We establish a relationship between the online mistake-bound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work

We study constructive and resource-bounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/poly-Turing reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rd-order scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turing-hard sets for ESPACE. 1

Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resource-bounded extension: if a class X has ∆-strong dimension less than 1, then X is ∆-meager. This has the applications of explaining some of Lutz’s simultaneous ∆-meager, ∆-measure 0 results and providing a new proof of a Gu’s strong dimension result on infinitely-often classes.

z. The dimensions of individual strings and sequences. Information and Computation, 187(1):49--79, 2003. [23] J. H. Lutz. E#ective fractal dimensions. In Computability and Complexity in Analysis, volume 302 of Informatik Berichte, pages 81--97. FernUniversitat in Hagen, August 2003. To appear in Mathematical Logic Quarterly. [24] E. Mayordomo. E#ective Hausdor# dimension. In Proceedings of Foundations of the Formal Sciences III. Kluwer Academic Press. To appear. [25] E. Mayordomo. A Kolmogorov complexity characterization of constructive Hausdor# dimension. Information Processing Letters, 84(1):1--3, 2002. [26] P. Moser. BPP has e#ective dimension at most 1/2 unless BPP = EXP. Technical Report TR03-029, Electronic Colloquium on Computational Complexity, 2003. [27] S. Reid. The classes of algorithmically random reals. Master's thesis, Victoria University of Wellington, 2003. [28] L. Staiger. Constructive dimension equals Kolmogorov complexity. Technical Report CDMTCS-210, Universi

Abstract Resource-bounded dimension is a notion of computational information density of in-finite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finite-state dimension.The pushdown dimension of any sequence is trivially bounded above by its finite-state dimension, since a pushdown gambler can simulate any finite-state gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finite-state dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewell-known fact that pushdown automata decide strictly more languages than finite-state

Abstract This paper develops new 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 \Delta E3, the third level of the exponential-time hierarchy. This generalresult is applied to simultaneously improve the results of Mayordomo (1994) on the measure on P/poly in \Delta E3 and of Lutz (2000) on the dimension of exponential-size circuit complexity classesin ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied inconjunction with time-bounded dimension. It is shown that rankable entropy rates give upper bounds for time-bounded dimensions. We use this to improve results of Lutz (1992) aboutpolynomial-size circuit complexity classes from resource-bounded measure to dimension.