Dimension, entropy rates, and compression

  • Authors:

  • John M. Hitchcock;N. V. Vinodchandran

  • Affiliations:

  • Department of Computer Science, University of Wyoming, WY, USA;Department of Computer Science and Engineering, University of Nebraska-Lincoln, NE, USA

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2006

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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 @D"3^E, the third level of the exponential-time hierarchy. This general result is applied to simultaneously improve the results of Mayordomo [E. Mayordomo, Contributions to the study of resource-bounded measure, PhD thesis, Universitat Politecnica de Catalunya, 1994] on the measure on P/poly in @D"3^E and of Lutz [J.H. Lutz, Dimension in complexity classes, SIAM J. Comput. 32 (5) (2003) 1236-1259] on the dimension of exponential-size circuit complexity classes in ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied in conjunction 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 [J.H. Lutz, Almost everywhere high nonuniform complexity, J. Comput. System Sci. 44 (2) (1992) 220-258] about polynomial-size circuit complexity classes from resource-bounded measure to dimension. Exact characterizations of the effective dimensions in terms of Kolmogorov complexity rates at the polynomial-space and higher levels have been established, but in the time-bounded setting no such equivalence is known. We introduce the concept of polynomial-time superranking as an extension of ranking. We show that superranking provides an equivalent definition of polynomial-time dimension. From this superranking characterization we show that polynomial-time Kolmogorov complexity rates give a lower bound on polynomial-time dimension.