Effective Strong Dimension, Algorithmic Information, and Computational Complexity

The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff [Math. Ann., 79 (1919), pp. 157-179], and packing dimension, developed independently by Tricot [Math. Proc. Cambridge Philos. Soc., 91 (1982), pp. 57-74] and Sullivan [Acta Math., 153 (1984), pp. 259-277]. Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz [Proceedings of the 15th IEEE Conference on Computational Complexity, Florence, Italy, 2000, IEEE Computer Society Press, Piscataway, NJ, 2000, pp. 158-169] 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 $\vec{\beta} = (\beta_0,\beta_1,\ldots)$ is a computable sequence of biases that are bounded away from 0 and $R$ is random with respect to $\vec{\beta}$, then the dimension and strong dimension of $R$ are the lower and upper average entropies, respectively, of $\vec{\beta}$. 2. For each pair of $\Delta^0_2$-computable real numbers $0 < \alpha \le \beta \le 1$, there exists $A \in {\rm E}$ such that the polynomial-time many-one degree of $A$ has dimension $\alpha$ in E and strong dimension $\beta$ in E. Our proofs of these theorems use a new large deviation theorem for self-information with respect to a bias sequence $\vec{\beta}$ that need not be convergent.