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The Randomness Deficiency Function and the Shift Operator

Samuel Epstein

TL;DR

The paper specializes a general result on algorithmic thermodynamic entropies to the Cantor space under the shift operator, using the uniform measure. It develops the framework of ${\mathbf K}$, ${\mathbf m}$, ${\mathbf D}$, and ${\mathbf I}$, and employs universal tests to relate transformations of infinite sequences to their randomness deficiencies. The main finding is that if $(\alpha,\beta)$ is Martin-Löf random and ${\mathbf I}((\alpha,\beta):{\mathcal{H}})<\infty$, then $\sup_n |{\mathbf D}(\sigma^{(n)}\alpha)-{\mathbf D}(\sigma^{(n)}\beta)|=\infty$, with the analogous almost-sure product-measure statement. This demonstrates an unbounded divergence of algorithmic randomness under repeated shifts, contributing to the understanding of randomness in dynamical systems and offering potential generalizations to other computable measures.

Abstract

Almost surely, the difference between the randomness deficiencies of two infinite sequences will be unbounded with respect to repeated iterations of the shift operator.

The Randomness Deficiency Function and the Shift Operator

TL;DR

The paper specializes a general result on algorithmic thermodynamic entropies to the Cantor space under the shift operator, using the uniform measure. It develops the framework of , , , and , and employs universal tests to relate transformations of infinite sequences to their randomness deficiencies. The main finding is that if is Martin-Löf random and , then , with the analogous almost-sure product-measure statement. This demonstrates an unbounded divergence of algorithmic randomness under repeated shifts, contributing to the understanding of randomness in dynamical systems and offering potential generalizations to other computable measures.

Abstract

Almost surely, the difference between the randomness deficiencies of two infinite sequences will be unbounded with respect to repeated iterations of the shift operator.
Paper Structure (4 sections, 4 theorems, 2 equations)

This paper contains 4 sections, 4 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Results
  4. Proof.

Key Result

Proposition 1

For partial computable $f:\{0,1\}^\infty\rightarrow\{0,1\}^\infty$, ${\mathbf I}(f(\alpha):{\mathcal{H}})<^+ {\mathbf I}(\alpha:{\mathcal{H}})+{\mathbf K}(f)$.

Theorems & Definitions (4)

  • Proposition 1
  • Theorem 1: EpsteinAnomalies24
  • Theorem 2: Vereshchagin21Levin74Geiger12
  • Theorem 3