Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the incomputability of computable dimension

Ludwig Staiger

TL;DR

The paper addresses the incomputability of computable dimension for computable ω-languages by constructing an iterative finite-tree framework that yields an ω-language $F$ with $\dim F=\liminf_{i\to\infty} q_i$. It shows that even when $F$ has a computable prefix language, the dimension value $\alpha$ can be incomputable or not computably approximable, separating Hausdorff, computable, and constructive dimensions. By comparing σ-(super)gales and martingales (via Schnorr-style orderings), the authors identify cases where martingale methods achieve the dimension precisely while supergales may fail to do so, and they highlight the superiority of the martingale framework in certain borderline scenarios. The results illuminate intrinsic limits of algorithmic dimension notions and underscore the nuanced relationships among the different dimensional concepts in Cantor space, with implications for computability and effective descriptive set theory.

Abstract

Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.

On the incomputability of computable dimension

TL;DR

The paper addresses the incomputability of computable dimension for computable ω-languages by constructing an iterative finite-tree framework that yields an ω-language with . It shows that even when has a computable prefix language, the dimension value can be incomputable or not computably approximable, separating Hausdorff, computable, and constructive dimensions. By comparing σ-(super)gales and martingales (via Schnorr-style orderings), the authors identify cases where martingale methods achieve the dimension precisely while supergales may fail to do so, and they highlight the superiority of the martingale framework in certain borderline scenarios. The results illuminate intrinsic limits of algorithmic dimension notions and underscore the nuanced relationships among the different dimensional concepts in Cantor space, with implications for computability and effective descriptive set theory.

Abstract

Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.

Paper Structure

This paper contains 12 sections, 13 theorems, 16 equations.

Table of Contents

  1. Notation
  2. Gales and Martingales
  3. Iterative Tree Construction
  4. Preliminaries
  5. Tree construction
  6. The infinite tree
  7. Monotone families $(q_{i})_{i\in\mathbb{N}}$
  8. Incomputable dimensions
  9. Hausdorff dimension
  10. Computable dimension
  11. Comparison of gales and martingales
  12. Concluding remark

Key Result

Proposition 2.2

Let $F\subseteq X^{\omega}$ be non-empty and satisfy the conditions Then $\dim F= \liminf\limits_{n\to\infty}\frac{\log_{|X|}|\mathbf{pref}(F)\cap X^{n}|}{n}$ .∎

Theorems & Definitions (16)

  • Definition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Lemma 3.3
  • Lemma 3.4
  • Corollary 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Proposition 3.8
  • Lemma 4.1
  • ...and 6 more