Algorithmic Dimensions via Learning Functions

TL;DR

This paper links the concept of algorithmic dimensions of infinite binary sequences to the limitations of constrained learning functions in pattern detection, extending randomness criteria with quantitative measures.

Contribution

It introduces a novel characterization of algorithmic dimensions using learning functions and extends randomness criteria with a quantitative pattern detection framework.

Findings

Characterizes algorithmic dimensions via learning functions.

Extends randomness criteria to a quantitative setting.

Uses gales and Kolmogorov complexity for proofs.

Abstract

We characterize the algorithmic dimensions (i.e., the lower and upper asymptotic densities of information) of infinite binary sequences in terms of the inability of learning functions having an algorithmic constraint to detect patterns in them. Our pattern detection criterion is a quantitative extension of the criterion that Zaffora Blando used to characterize the algorithmically random (i.e., Martin-L\"of random) sequences. Our proof uses Lutz's and Mayordomo's respective characterizations of algorithmic dimension in terms of gales and Kolmogorov complexity.