John M. Hitchcock

Fractal dimension and logarithmic loss unpredictability

We show that the Hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet. Using computable, feasible, and finite-state predictors, this equivalence also holds for the computable, feasible, and finite-state dimensions. Combining this with recent results of Fortnow and Lutz (Proc. 15th Ann. Conf. on Comput. Learning Theory (2002) 380), we have a tight relationship between prediction with respect to logarithmic loss and prediction with respect to absolute loss.

Effective Strong Dimension in Algorithmic Information and Computational Complexity

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.

Correspondence Principles for Effective Dimensions

Abstract We show that the classical Hausdorff and constructive dimensions of any union of

Entropy rates and finite-state dimension

\Pi^0_1

Extracting kolmogorov complexity with applications to dimension zero-one laws

-definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is

NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly

\Sigma^0_2

Small Spans in Scaled Dimension

-definable, then the computable dimension also equals the Hausdorff dimension. This second result is implicit in the work of Staiger (1998). Staiger also proved related results using entropy rates of decidable languages. We show that Staiger’s computable entropy rate provides an equivalent definition of computable dimension. We also prove that a constructive version of Staiger’s entropy rate coincides with constructive dimension.

Hardness hypotheses, derandomization, and circuit complexity

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 finite-state 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.

Dimension, entropy rates, and compression

We apply recent results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α, e> 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y) > (1–e)|y|. This result holds for both classical and space-bounded Kolmogorov complexity. We use the extraction procedure for space-bounded complexity to establish zero-one laws for polynomial-space strong dimension. Our results include: (i) If Dimpspace(E) > 0, then Dimpspace(E/O(1)) = 1. (ii) Dim(E/O(1) |ESPACE) is either 0 or 1. (iii) Dim(E/poly |ESPACE) is either 0 or 1. In other words, from a dimension standpoint and with respect to a small amount of advice, the exponential-time class E is either minimally complex or maximally complex within ESPACE.

Partial Bi-immunity, Scaled Dimension, and NP-Completeness

We show that hard sets S for NP must have exponential density, i.e. |S=n| ges 2nepsi for some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1-isin queries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n1-isin queries.