Entropy rates and finite-state dimension

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 ...

Finite-State Dimension

Finite-State Dimension and Lossy Decompressors

This paper examines information-theoretic questions regarding the difficulty of compressing data versus the difficulty of decompressing data and the role that information loss plays in this interaction. Finite-state compression and decompression are shown to be of equivalent difficulty, even when the decompressors are allowed to be lossy. Inspired by Kolmogorov complexity, this paper defines th...

Finite-State Dimension and Real Arithmetic

We use entropy rates and Schur concavity to prove that, for every integer k ≥ 2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and qα all have the same finite-state dimension and the same finitestate strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel...

Finite-State Dimension of Individual Sequences

I Classical Hausdorff Dimension (Fractal Dimension) I Lutz’s Effectivization of Hausdorff Dimension via gales [4] I The Cantor Space C I Restriction of gales to Complexity Classes endows sets with dimension within the complexity class. I Constructive Dimension, Resource Bounded Dimension I Zero-dimensional sets can have positive dimension I Lowest level: finite-state machines → Finite-State Dim...