Entropy rates and finite-state dimension

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 polynomialtime 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 6= 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 finitestate genericity of Ambos-Spies and Busse (2003) 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 (1978) on finite-state compressibility and the compressibility characterization of finite-state dimension by Dai, Lathrop, Lutz, and Mayordomo (2004). 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.