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Rauzy Dimension and Finite-state Dimension by Becher, Verónica; Carton, Olivier; Figueira, Santiago is a scholarly article available to read on EtoBox.

What is Rauzy Dimension and Finite-state Dimension about?

In a paper of 1976, Rauzy studied two complexity notions, $\underline{\beta}$ and $\overline{\beta}$, for infinite sequences over a finite alphabet. The function $\underline{\beta}$ is maximum exactly in the Borel normal sequences and $\overline{\beta}$ is minimum exactly in the sequences that, when added to any Borel normal sequence, the result is also Borel normal. Although the definition of $\underline{\beta}$ and $\overline{\beta}$ do not involve finite-state automata, we establish some connections between them and the lower $\underline{\rm dim}$ and upper $\overline{\rm dim}$ finite-state dimension (or other equivalent notions like finite-state compression ratio, aligned-entropy or cumulative log-loss of finite-state predictors). We show tight lower and upper bounds on $\underline{\rm dim}$ and $\overline{\rm dim}$ as functions of $\underline{\beta}$ and $\overline{\beta}$, respectively. In particular this implies that sequences with $\overline{\rm dim}$ zero are exactly the ones that that, when added to any Borel normal sequence, the result is also Borel normal. We also show that the finite-state dimensions $\underline{\rm dim}$ and $\overline{\rm dim}$ are essentially subadd

Author

Becher, Verónica; Carton, Olivier; Figueira, Santiago

Published

2024

Language

EN