Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithmic Dimensions via Learning Functions

Jack H. Lutz, Andrei N. Migunov

TL;DR

The paper unifies algorithmic dimension with learning-theoretic pattern detection by introducing $s$-learnability and showing that $\dim(X)$ equals the infimum of $s$ for which some learning function $s$-learns $X$. It leverages Blando’s uniform weak detection, gales, martingales, and Kolmogorov complexity to characterize both Hausdorff and packing-type dimensions in effective terms, including polynomial-time versions for randomness. The main contributions are (i) a dimension characterization via learning functions for Hausdorff dimension, (ii) a complete correspondence between algorithmic dimension and $s$-learnability, and (iii) analogous results for strong/packing dimensions, all within a unified framework that extends classical dimension theory to an algorithmic setting.

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-Löf random) sequences. Our proof uses Lutz's and Mayordomo's respective characterizations of algorithmic dimension in terms of gales and Kolmogorov complexity.

Algorithmic Dimensions via Learning Functions

TL;DR

The paper unifies algorithmic dimension with learning-theoretic pattern detection by introducing -learnability and showing that equals the infimum of for which some learning function -learns . It leverages Blando’s uniform weak detection, gales, martingales, and Kolmogorov complexity to characterize both Hausdorff and packing-type dimensions in effective terms, including polynomial-time versions for randomness. The main contributions are (i) a dimension characterization via learning functions for Hausdorff dimension, (ii) a complete correspondence between algorithmic dimension and -learnability, and (iii) analogous results for strong/packing dimensions, all within a unified framework that extends classical dimension theory to an algorithmic setting.

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-Löf random) sequences. Our proof uses Lutz's and Mayordomo's respective characterizations of algorithmic dimension in terms of gales and Kolmogorov complexity.
Paper Structure (7 sections, 13 theorems, 40 equations, 2 algorithms)

This paper contains 7 sections, 13 theorems, 40 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithmic randomness via learning functions
  4. Classical and algorithmic dimensions
  5. Hausdorff dimension via learning functions
  6. Algorithmic dimension via learning functions
  7. Strong algorithmic dimension via learning functions

Key Result

Theorem 1

Ville39 Let $\lambda(E)$ denote the Lebesgue measure of a set $E \subseteq \{0,1\}^\infty$. The following are equivalent: (1.) $\lambda(E) = 0$ (2.) There exists a martingale $d: \{0,1\}^* \rightarrow [0,\infty)$ that succeeds on $E$.

Theorems & Definitions (36)

  • Definition
  • Definition
  • Definition
  • Definition
  • Definition
  • Theorem 1
  • Theorem 2
  • Definition
  • Definition
  • Theorem 3
  • ...and 26 more