Entropy rates and finite-state dimension

  • Authors:

  • Chris Bourke;John M. Hitchcock;N. V. Vinodchandran

  • Affiliations:

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

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2005

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Abstract

The effective fractal dimensions at the polynomial-space level and above can all be equivalently defined as the C-entropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspace-dimension is equivalent to the PSPACE-entropy rate.At lower levels of complexity the equivalence proofs break down. In the polynomial-time case, the P-entropy rate is a lower bound on the p-dimension. Equality seems unlikely, but separating the P-entropy rate from p-dimension would require proving P ≠ NP.We show that at the finite-state level, the opposite of the polynomial-time case happens: the REG-entropy rate is an upper bound on the finite-state dimension. We also use the finite-state genericity of Ambos-Spies and Busse [Automatic forcing and genericity: On the diagonalization strength of finit automata, in: Proc. fourth Int. Conf. on Discrete Mathematics and Theoretical Computer Science, 2003, Springer, Berlin, pp. 97-108] to separate finite-state dimension from the REG-entropy rate.However, we point out that a block-entropy rate characterization of finite-state dimension follows from the work of Ziv and Lempel [Compression of individual sequences via variable rate coding, IEEE Trans. Inform. Theory 24 (1978) 530-536] on finite-state compressibility and the compressibility characterization of finite-state dimension by Dai et al. [Finite-state dimension, Theoret. Comput. Sci. 310(1-3) (2004) 1-33].As applications of the REG-entropy rate upper bound and the block-entropy rate characterization, we prove that every regular language has finite-state dimension 0 and that normality is equivalent to finite-state dimension 1.