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Complementing cappable degrees in the difference hierarchy

Rod Downey, Angsheng Li, Guohua Wu
2004 Annals of Pure and Applied Logic  
Consequently, a computably enumerable degree is cappable if and only if it can be complemented by a nonzero d.c.e. degree. This gives a new characterization of the cappable degrees.  ...  We prove that for any computably enumerable (c.e.) degree c, if it is cappable in the computably enumerable degrees, then there is a d.c.e. degree d such that c ∪ d = 0 and c ∩ d = 0.  ...  We are mainly interested in the complements of the c.e. degrees in the Ershov hierarchy.  ... 
doi:10.1016/j.apal.2003.10.002 fatcat:62kypw5qdzcb3i7b2e3m4y7hdy

Page 9544 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
difference hierarchy.  ...  The authors show that a computably enumerable (c.e.) degree is cappable if and only if it has a complement in the structure of the d.c.e. degrees. (A set D is d.c.e. if D = B—A forc.e. sets B and A.)  ... 

Author Index

2004 Annals of Pure and Applied Logic  
., An effective version of Wilkie's theorem of the complement and some effective o-minimality results (1-3) 43-74 Downey, R., Li, A. and Wu, G., Complementing cappable degrees in the difference hierarchy  ...  ., see Downey, R. (1-3) 101-118 ( The issue number is given in front of the page numbers.)  ... 
doi:10.1016/s0168-0072(04)00010-7 fatcat:uifws35bifawzomuaz2ykaiwhq

CAPPABLE CEA SETS AND RAMSEY'S THEOREM

Asher M. Kach, Manuel Lerman, Reed Solomon
2011 Proceedings of the 11th Asian Logic Conference  
We begin a search for degree-theoretic properties that might be used to separate Ramsey's Theorem for pairs from its stable version in the Reverse Mathematical sense.  ...  This paper introduces the notion of c-cappability and shows that this property cannot be used to obtain such a separation when combined with 2-CEA-ness.  ...  a given ∆ 0 2 set or its complement computable in that degree.  ... 
doi:10.1142/9789814360548_0007 fatcat:cyarjuwsbjaxhegpoac2vkamou

On the structure of the degrees of relative provability

Uri Andrews, Mingzhong Cai, David Diamondstone, Steffen Lempp, Joseph S. Miller
2015 Israel Journal of Mathematics  
of totality of computable functions are Π 0 2 -statements); and the structure of the degrees of provability can be viewed as the Lindenbaum algebra of true Π 0 2 -statements in first-order arithmetic.  ...  jump inversion as well as the corresponding high/low hierarchies, investigating the structure of true Π 0 1 -statements as a substructure, and connecting the degrees of provability to escape and domination  ...  In Section 6, we study the high/low hierarchy for both the hop and the jump. In Section 7, we study the cappable p-degrees. In Section 8, we show jump inversion for both the hop and the jump.  ... 
doi:10.1007/s11856-015-1182-8 fatcat:umzkztj7djganfrwdzghrzmsty

Bounding computably enumerable degrees in the Ershov hierarchy

Angsheng Li, Guohua Wu, Yue Yang
2006 Annals of Pure and Applied Logic  
In this paper, we study the c.e. predecessors of d.c.e. degrees, and prove that given a nonzero d.c.e. degree a, there is a c.e. degree b below a and a high d.c.e. degree d > b such that b bounds all the  ...  Wu, Isolation and the high/low hierarchy, Arch. Math. Logic 41 (2002) 259-266]; (2) there is a high d.c.e. degree bounding no minimal pairs [  ...  All three authors are partially supported by the International Joint Project No. 00310308 of NSFC of China.  ... 
doi:10.1016/j.apal.2005.10.004 fatcat:as3psv42jjakrbrhdetdqetwt4

Degrees of Unsolvability [chapter]

Klaus Ambos-Spies, Peter A. Fejer
2014 Handbook of the History of Logic  
We finish our history by quoting Gerald Sacks, who through his own work and the work of his students (including, Harrington, Robinson, Shore, Simpson, Slaman, and Thomason) shaped degree theory throughout  ...  Further n-CEA and ω-CEA operators are induced by the sets in the Ershov difference hierarchy or, to be more precise, by the corresponding operators as follows.  ...  Yates showed that not every incomplete degree is half of a minimal pair. In modern terminology, not every degree is cappable.  ... 
doi:10.1016/b978-0-444-51624-4.50010-1 fatcat:clf7varewrh6lc4jq2c2sp6pgy

A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES

ROD DOWNEY, NOAM GREENBERG
2018 Bulletin of Symbolic Logic  
The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.  ...  The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory.  ...  Maximality in the new hierarchy Remarkably, it turns out that the hierarchy we introduced gives new noncontinuity results in the c.e. degrees. Definition 7.1.  ... 
doi:10.1017/bsl.2017.41 fatcat:qa45hxbavfexbfcp5gn2zr5rhm

Page 1413 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
.; Wu, Guo Hua*) Complementing cappable degrees in the difference hierarchy. (English summary) Ann. Pure Appl. Logic 125 (2004), no. 1-3, 101-118.  ...  (Peter Cholak) 2004g:03069 03D30 — (with Wang, Yong*) A hierarchy for the plus cupping Turing degrees. (English summary) J. Symbolic Logic 68 (2003), no. 3, 972-988.  ... 

Strengthening prompt simplicity

David Diamondstone, Keng Meng Ng
2011 Journal of Symbolic Logic (JSL)  
However, it is a wtt-degree notion, and we show that it characterizes the degrees which satisfy a wtt-degree notion very close to the definition of superlow cuppability.  ...  Further, we study the strongly prompt c.e. sets in the context of other notions related promptness, superlowness, and cupping.  ...  We show that ω-c.e is the first place in the Ershov difference hierarchy where such a restriction on g makes sense.  ... 
doi:10.2178/jsl/1309952528 fatcat:7gi4c3jgsnctnhr4t7o52oy6fq

Page 2732 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(From the text) 2004j:03053 03E02 (03E55, 05A17) — (with Downey, Rodney G.; Li, Ang Sheng) Complementing cappable degrees in the difference hierarchy. (English summary) Ann. Pure Appl.  ...  (Michael Stob) 2004m:03151 03D25 (03D28) — On the density of the pseudo-isolated degrees. Proc. London Math. Soc. (3) 88 (2004), no. 2, 273-288.  ...