The typical Turing degree
type: Item
instance of: scholarly article
title: The typical Turing degree
zbMATH Open document ID: 1332.03009
author: George Barmpalias
author: Adam R. Day
author: Andrew E. M. Lewis
publication date: 2014-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.1064
Mathematics Subject Classification ID: 03D28
zbMATH DE Number: 6327940
zbMATH Keywords: 2-random degree
zbMATH Keywords: measure-risking
MaRDI profile type: Publication
OpenAlex ID: W2120390967
arXiv ID: 1111.1064
DOI: 10.1112/PLMS/PDT065
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published in: Proceedings of the London Mathematical Society. Third Series
review text: The paper continues the programme initiated by Yates, Martin, Jockusch and Kurtz of understanding the behaviour of ``most'' Turing degrees in terms of both measure and category. The intuition put forward is that many results which hold of 2-random degrees also hold of all predecessors of 2-random degrees.NEWLINENEWLINE The main results proved are:{\parindent=0.6cm\begin{itemize}\item[1.] Every degree bounded by a 2-random degree has a strong minimal cover.\item[2.] Every degree bounded by a 2-random degree satisfies the join property.\item[3.] Every degree bounded by a 2-random degree is the join of two 1-generic degrees.NEWLINENEWLINE\end{itemize}} The proofs are an elaboration on Paris's ``measure risking'' technique. The common features of these proofs are discussed in a methodological way. We remark that recently \textit{L. Bienvenu} and \textit{L. Patey} [``Diagonally non-computable functions and fireworks'', Inf. Comput. (to appear), \url{arXiv:1411.6846}] gave another general presentation of the measure-risking method.
reviewed by: Noam Greenberg
description: scientific article; zbMATH DE number 6327940