Dimension, Entropy Rates, and Compression

• Authors: John M. Hitchcock;N. V. Vinodchandran
• Venue: CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
• Year: 2004

This paper develops relationships between resource-boundeddimension, entropy rates, and compression. Newtools for calculating dimensions are given and used to improveprevious results about circuit-size complexity classes.Approximate counting of SpanP functions is used toprove that the NP-entropy rate is an upper bound for dimensionin \Delta _3^E, the third level of the exponential-time hierarchy.This general result is applied to simultaneously improve theresults of Mayordomo (1994) on the measure on P/poly in\Delta _3^E and of Lutz (2003) on the dimension of exponential-sizecircuit complexity classes in ESPACE.Entropy rates of efficiently rankable sets, sets that are optimallycompressible, are studied in conjunction with time-boundeddimension. It is shown that rankable entropy ratesgive upper bounds for time-bounded dimensions. We usethis to improve results of Lutz (1992) about polynomial-sizecircuit complexity classes from resource-bounded measureto dimension.Exact characterizations of the effective dimensions interms of Kolmogorov complexity rates at the polynomialspaceand higher levels have been established, but in thetime-bounded setting no such equivalence is known. Weintroduce the concept of polynomial-time superranking asan extension of ranking. We show that superranking providesan equivalent definition of polynomial-time dimension.From this superranking characterization we show thatpolynomial-time Kolmogorov complexity rates give a lowerbound on polynomial-time dimension.