we prove the following for 0pap1 and any polynomial qðnÞXn 2 :1. The class SIZEð2 an Þ and the time-and space-bounded Kolmogorov complexity classes KT q ð2 an Þ and KS q ð2 an Þ have first-order dimension a in ESPACE. 2. The classes SIZEð2 n a Þ; KT q ð2 n a Þ; and KS q ð2 n a Þ have second-order dimension a in ESPACE. 3. The classes KT q ð2 n ð1 À 2 Àan ÞÞ and KS q ð2 n ð1 À 2 Àan Þ have negative-first-order dimension a in ESPACE.