The typical Turing degree

TL;DR

This paper investigates the properties of the typical Turing degree, showing that it either almost surely satisfies or negates certain properties based on measure and category, and explores the role of randomness and genericity.

Contribution

It provides a detailed analysis of the typical Turing degree using measure and category, establishing foundational results in understanding its properties.

Findings

Typical Turing degree satisfies properties with measure 0 or 1

Randomness and genericity determine typicality in measure and category contexts

Results contribute to understanding the computational complexity of real numbers

Abstract

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the \emph{typical} degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. A similar analysis can be made in terms of Baire category, where a standard form of genericity now plays the role that randomness plays in the context of measure. We describe and prove a number of results in a programme of research which aims to establish the properties of the typical Turing degree, where typicality is gauged either in terms of Lebesgue measure or Baire category.