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Complementing cappable degrees in the difference hierarchy

2004
*
Annals of Pure and Applied Logic
*

Consequently, a computably enumerable

doi:10.1016/j.apal.2003.10.002
fatcat:62kypw5qdzcb3i7b2e3m4y7hdy
*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*. ...##
###
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.) ...

##
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Author Index

2004
*
Annals of Pure and Applied Logic
*

., An effective version of Wilkie's theorem of

doi:10.1016/s0168-0072(04)00010-7
fatcat:uifws35bifawzomuaz2ykaiwhq
*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.) ...##
###
CAPPABLE CEA SETS AND RAMSEY'S THEOREM

2011
*
Proceedings of the 11th Asian Logic Conference
*

We begin a search for

doi:10.1142/9789814360548_0007
fatcat:cyarjuwsbjaxhegpoac2vkamou
*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*. ...##
###
On the structure of the degrees of relative provability

2015
*
Israel Journal of Mathematics
*

of totality of computable functions are Π 0 2 -statements); and

doi:10.1007/s11856-015-1182-8
fatcat:umzkztj7djganfrwdzghrzmsty
*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. ...##
###
Bounding computably enumerable degrees in the Ershov hierarchy

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. ...

##
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Degrees of Unsolvability
[chapter]

2014
*
Handbook of the History of Logic
*

We finish our history by quoting Gerald Sacks, who through his own work and

doi:10.1016/b978-0-444-51624-4.50010-1
fatcat:clf7varewrh6lc4jq2c2sp6pgy
*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*. ...##
###
A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES

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. ...

##
###
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

2011
*
Journal of Symbolic Logic (JSL)
*

However, it is a wtt-

doi:10.2178/jsl/1309952528
fatcat:7gi4c3jgsnctnhr4t7o52oy6fq
*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. ...##
###
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. ...