John M. Hitchcock

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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(More)
We apply results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α, > 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− )|y|. This result holds for both unbounded and space-bounded Kolmogorov(More)
We show that the classical Hausdorff and constructive dimensions of any union of $\Pi^0_1$-definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is $\Sigma^0_2$-definable, then the computable dimension also equals the Hausdorff dimension. This second result is implicit in the work of Staiger (1998). Staiger(More)
This paper develops relationships between resource-bounded dimension, entropy rates, and compression. New tools for calculating dimensions are given and used to improve previous results about circuit-size complexity classes. Approximate counting of SpanP functions is used to prove that the NP-entropy rate is an upper bound for dimension in /spl Delta//sub(More)
Bibliography 85 ii Acknowledgments My foremost thanks go to my advisor Jack Lutz. Jack brought me into research in 1999 while I was still an undergraduate. In the four years since he has provided me with excellent research advice and I have thoroughly enjoyed working with him. thank them for collaborating with me. I also thank Pavan Aduri, Cliff Bergman,(More)
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 (2002), we have a(More)
We show that hard sets S for NP must have exponential density, i.e. |S=n| ≥ 2n for some > 0 and infinitely many n, unless coNP ⊆ NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1− queries. In addition we study the instance complexity of NP-hard problems and show that hard sets also have an exponential(More)
We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: The measure hypothesis: NP does not have p-measure 0. The pseudo-NP hypothesis: there is an NP language that can be distinguished from any DTIME $$(2^{{n}^\epsilon})$$ language by an NP refuter. The NP-machine(More)
Juedes and Lutz (1995) proved a small span theorem for polynomial-time many-one reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the upper span) is small in the sense of resource-bounded measure(More)