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Measure on small complexity classes, with applications for BPP
 In Proceedings of the 35th Symposium on Foundations of Computer Science
, 1994
"... We present a notion of resourcebounded measure for P and other subexponentialtime classes. This genemlization is based on Lutz’s notion of measure, but overcomes the limitations that cause Lptz’s definitions to apply only to classes at least as large as E. We present many of the basic properties ..."
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Cited by 48 (7 self)
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We present a notion of resourcebounded measure for P and other subexponentialtime classes. This genemlization is based on Lutz’s notion of measure, but overcomes the limitations that cause Lptz’s definitions to apply only to classes at least as large as E. We present many of the basic properties of this measure, and use it to ezplore the class of sets that are hard for BPP. Bennett and Gill showed that almost all sets are hard for BPP; Lutz improved this from Lebesgue measure to measure on ESPACE. We use OUT measure to improve this still further, showing that for all E> 0, almost every set in E, is hard for BPP, where E, = Us<rDTIME(2”6), which is the best that can be achieved without showing that BPP is properly contained in E. A number of related results are also obtained in this way. 1
FINITESTATE DIMENSION OF INDIVIDUAL SEQUENCES
, 2004
"... Classical Hausdorff dimension, popularly known as fractal dimension, has recently been effectivized by gales–functions that are essentially betting strategies that play against infinite binary sequences. Gales are a generalization of martingales and are sufficient for establishing fractal dimension ..."
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Classical Hausdorff dimension, popularly known as fractal dimension, has recently been effectivized by gales–functions that are essentially betting strategies that play against infinite binary sequences. Gales are a generalization of martingales and are sufficient for establishing fractal dimension on sets. When gales are restricted to functions computable in a certain complexity class, such a characterization endows sequences (or sets of sequences) with dimension within the complexity class. Countable sets and singletons, that would otherwise have Hausdorff dimension zero, may be given quantifiable positive dimension. In this thesis we restrict our examination to gale functions computable by finitestate machines and explore individual sequences within the topology of the Cantor space. We develop new concepts of periodicity, entropy and betting trees in terms of fixed “blocks ” of a sequence. We use these to establish that the entropy rate with respect to blocks is an upper bound to the dimension of any sequence. We also