The typical Turing degree

ISCAS OpenIR

	The typical Turing degree
	Barmpalias, George; Day, Adam R.; Lewis-Pye, Andy E. M.
	2014
发表期刊	PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
ISSN	0024-6115
卷号	109 页码:1-39
摘要	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 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. We describe and prove a large number of results in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree, and the level of randomness required in order to guarantee typicality. 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. This case has been fairly extensively examined in the previous literature. We analyse how our new results for the measure-theoretic case contrast with existing results for Baire category, and also provide some new results for the category-theoretic analysis.; 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 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. We describe and prove a large number of results in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree, and the level of randomness required in order to guarantee typicality. 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. This case has been fairly extensively examined in the previous literature. We analyse how our new results for the measure-theoretic case contrast with existing results for Baire category, and also provide some new results for the category-theoretic analysis.
收录类别	SCI
部门归属	[Barmpalias, George] Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Beijing 100190, Peoples R China. [Day, Adam R.] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA. [Lewis-Pye, Andy E. M.] Univ London London Sch Econ & Polit Sci, Dept Math, London WC2A 2AE, England.
语种	英语
WOS记录号	WOS:000339951600001
引用统计
内容类型	期刊论文
URI标识	http://ir.iscas.ac.cn/handle/311060/16848
专题	中国科学院软件研究所
推荐引用方式 GB/T 7714	Barmpalias, George,Day, Adam R.,Lewis-Pye, Andy E. M.. The typical Turing degree[J]. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY,2014,109:1-39.
APA	Barmpalias, George,Day, Adam R.,&Lewis-Pye, Andy E. M..(2014).The typical Turing degree.PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY,109,1-39.
MLA	Barmpalias, George,et al."The typical Turing degree".PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 109(2014):1-39.